High School Statistics — MGU Mathematics Series
Part II: Describing Data with Tables and Graphs
Dot Plots · Histograms · Box Plots · Shape · Outliers
Chapter 4 of 14
MGU Mathematics Series  ·  High School Statistics  ·  Library Textbook Edition

Chapter 4:
Displaying
Quantitative Data

Dot plots, stem-and-leaf plots, histograms, box plots, and cumulative displays — and how the shape of a distribution tells a story about variation

■ Part II: Describing Data with Tables and Graphs ■

■ 14 Sections  •  Chapter 4 of 14 ■

Follows Chapter 3: Organizing and Displaying Categorical Data  ·  Prepares students for Chapter 5: Summarizing Quantitative Data

Chapter 4 — Front Matter

Purpose and Overview

Numbers alone rarely reveal what data are trying to say. A list of one hundred test scores, the daily high temperatures for an entire year, or the finishing times of runners in a race may each be perfectly accurate as raw numbers — and yet each list is nearly impossible to interpret until it is organized, arranged, and displayed in a form that makes its patterns visible. That is exactly what this chapter is about.

Chapter 4 teaches you how to display quantitative data — data that take numerical values and that can be measured or counted along a scale. You will learn to construct and interpret dot plots, stem-and-leaf plots, histograms, box plots, and cumulative frequency displays. More importantly, you will learn how to read these graphs: how to describe the shape of a distribution, identify the center and spread, notice clusters and gaps, and recognize values that stand apart from the rest. These skills are not cosmetic. They are the foundation for every analysis that follows in this book.

This chapter also connects graphical thinking to geometric reasoning. The shape of a distribution — whether it is roughly symmetric, stretched to one side, or spread broadly and evenly — carries statistical meaning. Learning to see that shape, and to describe it precisely, is one of the most important skills a statistics student can develop.

Opening Question

A school records the number of minutes students spend on homework each night. Some students report 10 minutes; others report 180 minutes. Most cluster somewhere in between. How would you display this data so that a viewer could immediately see where most students fall, how spread out the values are, and whether any values seem unusually high or low? By the end of this chapter, you will have several tools to answer that question — and you will know when each tool is most appropriate.

14Sections
12Objectives
18Key Terms
12+Worked Examples

Learning Objectives

  1. 1Explain what quantitative data measure and how they differ from categorical data.
  2. 2Construct and interpret frequency tables for quantitative data using appropriate class intervals.
  3. 3Construct and interpret dot plots for small to moderate data sets.
  4. 4Construct and interpret stem-and-leaf plots, including back-to-back stem-and-leaf plots for comparison.
  5. 5Construct and interpret histograms, including relative frequency histograms.
  6. 6Choose appropriate class interval widths for histograms.
  7. 7Construct and interpret box plots using a five-number summary.
  8. 8Interpret cumulative frequency displays.
  9. 9Describe the shape of a distribution as symmetric, skewed left, skewed right, or roughly uniform.
  10. 10Identify center, spread, clusters, gaps, and possible outliers from a graph.
  11. 11Compare two or more distributions using appropriate displays and language.
  12. 12Recognize misleading displays of quantitative data and explain how they distort the picture.

Why This Chapter Matters

Data displays are not decoration. They are analytical tools. A well-constructed graph lets you see patterns that arithmetic alone hides. Scientists use histograms to detect anomalous measurements. Doctors use distribution plots to identify patients whose values fall far outside the typical range. Economists use displays to identify income inequality. Public health researchers use distributions to track how disease spreads across a population. Engineers use displays to monitor whether a manufacturing process is staying within acceptable bounds.

In every case, the display does the same fundamental job: it transforms a list of numbers into a visual picture of variation. Learning to construct these displays correctly, and to read them critically, is among the most practical skills that statistics offers.

Chapter 4 — Contents

Sections at a Glance

Sections 4.1 – 4.4
Quantitative Data: Frequency Tables, Dot Plots, and Stem-and-Leaf Plots
What Quantitative Data Measure · Frequency Tables · Dot Plots · Stem-and-Leaf Plots
Section 4.1

What Quantitative Data Measure

In Chapter 3, you worked with categorical data — data that sort individuals into named groups or categories, such as preferred music genre, political affiliation, or brand of phone. Categorical data answer the question "which kind?" or "which group?" and are displayed with bar graphs, pie charts, two-way tables, and segmented bars.

Quantitative data are different. They answer the question "how much?" or "how many?" They take numerical values along a meaningful scale, and arithmetic operations on those values are meaningful. The temperature outside is quantitative. So is a student's GPA, the distance a car travels on a tank of gas, the number of push-ups a person completes in one minute, or the time it takes water to boil.

Within quantitative data, statisticians distinguish two types. Discrete quantitative data take values from a countable set — whole numbers or specific integers. The number of children in a family is discrete: a family can have 0, 1, 2, or 3 children, but not 2.7. Continuous quantitative data can take any value within a range, including decimals and fractions. Height, weight, time, and temperature are typically treated as continuous because they can be measured to any degree of precision your instrument allows.

This distinction matters for display choices and for later analysis, but at the high school level the more important distinction is simply: does the variable take numerical values along a scale? If yes, it is quantitative, and the displays in this chapter apply.

TypeWhat it measuresExamplesDisplay tools
CategoricalGroup membership; named categoriesFavorite sport, eye color, grade levelBar graphs, pie charts, two-way tables
Discrete quantitativeCountable quantities; whole-number valuesNumber of siblings, goals scored, books readDot plots, histograms, bar charts (counts)
Continuous quantitativeMeasurable quantities; values on a continuumHeight, temperature, time, distanceDot plots, histograms, stem-and-leaf, box plots
Interpreting Quantitative Data in Context

Always identify the units when describing quantitative data. "The typical value is about 35" means nothing without knowing whether the unit is seconds, dollars, miles per hour, or milligrams. Units anchor quantitative data to the real-world situation they describe.

Practice 4.1Concept Check
  1. Classify each variable as categorical, discrete quantitative, or continuous quantitative. Explain your reasoning.

    a. The state in which a student was born.
    b. The number of text messages sent in one day.
    c. The time (in seconds) to run 100 meters.
    d. A person's blood type.
    e. The weight of a backpack in kilograms.
    f. The number of wins a team has in a season.
  2. A survey asks students to rate their stress level on a scale from 1 to 10. Is this variable best treated as categorical or quantitative? Defend your answer.
  3. Explain why the displays from Chapter 3 (bar graphs, pie charts) are not appropriate for displaying the distribution of heights in a class of 30 students.
Section 4.2

Frequency Tables for Quantitative Data

Before constructing any graph, you often begin by organizing quantitative data into a frequency table. Unlike a categorical frequency table — which simply lists group names and their counts — a quantitative frequency table must first group the numerical values into intervals called class intervals or bins. Each class interval spans a range of values, and the table records how many data values fall within each range.

To build a frequency table, you must decide on two things: where the intervals begin and end, and how wide each interval should be. These choices affect how the resulting table and graph will look. Good choices reveal the shape of the data; poor choices can hide important features or create a misleading picture.

Steps for Building a Frequency Table for Quantitative Data

Step 1: Find the minimum and maximum values in the data set.

Step 2: Calculate the range: range = maximum − minimum.

Step 3: Choose a number of class intervals — typically between 5 and 15 for most data sets. More data generally support more intervals.

Step 4: Divide the range by the number of intervals to estimate the interval width. Round to a convenient number.

Step 5: Set the left boundary of the first interval at or just below the minimum. Build each subsequent interval by adding the width.

Step 6: Tally the data values and record the frequency (count) in each interval. Add a relative frequency column (proportion or percent) if useful.

Example 4.2.1
Constructing a frequency table for homework times
  • ProblemTwenty students report their daily homework time in minutes: 20, 45, 30, 60, 75, 40, 55, 25, 80, 35, 50, 65, 90, 45, 70, 30, 55, 40, 60, 85. Construct a frequency table with class intervals of width 20, starting at 20.
  • ThinkThe minimum is 20 and the maximum is 90. With intervals of width 20 starting at 20, I get: [20, 40), [40, 60), [60, 80), [80, 100). I use the convention that the left boundary is included and the right boundary is excluded, except for the last interval which includes both.
  • Represent Interval Tally Frequency Relative Freq 20 to < 40 //// / 5 0.25 (25%) 40 to < 60 //// //// 9 0.45 (45%) 60 to < 80 //// 4 0.20 (20%) 80 to 100 // 2 0.10 (10%) Total 20 1.00 (100%)
  • Check5 + 9 + 4 + 2 = 20. ✓ The relative frequencies sum to 1.00. ✓
  • InterpretNearly half of students (45%) report between 40 and 60 minutes of homework. The distribution appears to be concentrated in the 40–60 minute range, with fewer students at either extreme.
Answer: See frequency table above. The most common range is 40 to <60 minutes (9 students, 45%).
Student Warning — Boundary Conventions

When a data value falls exactly on a class boundary, you must have a consistent rule for which interval it belongs to. The most common convention is to include the left boundary and exclude the right boundary: [20, 40) includes 20 but not 40; 40 belongs to [40, 60). Always state your convention, especially when values fall exactly on boundaries.

Practice 4.2Frequency Table Practice
  1. The resting heart rates (in beats per minute) for 15 athletes are: 58, 62, 55, 70, 64, 59, 67, 72, 61, 58, 65, 73, 60, 68, 56. Construct a frequency table using class intervals of width 5, starting at 55. Include a relative frequency column.
  2. A teacher records quiz scores (out of 20) for 18 students: 14, 18, 11, 20, 16, 15, 19, 13, 17, 12, 18, 16, 14, 20, 17, 11, 15, 19. Construct a frequency table with class intervals of width 3, starting at 11.
  3. Using the homework time data from Example 4.2.1, rebuild the frequency table using intervals of width 10. How does the new table compare to the one with width 20? Which do you think reveals more about the data?
Section 4.3

Dot Plots

A dot plot is one of the simplest displays for quantitative data. To construct a dot plot, you draw a number line that spans the range of the data, then place one dot directly above the number line for each data value. When values repeat, the dots stack vertically. The result is a picture of the distribution: you can immediately see where data values cluster, where there are gaps, and whether any values stand far apart from the others.

Dot plots are especially effective for small to moderate data sets — roughly 5 to 50 values. They preserve every data value (unlike a histogram, which groups values into intervals) and make it easy to count exact frequencies. For very large data sets, dot plots become cluttered and histograms or box plots are more practical.

Display Instruction — Dot Plot

Draw a horizontal number line. Mark a scale that begins slightly below the minimum data value and extends slightly above the maximum, using consistent, evenly spaced tick marks. For each data value, place a dot directly above the corresponding position on the number line. Stack dots vertically when values repeat, keeping them neatly aligned. Label the axis with the variable name and units. Give the plot a descriptive title.

Example 4.3.1
Constructing and interpreting a dot plot for quiz scores
  • ProblemTen students score the following on a 10-point quiz: 7, 8, 6, 9, 8, 10, 7, 8, 5, 9. Construct a dot plot and describe what it shows.
  • ThinkThe values range from 5 to 10. I will draw a number line from 5 to 10 and place dots for each value.
  • Represent Quiz Scores (out of 10) · · · · · · · · · ───────────────────── 5 6 7 8 9 10
  • InterpretThe distribution is centered around 8. The score 8 appears three times (the most common value). Scores of 9 and 7 each appear twice. The score 5 appears once and stands somewhat apart from the others — it may be worth investigating whether that student was absent for a key review session.
Answer: The distribution clusters between 7 and 9. A score of 5 appears once and is noticeably lower than the rest.
Dot Plots and Individual Values

One advantage of a dot plot over a histogram is that it preserves every individual value. You can count exactly how many times each value appears, identify which specific values are repeated, and trace any individual data point. This makes dot plots especially useful when you want to describe the data in detail rather than just its overall shape.

Practice 4.3Dot Plot Practice
  1. Construct a dot plot for the following data: the number of books read over the summer by 12 students: 3, 5, 2, 7, 4, 6, 3, 5, 8, 4, 5, 3. Describe the shape, center, and spread of the distribution.
  2. A dot plot shows the ages of participants in a community running club: values are 22, 25, 25, 28, 31, 31, 31, 34, 38, 42, 55. Describe what the dot plot would look like. Is there any value that stands noticeably apart from the others?
  3. A physical education teacher measures how many pull-ups each of 10 students can do: 2, 4, 4, 5, 6, 6, 6, 7, 8, 14. Construct a dot plot. What do you notice about the value of 14?
  4. When is a dot plot more useful than a histogram? Give one specific example where you would prefer a dot plot.
Section 4.4

Stem-and-Leaf Plots

A stem-and-leaf plot (sometimes called a stemplot) is a display that organizes quantitative data by splitting each value into two parts: a stem — the leading digit or digits — and a leaf — the final digit. Stems are listed vertically in order from least to greatest, and the leaves for each stem are written horizontally in a row to the right of the stem. The result is a kind of sideways histogram that also preserves the original data values.

For two-digit numbers, the tens digit is typically the stem and the ones digit is the leaf. For three-digit numbers, the hundreds digit (or the hundreds and tens together) becomes the stem. The key is to make the stem meaningful — it should capture the largest portion of each value while the leaf shows the fine detail.

Steps for Constructing a Stem-and-Leaf Plot

Step 1: Identify the stems — typically the leading digit(s). List them in order from least to greatest in a vertical column.

Step 2: For each data value, write the leaf (final digit) in the row corresponding to its stem. Initially, leaves may be written in the order the data appear.

Step 3: Reorder the leaves in each row from least to greatest. This makes the display easier to read.

Step 4: Add a title and a key. The key explains how to read one value. Example: 4 | 7 = 47 minutes.

Example 4.4.1
Constructing a stem-and-leaf plot for test scores
  • ProblemFifteen students earn the following scores on an exam: 72, 85, 91, 68, 77, 83, 95, 74, 88, 61, 79, 82, 90, 76, 84. Construct a stem-and-leaf plot.
  • ThinkThe scores range from 61 to 95. The tens digits (6, 7, 8, 9) will be the stems. The ones digits will be the leaves.
  • Represent Stem | Leaves 6 | 1 8 7 | 2 4 6 7 9 8 | 2 3 4 5 8 9 | 0 1 5 Key: 7 | 4 = 74 points
  • CheckCounting leaves: 2 + 5 + 5 + 3 = 15 values. ✓ The minimum leaf visible is 61 and the maximum is 95. ✓
  • InterpretThe distribution is roughly symmetric, centered in the 70s and 80s. Most scores fall between 72 and 88. The lowest scores (61, 68) and highest (91, 95) each appear in only one or two cases.
Answer: The stemplot shows a roughly mound-shaped distribution centered in the high 70s to mid-80s range.

Back-to-Back Stem-and-Leaf Plots

When you want to compare two groups, a back-to-back stem-and-leaf plot places the stems in the center column and the leaves for one group on the left and the leaves for the other group on the right. This layout makes direct visual comparison easy: you can see whether the two distributions have similar or different centers, similar or different spreads, and whether one group is shifted higher or lower than the other.

Example 4.4.2
Back-to-back stem-and-leaf plot comparing two groups
  • ProblemCompare the resting pulse rates (beats per minute) of 10 athletes and 10 non-athletes. Athletes: 52, 58, 60, 55, 63, 57, 61, 54, 59, 62. Non-athletes: 70, 68, 75, 72, 80, 66, 74, 78, 71, 76.
  • Represent Athletes Stem Non-Athletes 9 8 5 4 2 | 5 | 3 2 1 0 | 6 | 6 8 | 7 | 0 1 2 4 5 6 8 | 8 | 0 Key: 5|8 (Athletes) = 58 bpm 6|8 (Non-athletes) = 68 bpm
  • InterpretAthlete pulse rates cluster in the 50s and low 60s (center near 58–60 bpm). Non-athlete pulse rates cluster in the 70s (center near 72–74 bpm). Athletes have notably lower resting pulse rates, consistent with the cardiovascular efficiency that regular training produces. Neither group shows extreme outliers.
Answer: Athletes have lower resting pulse rates (centered around 58–60) compared to non-athletes (centered around 72–74).
Student Warning — Ordering Leaves

The leaves in each row of a stem-and-leaf plot must be ordered from least to greatest — or, in a back-to-back plot, ordered away from the stem. If leaves are left in the order data were collected, the plot cannot be easily read for shape or values. Always order the leaves before declaring the plot finished.

Practice 4.4Stem-and-Leaf Practice
  1. Construct a stem-and-leaf plot for the following data set of daily high temperatures (°F) recorded over two weeks: 88, 91, 84, 79, 93, 87, 82, 95, 90, 76, 85, 89, 92, 83. Include an appropriate key.
  2. A stem-and-leaf plot shows the following data. Reconstruct the original data values and describe the distribution.

    Stem | Leaves
    3 | 2 5 9
    4 | 0 1 4 7 8
    5 | 2 5 5 9
    6 | 1 3
    Key: 4|7 = 47
  3. Construct a back-to-back stem-and-leaf plot comparing the test scores of two classes: Class A: 71, 84, 78, 92, 65, 88, 75, 82, 90, 77. Class B: 68, 79, 83, 91, 74, 86, 70, 95, 62, 80. Compare the centers and spreads of the two classes.
  4. What is one advantage a stem-and-leaf plot has over a histogram? What is one disadvantage?
Sections 4.5 – 4.7
Histograms and Relative Frequency Displays
Histograms · Choosing Class Intervals · Relative Frequency Histograms
Section 4.5

Histograms

A histogram is the most widely used display for quantitative data. It looks similar to a bar graph, but there are important differences. In a bar graph for categorical data, the bars represent named categories, and there are gaps between bars because the categories are distinct groups. In a histogram, the bars represent class intervals along a continuous numerical scale, and the bars touch one another because the intervals are adjacent on the number line — there is no natural gap between them.

The height of each bar represents the frequency (or relative frequency) of data values in that interval. The width of each bar represents the width of the class interval. A critical point that many students miss: in a histogram, it is the area of each bar — not just its height — that is proportional to the number of data values in that interval. When all bars have equal width (the usual case at this level), height and area carry the same information. But this geometric interpretation of area becomes very important in Chapter 11 when you study normal distributions, where areas under a curve represent probabilities.

Key Differences: Bar Graph vs. Histogram

Bar graph — bars represent categories; gaps between bars; order may be arbitrary; height = frequency of category.

Histogram — bars represent class intervals on a number line; bars touch; order is fixed by the scale; height (or area) = frequency of values in interval.

Do not use a bar graph to display quantitative data. Do not put gaps between histogram bars.

Display Instruction — Histogram

Draw a horizontal axis labeled with the measurement variable and units. Mark the class interval boundaries as tick marks along the axis. Draw a vertical axis labeled "Frequency" (or "Count") with an appropriate scale. For each interval, draw a rectangle whose left and right sides align with the interval boundaries and whose height equals the frequency. Bars should touch — there are no gaps. Give the histogram a descriptive title.

Example 4.5.1
Constructing a histogram for homework times
  • ProblemUse the frequency table from Example 4.2.1 to construct a histogram for the homework time data.
  • ThinkThe intervals are [20, 40), [40, 60), [60, 80), [80, 100] with frequencies 5, 9, 4, 2. I will draw bars with these heights.
  • Represent Frequency 10 | 9 | ████ 8 | ████ 7 | ████ 6 | ████ 5 | ████ ████ 4 | ████ ████ ████ 3 | ████ ████ ████ 2 | ████ ████ ████ ████ 1 | ████ ████ ████ ████ ────────────────────────────── 20–40 40–60 60–80 80–100 Minutes of Homework per Night
  • InterpretThe tallest bar (40–60 minutes) shows that most students spend between 40 and 60 minutes on homework. The distribution has a slight right skew: the bars decrease from the peak but the right tail (60–100 minutes) is longer than the left tail (20–40 minutes).
Answer: The histogram shows the distribution is peaked in the 40–60 minute range with a slight skew to the right.
Reading a Histogram

When reading a histogram, look for four things: (1) the location of the tallest bar — that is roughly where the center lies; (2) how wide the bars are spread — that tells you about the spread; (3) whether the histogram is roughly symmetric, or whether one tail is longer; and (4) whether any bars are isolated by a gap — those may indicate unusual values.

Practice 4.5Histogram Practice
  1. Using the resting heart rate frequency table you built in Practice 4.2, Problem 1, construct a histogram. Label both axes. Describe the shape of the distribution.
  2. A histogram shows bars at the following heights for intervals [0, 10), [10, 20), [20, 30), [30, 40), [40, 50]: heights are 3, 8, 14, 6, 2. Describe the distribution: shape, approximate center, and spread.
  3. Why do the bars in a histogram touch but the bars in a bar graph have gaps? Explain in terms of what each type of bar represents.
  4. A student constructs a histogram of exam scores and puts gaps between the bars. What does this suggest the student misunderstood about the display?
Section 4.6

Choosing Class Intervals

One of the most consequential decisions in constructing a histogram is the choice of class interval width. A poor choice can dramatically distort the picture of the data — not through dishonesty, but simply by grouping data in a way that hides variation or creates false patterns.

If you choose intervals that are too wide, the histogram will have very few bars and will look flat and uninformative. All the variation within each interval is hidden. You lose the ability to see where values cluster within a broad range, and important features like bimodal patterns (two peaks) may disappear entirely.

If you choose intervals that are too narrow, the histogram will have many thin bars, most of which may contain only one or zero values. The display looks jagged and noisy rather than smooth. The overall shape of the distribution is hard to discern because every small fluctuation in the data creates a visible spike or gap.

A good choice of interval width produces a histogram with enough bars to show the shape of the distribution — typically somewhere between 5 and 15 bars for most classroom data sets — while smoothing out small random fluctuations. There is no single formula, but a common starting rule is to aim for roughly the square root of the number of data values as the number of bars. From there, adjust using judgment and by trying a few alternatives.

Example 4.6.1
The effect of interval width on a histogram's appearance
  • ProblemThe ages (in years) of 30 museum visitors are collected. The minimum is 8 and the maximum is 72. Compare histograms using interval widths of 5, 10, and 20. Describe what each reveals and hides.
  • ThinkWith width 5, there are about (72−8)/5 ≈ 13 bars. With width 10, about 6–7 bars. With width 20, about 3–4 bars.
  • Analyze
    • Width = 5 (many narrow bars): The display is detailed but jagged. Small random variations between adjacent bars look like real patterns. It may be difficult to see the overall shape.
    • Width = 10 (moderate): The shape is clear — perhaps a concentration in the 20–40 range, with fewer visitors at young and old extremes. This is likely the most informative choice for 30 data values.
    • Width = 20 (very wide): The histogram has only 3–4 bars. All variation within each 20-year range is hidden. You cannot tell whether most visitors in the 20–40 range are in their 20s or 30s.
  • InterpretThe width-10 histogram is the best choice here. It reveals the shape without excessive noise and uses a manageable number of bars for 30 data values.
Answer: Width = 10 gives the clearest picture for this data set. Wider intervals hide variation; narrower intervals produce jagged, noisy displays.
Data Ethics Note — Interval Width and Misleading Histograms

Someone who wants to make a distribution look more spread out, more concentrated, or more uniform than it really is can manipulate the interval width of a histogram to achieve that appearance — without altering the data. Be skeptical of histograms where the interval width is very unusual or not clearly labeled. Always check the axis scale and interval width before drawing conclusions from any histogram you did not construct yourself.

Practice 4.6Interval Width Practice
  1. A data set of 40 values ranges from 12 to 94. If you want approximately 8 class intervals, what width should each interval be? What if you want approximately 5 intervals?
  2. A student uses class intervals of width 1 to display the ages of 200 participants (ranging from 18 to 65) in a study. What problem will this create? What would be a better choice?
  3. A student uses class intervals of width 50 to display a data set that ranges from 0 to 100. What problem will this create? What would be a better choice?
  4. Explain, in your own words, why choosing interval width is a judgment call rather than a mechanical rule. What trade-offs are involved?
Section 4.7

Relative Frequency Histograms

A relative frequency histogram is identical to a regular histogram in construction, except that the vertical axis shows relative frequencies — proportions or percentages — rather than raw counts. The height of each bar tells you what fraction (or what percent) of all data values fall in that interval.

Relative frequency histograms are especially valuable when you want to compare two distributions that have different total numbers of data values. If one class has 20 students and another has 35, a frequency histogram will make the larger class look more prominent simply because its bars are taller. A relative frequency histogram removes this size effect and lets you compare the shapes and centers of the two distributions directly.

Relative frequency histograms also connect naturally to probability. If you treat the relative frequency as the probability that a randomly selected value from the data set falls in a given interval, the histogram begins to look like a probability model. This connection will become very important in Chapter 11 when you study normal distributions.

Example 4.7.1
Comparing two classes using relative frequency histograms
  • ProblemClass A (20 students) and Class B (35 students) both take the same test. Their score distributions are summarized in the table below. Compare them using relative frequency histograms.
  • Represent Score Range Class A Freq Class A Rel.Freq. Class B Freq Class B Rel.Freq. 60 to < 70 2 0.10 (10%) 4 0.11 (11%) 70 to < 80 7 0.35 (35%) 12 0.34 (34%) 80 to < 90 8 0.40 (40%) 14 0.40 (40%) 90 to 100 3 0.15 (15%) 5 0.14 (14%) Total 20 1.00 (100%) 35 1.00 (100%)
  • InterpretAlthough Class B has more students, the relative frequency histograms have nearly identical shapes. Both classes have about 10–11% scoring in the 60s, about 35% in the 70s, 40% in the 80s, and 14–15% in the 90s. The distributions are very similar. Using counts alone would make Class B's bars look much taller, which could suggest Class B performed "better" — but the relative frequencies reveal that both classes performed almost identically.
Answer: The two classes have nearly identical relative frequency distributions despite their different sizes. Using proportions removes the size difference and makes fair comparison possible.
Practice 4.7Relative Frequency Histogram Practice
  1. A hospital records the lengths of patient stays (in days). Group 1 (general ward, 50 patients): [1,3) → 10, [3,5) → 18, [5,7) → 14, [7,9) → 8. Group 2 (surgical ward, 30 patients): [1,3) → 3, [3,5) → 9, [5,7) → 12, [7,9) → 6. Construct relative frequency histograms for both groups and compare them. Which group tends to have longer stays?
  2. A relative frequency histogram shows bars with heights 0.05, 0.20, 0.35, 0.30, 0.10 for intervals [0,10), [10,20), [20,30), [30,40), [40,50]. If the data set has 80 values, how many values fall in each interval? What is the total?
  3. Explain why a relative frequency histogram is more informative than a frequency histogram when comparing distributions of different sizes. Give a real-world example where this would matter.
Sections 4.8 – 4.9
Box Plots and Cumulative Frequency Displays
Box Plots from Five-Number Summaries · Ogives and Cumulative Percentages
Section 4.8

Box Plots

A box plot (also called a box-and-whisker plot) is a display that summarizes a quantitative distribution using five key values, called the five-number summary: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. These five numbers divide the data into four roughly equal-sized groups: the bottom 25%, the lower middle 25%, the upper middle 25%, and the top 25%.

In a box plot, a rectangular box is drawn from Q1 to Q3. A line inside the box marks the median. Two lines — called whiskers — extend from the box outward to the minimum and maximum values (or, when outliers are marked separately, to the furthest non-outlier values). The box spans the middle half of the data; its width is called the interquartile range (IQR). The whiskers show how far the lower and upper quarters of data extend. Outliers, when present, are typically plotted as individual dots beyond the whiskers.

Box plots are especially powerful for comparing two or more distributions side by side, because multiple box plots can be drawn on the same axis. At a glance, you can compare centers (where each median line falls), spreads (how wide each box is, how long the whiskers are), and whether the distributions are symmetric or skewed.

Steps for Constructing a Box Plot

Step 1: Arrange the data from least to greatest.

Step 2: Find the five-number summary: minimum, Q1, median, Q3, maximum. (Quartiles will be reviewed in detail in Chapter 5.)

Step 3: Draw a number line scaled to span the data range.

Step 4: Draw a rectangular box from Q1 to Q3. Mark the median with a vertical line inside the box.

Step 5: Draw whiskers from the left side of the box to the minimum, and from the right side of the box to the maximum. (In the version with outlier identification, draw whiskers only to the last non-outlier value and plot outliers separately. This rule will be covered in Chapter 5 Section 5.9.)

Step 6: Label the number line axis with the variable name and units.

Example 4.8.1
Constructing a box plot for exam scores
  • ProblemConstruct a box plot for the following 15 exam scores (already ordered): 61, 68, 72, 74, 76, 77, 79, 82, 83, 84, 85, 88, 90, 91, 95.
  • ThinkWith 15 values, the median is the 8th value. Q1 is the median of the lower 7 values. Q3 is the median of the upper 7 values.
  • Calculate Minimum = 61 Q1 = 74 (median of the lower 7 values: 61, 68, 72, 74, 76, 77, 79) Median = 82 (8th value) Q3 = 88 (median of the upper 7 values: 83, 84, 85, 88, 90, 91, 95) Maximum = 95 IQR = Q3 − Q1 = 88 − 74 = 14
  • Represent Box Plot: Exam Scores |──────[─────────|────────]──────────| 60 65 70 75 80 85 90 95 100 ↑ ↑ ↑ Q1=74 Med=82 Q3=88 Min=61 Max=95
  • InterpretThe box (middle 50% of data) spans from 74 to 88, an IQR of 14 points. The median is at 82. The left whisker (61 to 74) is slightly longer than the right whisker (88 to 95), suggesting a slight left skew — a small number of lower scores pull the distribution down. The score of 61 is the lowest and farthest from the box.
Answer: Five-number summary: 61, 74, 82, 88, 95. The distribution has a slight left skew with the middle 50% of scores between 74 and 88.
What Box Plots Show and What They Hide

A box plot efficiently shows the five-number summary and the spread of the distribution. However, it hides information that a histogram or dot plot preserves: you cannot see individual values, you cannot detect bimodal patterns (two distinct peaks), and you cannot see clusters within each quartile. A box plot tells you about spread and center but not about the fine-grained shape inside the box. Use a histogram or dot plot when shape matters most; use a box plot when comparison matters most.

Practice 4.8Box Plot Practice
  1. The daily high temperatures (°F) for 11 days are: 72, 68, 75, 80, 65, 78, 83, 70, 77, 74, 90. Order the data, find the five-number summary, and construct a box plot.
  2. A box plot has the following five-number summary: Min = 12, Q1 = 28, Median = 35, Q3 = 44, Max = 70. Describe the distribution. Is it symmetric? Is there evidence of skewness? Which whisker is longer?
  3. Two box plots are drawn on the same axis. Box Plot A has: Min = 10, Q1 = 22, Median = 30, Q3 = 38, Max = 50. Box Plot B has: Min = 20, Q1 = 35, Median = 45, Q3 = 55, Max = 65. Compare the two distributions in terms of center, spread, and overlap.
  4. A student says "since the box is in the center of the whiskers, the distribution must be symmetric." Explain what is wrong with this reasoning.
Section 4.9

Cumulative Frequency Displays

A cumulative frequency display (also called an ogive, pronounced "OH-jive") is a graph that shows the running total of frequencies — or relative frequencies — as you move from left to right across the data range. At each class boundary, the graph plots the cumulative frequency up to that point. The resulting curve always rises from left to right and eventually reaches either the total number of data values (for a cumulative frequency ogive) or 100% (for a cumulative relative frequency ogive).

Ogives are useful for reading off percentiles — the value below which a given percentage of the data falls. To find the percentile value for 60%, for example, you draw a horizontal line at the 60% mark on the vertical axis, then drop a vertical line down to the horizontal axis to read off the corresponding data value. Ogives are also useful for estimating the median (the value at 50%) and the quartiles (at 25% and 75%) directly from the graph.

Display Instruction — Cumulative Frequency Ogive

From your frequency table, add a cumulative frequency column by summing the frequencies from top to bottom: each row's cumulative frequency is the count of all values up to and including that interval. Divide by total n to get cumulative relative frequencies (proportions). Plot points at each class upper boundary with the cumulative relative frequency on the vertical axis. Connect the points with straight line segments. The ogive begins at the lower boundary of the first interval at 0% and ends at the upper boundary of the last interval at 100%.

Example 4.9.1
Constructing and reading a cumulative relative frequency ogive
  • ProblemUsing the homework time data from Example 4.2.1, construct a cumulative relative frequency ogive and use it to estimate the 75th percentile.
  • Represent Upper Boundary Cumulative Freq Cumulative Rel. Freq. 40 5 25% 60 14 70% 80 18 90% 100 20 100%
  • AnalyzeThe ogive rises steeply between 40 and 60 minutes (where most data fall), then flattens as fewer values appear in the upper ranges. To estimate the 75th percentile, draw a horizontal line at 75% and read across to the curve, then down to the horizontal axis. The curve passes through 70% at 60 minutes and 90% at 80 minutes. By interpolation, 75% falls at approximately 65 minutes.
  • InterpretThe 75th percentile is approximately 65 minutes. This means 75% of students in this sample spend 65 minutes or less on homework each night.
Answer: The 75th percentile is approximately 65 minutes — about three-quarters of students report 65 minutes or less of nightly homework.
Practice 4.9Cumulative Display Practice
  1. A frequency table for quiz scores has intervals [10,13), [13,16), [16,19), [19,22] with frequencies 4, 9, 11, 6 out of 30 total. Build the cumulative relative frequency table. Sketch the ogive and estimate the median from the graph.
  2. A cumulative relative frequency ogive for a data set passes through these points: (50, 0%), (60, 15%), (70, 40%), (80, 75%), (90, 95%), (100, 100%). Estimate: (a) the 25th percentile, (b) the median, (c) the 90th percentile.
  3. Why does a cumulative frequency ogive always start at 0% on the left and end at 100% on the right? Could it ever decrease? Explain.
Sections 4.10 – 4.14
Shape, Comparison, and Critical Thinking
Shape · Center · Spread · Clusters · Outliers · Comparing · Misleading Graphs · Common Mistakes
Section 4.10

Shape: Symmetric, Skewed, and Uniform

The single most informative thing you can say about a distribution's graph is its shape. Shape tells you how data values are arranged across the range — whether they pile up in the middle, lean toward one end, or spread evenly from bottom to top. Different shapes carry different statistical implications, and shape determines which summary statistics (mean, median, standard deviation, IQR) are most appropriate to use — a distinction that Chapter 5 will develop in detail.

Symmetric Distributions

A distribution is symmetric if the left half of the graph is roughly a mirror image of the right half. If you were to fold the histogram along the tallest bar, the two sides would align approximately. In a perfectly symmetric, mound-shaped distribution, the mean, median, and mode all coincide at the center. Many naturally occurring measurements — heights of adults in a population, errors in precision instruments, scores on well-designed tests — tend to produce symmetric or nearly symmetric distributions.

Symmetric (Bell-shaped): · ··· ····· ······· ············· ───────────── Low → High The peak is in the center. Both sides taper off at roughly equal rates.

Skewed Distributions

A distribution is skewed when one tail is longer than the other. The direction of skew is the direction of the longer tail — not the direction of the bulk of the data, which often confuses students.

In a right-skewed (or positively skewed) distribution, the right tail is longer. Most values cluster toward the left side, but a few unusually large values pull the distribution's tail to the right. Examples include incomes in a population (most people earn moderate incomes, but a small number of very high earners create a long right tail), real estate prices, and the wait times at an emergency room.

In a left-skewed (or negatively skewed) distribution, the left tail is longer. Most values cluster toward the right side, but a few unusually small values pull the tail to the left. Examples include scores on a very easy test (where most students score near the top but a few score very low) or ages at retirement (most people retire in their 60s or 70s, but a few retire very early).

Right-Skewed (Positive Skew): Left-Skewed (Negative Skew): · · ··· ··· ····· ····· ······· ······· ·············· · · · · ·············· ────────────────── ────────────────── Low → High Low → High Bulk of data is on the LEFT. Bulk of data is on the RIGHT. Long tail extends to the right. Long tail extends to the left.

Uniform Distributions

In a uniform distribution, data values are spread roughly evenly across the entire range. The histogram looks approximately flat — no bar is dramatically taller than the others. A perfectly uniform distribution would have equal frequency in every class interval. In practice, truly uniform distributions are uncommon with real data, but some situations produce approximately uniform distributions: the outcome of rolling a fair die many times, random number generation, or certain waiting times.

Shape and the Mean vs. Median

Shape directly affects the relationship between the mean and median. In a right-skewed distribution, a few large values pull the mean to the right (higher) while the median remains more centrally placed. This is why income data is typically described using the median rather than the mean — a small number of very high incomes can push the mean far above what a "typical" person earns. You will study this connection quantitatively in Chapter 5.

Practice 4.10Shape Practice
  1. For each situation, predict whether the distribution of the variable would likely be symmetric, right-skewed, or left-skewed. Explain your reasoning.

    a. The number of minutes a customer waits at a fast-food drive-through.
    b. The heights of adult men in the United States.
    c. The scores on an extremely difficult exam where most students struggle.
    d. The annual salaries of employees at a company, including the CEO.
    e. The ages of cars in a used-car lot (most are 2–8 years old, but a few are very old).
  2. A histogram has bars with the following heights (left to right): 2, 4, 8, 12, 8, 4, 2. What is the shape of this distribution? Where does the center appear to be?
  3. A histogram has bars with heights: 14, 10, 7, 4, 2, 1. What is the shape? Which tail is longer?
  4. Sketch (describe in words) what a dot plot for a roughly uniform distribution would look like for whole-number values from 1 to 6. How is this connected to rolling a fair die many times?
Section 4.11

Center, Spread, Clusters, Gaps, and Outliers

Describing a distribution completely requires more than identifying its shape. A full description of a distribution should always address five features: center, spread, clusters, gaps, and outliers. These five features are what statistical graphs are designed to reveal, and they are what you should systematically look for every time you examine a quantitative display.

Center

The center of a distribution is a value that represents where the data are concentrated — a typical value. From a graph, you can estimate the center visually as the value that divides the distribution into two roughly equal areas, or the value at which the distribution peaks. The center is not always at the arithmetic middle of the range; it is the value where the bulk of the data lies.

Spread

The spread of a distribution describes how variable the data are — how widely the values are dispersed from the center. A narrow distribution with bars concentrated over a small range has little spread. A distribution whose bars are spread broadly across a wide range has a lot of spread. Spread is related to the concept of variability: high spread means values differ substantially from one another; low spread means they tend to be similar. From a graph, you can estimate spread visually by looking at how wide the distribution is from the leftmost to the rightmost data values and whether the bars are tightly bunched or widely spread.

Clusters and Gaps

A cluster is a concentration of data values within a narrow portion of the range — a place where many observations fall close together. A gap is an interval within the range where no (or very few) data values appear. Clusters and gaps are often the most statistically interesting features of a distribution because they may signal that the population is composed of distinct subgroups, or that a measuring instrument has a limited number of possible values.

For example, a distribution of speeds recorded at a highway checkpoint might cluster around 65 mph (the speed limit) and around 80 mph (those exceeding the limit), with a gap between the two clusters. This bimodal pattern — two distinct clusters — would suggest two different types of drivers, and would be completely hidden by a simple average.

Outliers

An outlier is a data value that falls far apart from the bulk of the distribution — either much larger or much smaller than most other values. Outliers appear visually as isolated dots in a dot plot, isolated leaves separated from the main body of a stemplot, isolated bars separated by gaps in a histogram, or as individual dots plotted beyond the whiskers of a box plot.

Outliers deserve careful attention. They may represent genuine extremes in the data (a student who is unusually talented or unusually unprepared), recording errors or measurement mistakes, or members of a different population than the one you intended to study. Before dismissing or removing an outlier, always investigate its source. You will learn a formal rule for identifying outliers (the 1.5 IQR rule) in Chapter 5.

Example 4.11.1
Writing a complete description of a distribution
  • ProblemA dot plot shows the number of hours per week that 20 high school students spend on social media. The values are: 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 9, 10, 11, 14, 18, 25. Describe the distribution completely.
  • ThinkI should address shape, center, spread, clusters, gaps, and outliers.
  • Analyze
    • Shape: The distribution is strongly right-skewed. Most values cluster on the left side (low hours), but a few unusually high values pull a long right tail toward 25 hours.
    • Center: The center appears to be around 6–7 hours per week, where values are most concentrated.
    • Spread: The distribution is quite spread out, ranging from 2 to 25 hours — a range of 23 hours. Most values fall between 2 and 11 hours.
    • Clusters: There is a clear cluster from 2 to 10 hours, where 17 of the 20 values appear.
    • Gaps: There is a visible gap between 11 and 14 hours, then between 14 and 18, and between 18 and 25.
    • Outliers: The values 18 and 25 are separated from the main cluster by a noticeable gap. The value 25 is especially far from the rest and may be an outlier worth investigating.
  • InterpretMost students in this sample spend between 2 and 11 hours per week on social media, with a typical value near 6–7 hours. A small number of students spend dramatically more time — up to 25 hours — pulling the distribution strongly to the right.
Answer: Shape: strongly right-skewed. Center: approximately 6–7 hours. Spread: 2 to 25 hours. Cluster: 2–10 hours. Gap: 11 to 14 and beyond. Possible outliers: 18 and 25 hours.
SOCS — A Framework for Describing Distributions

Use the acronym SOCS to remember what to address when describing a distribution from a graph:

S — Shape (symmetric, right-skewed, left-skewed, uniform, bimodal)

O — Outliers (any values that stand noticeably apart from the rest?)

C — Center (where does the distribution peak or balance? What is the typical value?)

S — Spread (how wide is the distribution? Are values tightly clustered or broadly dispersed?)

Always interpret SOCS in context. Use the variable name and units in your description.

Practice 4.11Description Practice
  1. A histogram of the number of minutes spent exercising per week for 40 adults shows bars with the following heights for intervals [0,60), [60,120), [120,180), [180,240), [240,300], [300+]: heights are 3, 8, 14, 9, 4, 2. Write a complete description of this distribution using SOCS.
  2. A dot plot shows the prices (in dollars) of 15 used textbooks: 5, 8, 10, 10, 12, 14, 15, 15, 18, 20, 22, 25, 45, 60, 95. Describe the distribution using SOCS. Identify any possible outliers.
  3. Two histograms are described: Distribution A is narrow with bars tightly clustered around 50, ranging from 40 to 60. Distribution B is wide with bars spread from 10 to 90. Which distribution has more variability? Which would have a larger standard deviation? (You do not need to calculate — just reason from the graphs.)
  4. Why is it not sufficient to describe a distribution only by its center? Give an example of two distributions that have the same center but very different spreads.
Section 4.12

Comparing Distributions Visually

Statistics often asks us to compare groups — not just to describe one distribution, but to ask whether two or more distributions differ in their centers, spreads, shapes, or whether one group consistently produces higher or lower values than another. Visual comparison is frequently the first and most powerful step in this analysis.

Three displays are especially useful for comparing distributions: side-by-side box plots (placed on the same numerical axis), back-to-back stem-and-leaf plots (already seen in Section 4.4), and histograms with a common scale (placed one above the other or overlaid with transparency). Each has strengths and weaknesses. Box plots show five-number summaries and medians clearly but hide shape within each quartile. Back-to-back stemplots preserve every value but only work well for small data sets. Histograms show overall shape and density but can be harder to compare when data sets differ in size (use relative frequency histograms to solve this problem).

When comparing two distributions, always address the same SOCS features for each group and use explicit comparison language: "Group A has a higher center than Group B," "Group B is more spread out," "Group A appears symmetric while Group B is right-skewed."

Example 4.12.1
Comparing two groups with side-by-side box plots
  • ProblemCompare the distribution of commute times (in minutes) for workers in two cities using the following five-number summaries. City A: Min = 5, Q1 = 18, Median = 30, Q3 = 45, Max = 80. City B: Min = 10, Q1 = 25, Median = 42, Q3 = 60, Max = 95.
  • Represent Side-by-Side Box Plots — Commute Times (minutes) City A: |──[──────|──────]──────| City B: |──[────────|─────────]────────| ────────────────────────────────────── 0 20 40 60 80 100 Minutes
  • Interpret
    • Center: City B has a higher median commute (42 min) than City A (30 min). Typical commuters in City B spend about 12 more minutes commuting.
    • Spread: Both cities have similar IQRs (City A: 27 min; City B: 35 min), but City B's IQR is wider, indicating more variability among the middle 50% of commuters. City B also has a longer overall range (85 min vs. 75 min).
    • Shape: Both distributions appear somewhat right-skewed — the right whisker is longer than the left in each case, suggesting a few very long commutes pull each distribution's tail upward.
    • Outliers: Neither box plot flags specific outliers at this level of analysis, though both have long right whiskers worth investigating.
Answer: City B has longer, more variable commute times. The median commute in City B (42 min) is about 12 minutes longer than in City A (30 min), and City B's middle 50% spans a wider range.
Practice 4.12Comparison Practice
  1. Two statistics classes take the same exam. Class A five-number summary: 55, 70, 78, 86, 98. Class B five-number summary: 62, 75, 84, 90, 100. Describe and compare the two distributions. Which class performed better on this exam? How do you know?
  2. A physical education study measures the maximum number of push-ups students can do before and after a 6-week training program. Before: 5, 8, 10, 10, 12, 14, 15, 15, 16, 18. After: 12, 14, 16, 16, 18, 20, 20, 22, 24, 30. Construct dot plots or back-to-back stem-and-leaf plots for both data sets and compare the distributions.
  3. A relative frequency histogram is used to compare the number of hours per week spent reading by high school students and college students. Why would a relative frequency histogram be more appropriate here than a frequency histogram?
Section 4.13

Misleading Displays of Quantitative Data

Graphs of quantitative data can be manipulated — deliberately or accidentally — to create a false impression. Unlike calculation errors, which can often be checked by recomputing, misleading graphs require the viewer to notice that something about the display has been distorted. Critical reading of statistical graphs is a skill that protects you from being misled by data presented in media, advertising, political campaigns, and business reports.

Truncated Vertical Axis

One of the most common manipulations is starting the vertical axis at a value other than zero. If a histogram or time plot shows only the range from, say, 85 to 100, small differences between bars look enormous. A difference that is actually 3 units on a 0-to-100 scale fills nearly a third of the visible graph area when the axis starts at 88. This technique is frequently used in news graphics and product advertisements to make modest changes appear dramatic.

Inconsistent Class Intervals

In a histogram, all bars must have equal width — unless you explicitly label each bar's width and show proportional areas. When bars have different widths but equal heights, the wider bars represent more data values, but the graph makes every interval look equally common. This creates a severely distorted picture of the distribution. Always check that class intervals are equal-width (or that area is correctly proportional) before trusting a histogram.

Missing Labels and Missing Context

A graph without labeled axes, missing units, or no title forces you to guess what the data represent and what scale the values are on. This omission may be innocent carelessness, or it may be a way to hide information that would make an interpretation less impressive. Always ask: What variable is shown? What are the units? What is the scale? Who collected this data and why?

Data Ethics — Graphs in Media and Policy

Misleading graphs appear regularly in news articles, social media posts, campaign materials, and corporate earnings presentations. When examining any graph you did not construct yourself, ask: Does the vertical axis start at zero? Are all class intervals equal? Are there labels, units, and a source? What impression is the graph trying to create — and is that impression accurate? Developing the habit of critical graph-reading is one of the most practically useful outcomes of a statistics course.

Practice 4.13Misleading Display Practice
  1. A company's annual profit histogram shows bars with heights 47, 49, 50, and 52 (in millions of dollars) for years 2021–2024. The vertical axis starts at 45 rather than 0. Describe what impression the graph gives. What would it look like if the axis started at 0? Calculate the actual percentage increase from 2021 to 2024.
  2. A histogram is shown with three bars: one bar covers the range 0–20 (height 6), one covers 20–22 (height 8), and one covers 22–100 (height 4). What is misleading about this display? What should the histogram look like if the class intervals are equal-width?
  3. You see a graph in a news article with no axis labels, no units, and no source. What questions would you ask before drawing any conclusions from this graph? List at least four.
Section 4.14

Common Quantitative Graph Mistakes

Mistake 1 — Drawing gaps between histogram bars
Drawing a histogram with gaps between the bars, as if it were a bar graph for categorical data.

Correct approach: In a histogram, bars represent adjacent intervals on a continuous number line. They touch because there is no natural gap between, say, 40–50 and 50–60. A gap in a histogram signals an interval with zero frequency — a real gap in the data. Drawing gaps where none exist misrepresents the data.

Mistake 2 — Confusing direction of skew
"The distribution is left-skewed because most of the data is on the left side."

Correct approach: The direction of skew is determined by the tail, not by where most data lie. A right-skewed distribution has most data on the left and a long tail extending to the right. A left-skewed distribution has most data on the right and a long tail extending to the left. "Skewed right" means the tail points right.

Mistake 3 — Describing shape without context
"The histogram is shaped like a hill." (No further description.)

Correct approach: A complete description of shape connects the graph to the data. "The distribution of homework times is approximately symmetric and mound-shaped, centered near 50 minutes, with most students spending between 35 and 65 minutes." Use the variable name, units, and context in every description.

Mistake 4 — Using a bar graph for quantitative data
Constructing a bar graph with one bar for each individual value of a quantitative variable (e.g., separate bars for every score from 50 to 100).

Correct approach: When the number of possible values is large, group quantitative data into class intervals and use a histogram. Constructing one bar per value produces a cluttered, unreadable display that does not reveal the distribution's shape.

Mistake 5 — Concluding from shape without addressing outliers
"The data are right-skewed." (Description ends without noting the isolated value at 95 that is separated by a large gap.)

Correct approach: A complete description notes both the overall shape and any specific outliers or unusual features. Outliers can affect both the apparent shape and later calculations. They deserve to be named and investigated, not ignored.

Mistake 6 — Treating the center as the only important feature
"Both distributions have the same center, so they are basically the same."

Correct approach: Two distributions with identical centers can look very different — one may be tightly concentrated and the other extremely spread out. Always report spread, shape, and unusual features in addition to center. A center alone does not describe a distribution.

Practice 4.14Error Analysis
  1. A student constructs a display for exam scores (all values between 50 and 100) by drawing a separate bar for each whole-number score, with gaps between bars. List two things wrong with this display and explain how to fix each one.
  2. A student describes a histogram: "The distribution is right-skewed because most of the bars are on the right side." Identify the error and write a corrected description.
  3. Two dot plots have identical medians (both centered at 20) but very different appearances: Dot Plot A shows all values between 18 and 22. Dot Plot B shows values ranging from 5 to 40. A student concludes they are "basically the same distribution." What important feature has the student ignored?
  4. A stem-and-leaf plot has leaves that were not ordered. What problem does this create for someone reading the display? How should it be corrected?
  5. Identify the mistake: A box plot is constructed for a data set and the whisker on the right extends to 95, while the whisker on the left ends at 42. A student says "the distribution must be right-skewed because the right whisker is so long." Is this always correct? What else would you need to check?
Chapter 4 — Back Matter

Chapter Summary

Key Terms Review

quantitative data
Data that take numerical values along a meaningful scale and can be measured or counted. Arithmetic operations on the values are meaningful.
discrete data
Quantitative data that take values from a countable set, typically whole numbers. Example: number of children in a family.
continuous data
Quantitative data that can take any value within a range, including decimals. Example: height, time, temperature.
class interval
A range of values that defines one row of a frequency table or one bar of a histogram. Also called a bin.
frequency table
A table that lists class intervals and the count (frequency) of data values that fall within each interval.
relative frequency
The proportion or percentage of data values in a given interval or category: relative frequency = frequency ÷ total n.
dot plot
A display in which each data value is represented by a dot above a number line. Repeated values stack vertically. Ideal for small data sets.
stem-and-leaf plot
A display that splits each value into a stem (leading digit[s]) and a leaf (final digit), preserving original values while showing shape.
histogram
A display for quantitative data using touching rectangular bars, where bar height represents the frequency of values in each class interval.
relative frequency histogram
A histogram with proportions (not counts) on the vertical axis. Used to compare distributions of different sizes.
box plot
A display of the five-number summary (minimum, Q1, median, Q3, maximum) using a box from Q1 to Q3, a median line, and whiskers.
five-number summary
The minimum, first quartile (Q1), median, third quartile (Q3), and maximum of a data set.
ogive
A cumulative frequency (or relative frequency) graph that plots running totals against class upper boundaries. Used to estimate percentiles.
symmetric distribution
A distribution whose left half is roughly a mirror image of the right half. Mean, median, and mode coincide in a perfectly symmetric distribution.
skewed right
A distribution with a long right tail. Most values cluster toward the lower end, but a few large values extend the right tail. Also called positively skewed.
skewed left
A distribution with a long left tail. Most values cluster toward the upper end, but a few small values extend the left tail. Also called negatively skewed.
outlier
A data value that falls far apart from the bulk of the distribution. May represent a genuine extreme, a data error, or a member of a different population.
SOCS
An acronym for the four features to address when describing a distribution: Shape, Outliers, Center, and Spread.

Concept Review Questions

  1. What is the key difference between categorical data and quantitative data? Give one example of each.
  2. Explain why bars in a histogram must touch, while bars in a categorical bar graph have gaps between them.
  3. What is the difference between a frequency histogram and a relative frequency histogram? When is each preferred?
  4. What are the five numbers in a five-number summary? What does each number tell you about the data?
  5. What is a cumulative frequency ogive, and what type of question can you answer with it?
  6. Define right-skewed and left-skewed distributions. For each, describe what the histogram looks like and give a real-world example.
  7. What does SOCS stand for? Why is it useful when describing a distribution?
  8. Why is it important to investigate outliers rather than simply removing them?
  9. Explain one way in which a histogram can mislead a viewer without changing the underlying data.
  10. What information does a box plot show clearly that a histogram does not? What information does a histogram show that a box plot hides?

Skill Practice

Part AFrequency Tables

Construct a frequency table (with relative frequencies) for each data set using the interval width indicated.

  1. Heights (in inches) of 20 students: 62, 64, 67, 70, 63, 68, 65, 72, 66, 69, 71, 64, 67, 68, 63, 70, 65, 66, 73, 61. Use width = 3, starting at 61.
  2. Time (in minutes) for 18 students to complete an assignment: 8, 15, 22, 12, 30, 18, 25, 10, 16, 28, 20, 14, 32, 11, 19, 24, 17, 26. Use width = 5, starting at 5.
Part BDot Plots and Stem-and-Leaf Plots
  1. Construct a dot plot for the number of pages read per day by 12 students: 15, 22, 18, 30, 22, 25, 18, 15, 40, 22, 28, 20. Describe what the dot plot reveals.
  2. Construct a stem-and-leaf plot for the following ages of participants in a survey: 23, 31, 27, 45, 38, 29, 34, 42, 26, 33, 48, 25, 37, 41, 30. Include an ordered stem-and-leaf plot and a key.
  3. Construct a back-to-back stem-and-leaf plot for the following finishing times (in minutes) for a 5K run. Male runners: 22, 28, 31, 25, 34, 29, 26, 33, 24, 30. Female runners: 26, 30, 35, 28, 32, 27, 34, 29, 31, 25. Compare the two distributions.
Part CHistograms
  1. A frequency table shows the following data for hours of screen time per week among 30 teenagers. Construct a histogram and describe the distribution.

    [0, 5): 2     [5,10): 5     [10,15): 10     [15,20): 8     [20,25): 3     [25,30]: 2
  2. Two groups each have 30 participants. Group X has frequencies 4, 8, 12, 5, 1 for intervals [50,60), [60,70), [70,80), [80,90), [90,100]. Group Y has frequencies 2, 6, 10, 8, 4 for the same intervals. Construct relative frequency histograms for both groups on a common scale and compare the distributions.
Part DBox Plots
  1. Find the five-number summary and construct a box plot for the following data set: 14, 19, 23, 27, 31, 35, 38, 42, 47, 55, 62.
  2. A box plot has Min = 25, Q1 = 40, Median = 52, Q3 = 65, Max = 80. Compute the IQR. Describe the shape of the distribution based on the relative positions of the median, quartiles, and whiskers.
  3. Construct side-by-side box plots for these two data sets and compare them. Group A: 45, 52, 58, 62, 66, 70, 74, 78, 83, 90. Group B: 30, 48, 55, 60, 64, 68, 71, 76, 82, 95.
Part EShape and Description
  1. Describe each distribution in terms of SOCS (shape, outliers, center, spread). Use context and units.

    a. A histogram of annual salaries (in thousands of dollars) for 50 employees: bars are highest between $40k and $60k, with a long tail extending to $250k.

    b. A dot plot of scores (out of 100) on an easy quiz: most values between 85 and 100, with a few values around 60–70.

    c. A histogram of rainfall (in inches) per month over 5 years: bars are evenly distributed from 1 to 8 inches with no clear peak.

Data Interpretation Problems

For each problem, interpret the display in context. Use the variable name and units.

  1. A school nurse records the body mass index (BMI) of 200 students. A histogram of the data shows a roughly symmetric distribution centered near 22, with most values between 17 and 27 and a few isolated values above 35. Write a complete contextual description of this distribution. What do the isolated high values suggest? What would you want to know before drawing health conclusions?
  2. A traffic study records vehicle speeds (mph) at two locations on the same highway. Location A box plot: Min = 45, Q1 = 58, Median = 62, Q3 = 68, Max = 78. Location B box plot: Min = 35, Q1 = 55, Median = 65, Q3 = 75, Max = 95. Compare the distributions of speeds at the two locations. Which location has more consistent speeds? Which has a higher typical speed? What might explain the differences?
  3. A cumulative relative frequency ogive for student GPA at a college passes through: (1.0, 0%), (1.5, 4%), (2.0, 12%), (2.5, 28%), (3.0, 55%), (3.5, 80%), (4.0, 100%). Estimate the median GPA. What percentage of students have a GPA above 3.0? What percentage fall below 2.0? Interpret each answer in context.
  4. A dot plot of the number of absences per student over a semester shows: most students with 0–4 absences, several students with 5–8 absences, and one student with 18 absences. Write a complete description. Should the value of 18 be removed from the analysis? Explain your reasoning.

Real-World Applications

  1. Health and Medicine: A hospital records the time (in minutes) from symptom onset to emergency room treatment for 30 heart attack patients. The times range from 40 to 420 minutes. Explain why this data is likely right-skewed. What display would you use to communicate this to a medical team? What feature of the distribution is most important for patient outcomes?
  2. Finance: A financial analyst records the daily percentage changes in a stock's price over one year (approximately 250 trading days). The values range from −4.5% to +5.2%, with most values near 0%. What shape would you expect for this distribution? What display would best reveal whether extreme gains and extreme losses are symmetric?
  3. Public Policy: A city is comparing household income distributions across two neighborhoods. Neighborhood A has a median income of $52,000 and Neighborhood B has a median of $58,000. A journalist reports that Neighborhood B residents are "better off." What additional information would you need — and what additional displays would you want to see — before accepting this conclusion?
  4. Education: A school district tracks the number of students per classroom across 40 schools. The data ranges from 18 to 38 students, with most classes having 24–28 students. Construct a frequency table (width = 4, starting at 18). Describe what shape you would expect for the histogram and explain your reasoning.
  5. Sports: A coach records the distances (in feet) of 25 baseball throws from each of two outfielders. Outfielder A: Min = 180, Q1 = 215, Median = 230, Q3 = 245, Max = 280. Outfielder B: Min = 170, Q1 = 200, Median = 235, Q3 = 265, Max = 310. Construct side-by-side box plots and compare the two players. Which player is more consistent? Which player has a higher typical throwing distance?

Error Analysis

Each problem shows a student's error or misconception. Identify the error and provide the correct reasoning or display.

Problem 1
A student constructs a histogram and draws gaps between every bar because "that's how bar graphs work."

Identify the error. How does leaving gaps between histogram bars change the message of the graph? What does a gap in a histogram actually indicate?

Problem 2
A distribution shows bars like this (left to right): tall, medium, short, very short, nearly zero. A student says: "The data is left-skewed because most of the bars are on the left."

Identify the confusion. What is the actual shape of the distribution described? What does it mean for a distribution to be right-skewed vs. left-skewed?

Problem 3
A student compares two groups using frequency histograms: Group X has 25 students and Group Y has 100 students. The student says Group Y has "much more" data in every interval, so the distributions must be different.

What is wrong with this conclusion? What type of histogram should the student use for a fair comparison? Why?

Problem 4
A student looks at a box plot and says: "The median is at 45 and the box goes from 30 to 60, so the data must be symmetric because 45 is exactly in the middle of the box."

What additional information would you need to determine whether the distribution is symmetric? Can a box plot always reveal whether a distribution is symmetric? Explain.

Problem 5
A stem-and-leaf plot was constructed from data on student ages in years. The leaves for the stem "2" are written as: 2 | 9 5 3 7 1 4 (in the order collected, not ordered).

What is wrong with this display? Reorder the leaves correctly and explain why ordering matters.

Chapter 4 Checkpoint

Chapter 4 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 4
Section 1 — Vocabulary and Concepts
  1. Explain the difference between discrete and continuous quantitative data. Give one example of each.
  2. What is a class interval? Why must class intervals in a frequency table be non-overlapping and exhaustive?
  3. Why do histogram bars touch, while categorical bar graph bars do not?
  4. What does SOCS stand for? Name one thing you should address under each letter when describing a distribution.
Section 2 — Constructing Displays
  1. The following data are the weights (in pounds) of 10 dogs at a veterinary clinic: 12, 18, 22, 8, 35, 45, 18, 25, 60, 15.
    a. Construct a dot plot for the data.
    b. Identify any possible outliers.
    c. Describe the shape of the distribution.
  2. Construct a stem-and-leaf plot for the following test scores and write a key: 73, 82, 91, 68, 75, 88, 95, 80, 72, 87, 64, 93, 79, 85, 71.
  3. A frequency table has intervals [20,30), [30,40), [40,50), [50,60] with frequencies 3, 7, 12, 8 from a sample of 30 values. Construct a relative frequency histogram. Label both axes completely.
  4. The following data represent the number of minutes 11 students spent exercising on a given day (ordered): 10, 20, 30, 30, 40, 45, 60, 65, 75, 80, 120.
    a. Find the five-number summary.
    b. Construct a box plot.
    c. Compute the IQR.
    d. Describe the shape of the distribution based on the box plot.
Section 3 — Reading and Interpreting Displays
  1. A histogram of annual reading hours for 50 adults shows: [0,20) frequency 3; [20,40) frequency 6; [40,60) frequency 14; [60,80) frequency 18; [80,100] frequency 9. Describe this distribution using SOCS.
  2. A cumulative relative frequency ogive passes through these points: (0, 0%), (25, 10%), (50, 30%), (75, 65%), (100, 90%), (125, 100%). Estimate: (a) the median, (b) the 25th percentile, (c) what percentage of values are below 75.
Section 4 — Comparison
  1. Two data sets have the following five-number summaries. Data Set P: 10, 25, 40, 55, 80. Data Set Q: 30, 42, 50, 60, 72. Compare the two distributions. Which has a higher center? Which has more spread? Which appears more symmetric?
Section 5 — Error Analysis
  1. A student describes a histogram as "right-skewed because the bars on the right are taller." Is this correct? Explain, and write a correct description of what right-skewed means.
  2. A student creates a histogram comparing 20 responses from Group A and 80 responses from Group B using raw frequency counts. Group B's bars are all taller. The student concludes: "Group B has more data in every range, so the distributions are different." What is wrong with this conclusion? What display should be used instead?
Looking Ahead
Chapter 5: Summarizing Quantitative Data

Chapter 4 gave you the visual tools to see a distribution — to look at its shape, estimate its center, observe its spread, and identify unusual values. Chapter 5 will give you the numerical tools to measure those same features precisely. You will learn to calculate the mean, median, mode, range, interquartile range, variance, and standard deviation. You will also learn to compute the five-number summary formally and to find percentiles and z-scores. Most importantly, you will learn when to use each measure — and the answer will depend directly on what you learned about shape in Chapter 4. For a symmetric distribution, the mean and standard deviation are appropriate; for a skewed distribution or one with outliers, the median and IQR provide more honest summaries. The graphs you have learned to read in Chapter 4 will guide every calculation you make in Chapter 5.

MGU Library Connection
  • Graph and Display Reference — Appendix E of this textbook
  • Summary Statistics Formula Sheet — Appendix F of this textbook (for Chapter 5 preparation)
  • Calculator and Spreadsheet Skills — Appendix O: constructing histograms and box plots with technology
  • Chapter 3: Organizing and Displaying Categorical Data — for contrast with quantitative displays
  • Chapter 5: Summarizing Quantitative Data — numerical measures of center, spread, and position
  • Chapter 11: Normal Distributions — area under a density curve as probability, building on histogram shape
  • Geometry, Chapter on Area — geometric reasoning about area and proportion, applied to histograms and density curves
  • MGU Dictionary entries: histogram, dot plot, stem-and-leaf plot, box plot, five-number summary, quartile, symmetric distribution, skewness, outlier, ogive