Chapter 5 — Front Matter
Purpose and Overview
Data without summary is overwhelming. A list of 200 test scores, a column of 500 reaction times, or a spreadsheet of daily temperatures across a year contains real information, but that information is buried. Numerical summaries compress data into a few meaningful numbers that describe the center of a distribution, the spread around that center, and the relative position of any individual value within the distribution.
Chapter 5 develops the principal tools of numerical summary for quantitative data: measures of center (mean, median, mode), measures of spread (range, interquartile range, standard deviation), the five-number summary, the 1.5 IQR rule for identifying outliers, an introduction to variance, percentiles, and z-scores. The chapter also teaches students to choose the right summary tools based on what a distribution looks like, and to understand clearly what is gained — and what is lost — when a whole distribution is reduced to a handful of numbers.
Opening Question
A school nurse records the resting heart rates (in beats per minute) of 30 students: 68, 72, 75, 64, 80, 77, 70, 66, 74, 83, 71, 69, 78, 62, 76, 79, 73, 67, 81, 70, 65, 72, 74, 68, 77, 75, 63, 82, 71, 69. What is a typical resting heart rate in this group? How much do students vary from that typical value? Is any heart rate unusually high or low? These are questions of summary. By the end of this chapter, you will be able to answer all of them with precision and explain what your numbers mean in context.
15Sections
10Objectives
17Key Terms
12+Worked Examples
Why This Chapter Matters
Statistics is frequently mistaken for calculation. But before you can calculate sensibly, you must know what you are trying to measure. Is the question about the typical value? About how widely spread the data are? About whether a particular person is unusual compared to the group? Each of these questions calls for a different numerical summary. Numerical summaries are used everywhere that data appear — in medicine, education, finance, manufacturing, and sports analytics. Every profession that uses data depends on the ability to summarize distributions clearly and honestly.
Learning Objectives
- 1Calculate and interpret the mean, median, and mode of a data set.
- 2Explain why mean and median can differ, and when each is more informative.
- 3Calculate and interpret the range, quartiles, interquartile range, and five-number summary.
- 4Apply the 1.5 IQR rule to identify potential outliers.
- 5Calculate and interpret variance and standard deviation.
- 6Find and interpret percentiles for individual values.
- 7Calculate and interpret z-scores.
- 8Choose between the mean/standard deviation pair and the median/IQR pair based on distribution shape.
- 9Explain what summary statistics reveal and what they conceal.
- 10Recognize and avoid common mistakes in summarizing quantitative data.
Sections 5.1 – 5.4
Measures of Center: Mean, Median, and Mode
Why Summaries Matter · Mean · Median · Mode
Section 5.1
Why Numerical Summaries Matter
A distribution is a complete picture of all the values in a data set and how often each value occurs. You can see a distribution in a dot plot, histogram, or box plot. But sometimes you need a compact answer to a compact question: What is the center? How much do values spread out? How unusual is this particular value?
Numerical summaries answer those questions with single numbers or small sets of numbers. They are useful precisely because they are compact. The cost of that compactness is information loss. When you reduce 200 scores to a single average, you lose the shape of the distribution: whether it was symmetric or skewed, whether there were clusters or gaps, whether some values were extreme. A summary is not a replacement for a graph; it is a companion to one.
Throughout this chapter, keep asking: What does this number actually measure? What would a different data set that produced the same summary look like? What can this number not tell me?
Section 5.2
Mean
The mean of a data set is the arithmetic average of all the values. It is the most widely used measure of center, and also the one most sensitive to extreme values. The mean is the balancing point of the distribution — if you imagine the number line as a physical beam and each data value as a weight placed at its location, the mean is the fulcrum where the beam would balance.
Example 5.2.1
Calculating and Interpreting the Mean
- ProblemSeven students reported the following hours spent studying last week: 4, 6, 5, 8, 3, 7, 5. Find the mean and interpret it.
- ThinkThe mean summarizes the center. Add all values and divide by the count.
- Represent$\bar{x} = \dfrac{\sum x_i}{n}$
- Calculate$$\bar{x} = \frac{4 + 6 + 5 + 8 + 3 + 7 + 5}{7} = \frac{38}{7} \approx 5.43 \text{ hours}$$
- CheckThe values range from 3 to 8. A mean of approximately 5.43 falls within that range and near the middle — reasonable.
- InterpretOn average, these seven students studied approximately 5.4 hours last week. This does not mean every student studied 5.4 hours; it means 5.4 is the balancing point of the distribution.
Answer: $\bar{x} \approx 5.43$ hours
Example 5.2.2
How an Extreme Value Affects the Mean
- ProblemEight employees earn the following annual salaries (in thousands of dollars): 38, 41, 40, 39, 42, 44, 37, 210. Find the mean and comment on whether it represents the typical salary.
- ThinkOne value, 210, is far larger than the rest. The mean will be pulled upward significantly.
- Calculate$$\bar{x} = \frac{38 + 41 + 40 + 39 + 42 + 44 + 37 + 210}{8} = \frac{491}{8} = 61.375 \text{ thousand dollars}$$
- CheckSeven of the eight employees earn between $37,000 and $44,000. The mean of $61,375 is far above six of the eight salaries.
- InterpretThe mean salary of $61,375 is inflated by the one very high earner. It does not represent the typical worker's pay. The median would give a fairer picture here.
Answer: $\bar{x} = \$61,375$ — not representative of most employees
Student Warning — Outliers and the Mean
When a distribution is strongly skewed or contains an outlier, the mean can be pulled far from the bulk of the data. Always look at the shape of the distribution before deciding whether the mean is an appropriate summary of center.
Section 5.3
Median
The median is the middle value in an ordered data set. It divides the distribution so that half the values are below it and half are above it. The median is a resistant measure: it is not strongly affected by outliers or skewness.
Finding the Median — Two-Step Method
Step 1. Arrange all values in order from smallest to largest.
Step 2a. If $n$ is odd, the median is the middle value at position $\dfrac{n+1}{2}$.
Step 2b. If $n$ is even, the median is the mean of the two middle values at positions $\dfrac{n}{2}$ and $\dfrac{n}{2} + 1$.
Example 5.3.1
Median with an Odd Number of Values
- ProblemFind the median of: 4, 6, 5, 8, 3, 7, 5 (the study hours from Example 5.2.1).
- ThinkOrder the values, then find the single middle one. $n = 7$ (odd).
- Calculate
Ordered: 3, 4, 5, [5], 6, 7, 8Middle position: (7+1)/2 = 4th value = 5
- InterpretThe median is 5 hours. Half the students studied fewer than 5 hours; half studied more.
Answer: Median = 5 hours
Example 5.3.2
Median with an Even Number of Values
- ProblemEight quiz scores: 72, 85, 68, 91, 74, 88, 65, 79. Find the median.
- Calculate
Ordered: 65, 68, 72, [74, 79], 85, 88, 91$$\text{Median} = \frac{74 + 79}{2} = \frac{153}{2} = 76.5$$
- InterpretThe median quiz score is 76.5. Half the students scored below 76.5 and half scored above.
Answer: Median = 76.5
Example 5.3.3
Mean vs. Median with an Extreme Value
- ProblemReturn to the salary data: 37, 38, 39, 40, 41, 42, 44, 210 (ordered). Compare mean and median.
- Calculate$$\text{Median} = \frac{40 + 41}{2} = 40.5 \text{ thousand dollars}$$Mean from Example 5.2.2 = $61,375
- InterpretThe median salary of $40,500 is much closer to what most employees actually earn. The mean of $61,375 is inflated by the one very high salary. For skewed data, the median gives a better description of the typical value.
Answer: Median = $40,500 (more representative); Mean = $61,375 (inflated by outlier)
Key Idea — Mean vs. Median
The mean is sensitive to every value, including outliers. The median is resistant. When a distribution is symmetric and has no outliers, mean and median will be close. When a distribution is skewed or contains outliers, the median is usually the better measure of center.
Section 5.4
Mode
The mode is the value (or values) that appears most frequently in a data set. It is the only measure of center that can also be used for categorical data. A data set may be unimodal (one mode), bimodal (two modes tied for most frequent), or have no mode (all values appear exactly once).
Example 5.4.1
Finding the Mode
- ProblemTwelve coffee shop customers ordered drinks: 12, 16, 20, 16, 12, 16, 20, 12, 16, 8, 12, 16 oz. Find the mode.
- Calculate
8 oz: 1 time · 12 oz: 4 times · 16 oz: 5 times · 20 oz: 2 times
- InterpretThe mode is 16 ounces — the most commonly ordered size.
Answer: Mode = 16 oz
When to Use the Mode
The mode is most useful for categorical data or for identifying the most popular value. For a symmetric, unimodal quantitative distribution, the mode, median, and mean are all close together. The mode is least informative when values are spread across many different numbers, as often happens with continuous measurements.
- Find the mean: 5, 8, 6, 7, 9, 8, 10, 7, 6, 8
- Find the median: 14, 17, 15, 14, 16, 98, 15, 13, 14, 16
- Find the mode: 3, 5, 7, 5, 3, 8, 5, 3, 7, 5
- A data set has mean 50 and median 38. What does this suggest about the shape?
- Ages: 14, 14, 15, 15, 15, 16, 16, 42. Find mean and median. Which is more representative?
- Why can the median but not the mean be used for categorical data?
Sections 5.5 – 5.11
Measures of Spread: Range, IQR, Variance, and Standard Deviation
Range · Quartiles · IQR · Five-Number Summary · Outliers · Variance · Standard Deviation
Section 5.5
Range
A measure of center tells you where the data are located; a measure of spread tells you how widely scattered the values are around that center. The simplest measure of spread is the range.
Example 5.5.1
Calculating the Range
- ProblemAges of ten fitness participants: 23, 19, 31, 44, 27, 35, 22, 41, 29, 38. Find the range.
- Calculate$$\text{Range} = 44 - 19 = 25 \text{ years}$$
- InterpretThe ages span a range of 25 years, from 19 to 44.
Answer: Range = 25 years
Limitation of the Range
The range uses only the two most extreme values. A single outlier dramatically increases the range without reflecting what is typical for most values. Always use the IQR or standard deviation as a companion spread measure.
Section 5.6
Quartiles
Quartiles divide a distribution into four roughly equal parts. The three quartiles are Q1 (lower quartile, 25th percentile), Q2 (the median, 50th percentile), and Q3 (upper quartile, 75th percentile).
Finding Quartiles
Step 1. Order the data from smallest to largest.
Step 2. Find the median (Q2), dividing the data into a lower half and an upper half.
Step 3. Q1 = median of the lower half. Q3 = median of the upper half.
When $n$ is odd, exclude the median value itself before finding Q1 and Q3.
Example 5.6.1
Finding Quartiles
- ProblemTwelve test scores (ordered): 58, 63, 67, 70, 74, 76, 79, 82, 85, 88, 92, 97. Find Q1, Q2, and Q3.
- Think$n = 12$ (even). Divide into two halves of 6 values each.
- Calculate
Lower half: 58, 63, 67, 70, 74, 76 → Q1 = (70 + 74)/2 = 72
Upper half: 79, 82, 85, 88, 92, 97 → Q3 = (85 + 88)/2 = 86.5
Q2 (median of all 12): (76 + 79)/2 = 77.5
- InterpretAbout 25% of scores fall at or below 72. About 50% fall at or below 77.5. About 75% fall at or below 86.5.
Answer: Q1 = 72, Q2 = 77.5, Q3 = 86.5
Section 5.7
Interquartile Range
The interquartile range (IQR) is the distance between Q1 and Q3. It measures the spread of the middle 50 percent of the data. Unlike the range, it is not affected by the most extreme values, making it a resistant measure of spread.
Example 5.7.1
Calculating the IQR
- CalculateUsing Q1 = 72 and Q3 = 86.5 from Example 5.6.1:$$\text{IQR} = 86.5 - 72 = 14.5$$
- InterpretThe middle 50 percent of test scores span a range of 14.5 points.
Answer: IQR = 14.5 points
Section 5.8
Five-Number Summary
The five-number summary condenses an entire distribution into five key values that can be used to construct a box plot and to describe the distribution's shape.
Example 5.8.1
Constructing a Five-Number Summary
- ProblemFind the five-number summary for the twelve test scores from Section 5.6.
- Calculate
| Value | Result |
|---|
| Minimum | 58 |
| Q1 | 72 |
| Median (Q2) | 77.5 |
| Q3 | 86.5 |
| Maximum | 97 |
- InterpretThe distribution spans 58 to 97. The middle half of scores falls between 72 and 86.5. The median of 77.5 is slightly below the midpoint of the box, suggesting slight left skew in the center.
Answer: 58 | 72 | 77.5 | 86.5 | 97
Reading Shape from the Five-Number Summary
If the median is near the center of the box (Q1 to Q3), the middle of the distribution is roughly symmetric. If the median is closer to Q1, the center is right-skewed. If the median is closer to Q3, the center is left-skewed. A much longer whisker on one side indicates a longer tail in that direction.
Section 5.9
Outliers and the 1.5 IQR Rule
An outlier is a value that lies unusually far from the bulk of the data. The 1.5 IQR rule provides a consistent, objective method to identify potential outliers.
The 1.5 IQR Rule — Outlier Fences
A value is a potential outlier if it falls:
Below the lower fence: $Q1 - 1.5 \times \text{IQR}$
Above the upper fence: $Q3 + 1.5 \times \text{IQR}$
Example 5.9.1
Applying the 1.5 IQR Rule — No Outliers
- ProblemUse the twelve test scores (Q1 = 72, Q3 = 86.5, IQR = 14.5). Find fences and check for outliers.
- Calculate
Lower fence = 72 − 1.5(14.5) = 72 − 21.75 = 50.25
Upper fence = 86.5 + 1.5(14.5) = 86.5 + 21.75 = 108.25
- CheckMin = 58 and Max = 97 both fall within the fences (50.25 to 108.25).
- InterpretNo potential outliers. All twelve values fall within the expected range.
Answer: No potential outliers
Example 5.9.2
Identifying a Potential Outlier
- ProblemA running club records 5K finish times (minutes) for 9 members: 18, 22, 24, 25, 26, 27, 28, 30, 54. Identify any potential outliers.
- Calculate
Median (5th value) = 26
Lower half: 18, 22, 24, 25 → Q1 = (22+24)/2 = 23
Upper half: 27, 28, 30, 54 → Q3 = (28+30)/2 = 29
IQR = 29 − 23 = 6
Lower fence: 23 − 1.5(6) = 14 Upper fence: 29 + 1.5(6) = 38
- InterpretThe time of 54 minutes exceeds the upper fence of 38. It is a potential outlier — substantially slower than the rest of the group. This may warrant investigation rather than deletion.
Answer: 54 minutes is a potential outlier (above upper fence of 38)
Outliers Are Not Automatic Errors
The 1.5 IQR rule flags potential outliers. It does not automatically mean those values are errors or should be removed. Every outlier should be investigated in context. Some outliers represent genuine, important variation — they may be the most interesting data points in the set.
Section 5.10
Variance as an Introduction
The range and IQR measure spread by looking at specific gaps. A different approach measures how far, on average, each value is from the mean. This approach leads to variance and standard deviation, the most commonly reported measures of spread in statistics.
The deviations above and below the mean cancel out when added directly — the sum of all deviations from the mean is always zero. Squaring the deviations makes them all positive before averaging, which is why the variance formula involves squared deviations.
Example 5.10.1
Calculating Variance
- ProblemFive noon temperatures (°F): 68, 72, 70, 65, 75. Calculate the sample variance.
- Calculate
Step 1 — Mean: $\bar{x} = \dfrac{68+72+70+65+75}{5} = \dfrac{350}{5} = 70$
| $x_i$ | $x_i - \bar{x}$ | $(x_i - \bar{x})^2$ |
|---|
| 68 | −2 | 4 |
| 72 | +2 | 4 |
| 70 | 0 | 0 |
| 65 | −5 | 25 |
| 75 | +5 | 25 |
| Sum | 0 | 58 |
$$s^2 = \frac{58}{5-1} = \frac{58}{4} = 14.5 \text{ degrees}^2$$
- InterpretThe variance is 14.5 square degrees. Because it is in squared units, it is harder to interpret directly — which is why we take the square root to get the standard deviation.
Answer: $s^2 = 14.5 \text{ °F}^2$
Section 5.11
Standard Deviation
The standard deviation is the square root of the variance. It returns the measure of spread to the original units of the data and measures a typical distance from the mean. The sample standard deviation is denoted $s$; the population standard deviation is denoted $\sigma$ (Greek letter sigma).
Example 5.11.1
Calculating Standard Deviation
- CalculateContinuing from Example 5.10.1 ($s^2 = 14.5$):$$s = \sqrt{14.5} \approx 3.81 \text{ °F}$$
- InterpretThe five noon temperatures are, on average, about 3.8°F away from the mean of 70°F.
Answer: $s \approx 3.81$ °F
Example 5.11.2
Comparing Two Classes Using Standard Deviation
- ProblemTeacher A: $\bar{x} = 76$, $s = 4.2$. Teacher B: $\bar{x} = 76$, $s = 14.7$. Both have the same mean. What does the standard deviation reveal?
- InterpretBoth classes average 76, but in Teacher A's class most students scored within about 4 points of the mean — very consistent. In Teacher B's class, scores are far more spread out, with some students scoring very high and some very low. The standard deviation reveals what the mean alone cannot: how consistently students are performing.
Answer: Same center, very different spread — standard deviation captures the difference
Example 5.11.3
Standard Deviation in Quality Control
- ProblemA quality control manager samples 8 juice bottles (oz): 16.1, 15.9, 16.0, 16.2, 15.8, 16.1, 16.0, 15.9. Calculate the mean and standard deviation.
- Calculate
$\bar{x} = \dfrac{128.0}{8} = 16.0$ oz
| $x_i$ | $x_i - 16.0$ | $(x_i - 16.0)^2$ |
|---|
| 16.1 | +0.1 | 0.01 |
| 15.9 | −0.1 | 0.01 |
| 16.0 | 0 | 0 |
| 16.2 | +0.2 | 0.04 |
| 15.8 | −0.2 | 0.04 |
| 16.1 | +0.1 | 0.01 |
| 16.0 | 0 | 0 |
| 15.9 | −0.1 | 0.01 |
| Sum | 0 | 0.12 |
$$s^2 = \frac{0.12}{7} \approx 0.0171 \text{ oz}^2 \qquad s = \sqrt{0.0171} \approx 0.131 \text{ oz}$$
- InterpretThe mean fill is exactly 16.0 oz with a standard deviation of about 0.13 oz. The bottles are filled very consistently — most are within about 0.13 oz of the target. This is a tight, high-quality process.
Answer: $\bar{x} = 16.0$ oz, $s \approx 0.131$ oz
Properties of Standard Deviation
Always ≥ 0. Standard deviation equals zero only when every value is identical.
Same units as data. If heights are in cm, $s$ is in cm.
Not resistant. Like the mean, $s$ is sensitive to outliers.
Paired with the mean. When you report $\bar{x}$, report $s$ alongside it.
Student Warning — Always Report the Mean with Standard Deviation
A standard deviation of 10 tells you nothing useful unless you know what mean it measures spread around. A data set with $\bar{x} = 50$ and $s = 10$ is very different from a data set with $\bar{x} = 500$ and $s = 10$. Never report standard deviation without its corresponding mean.
Technology Note
Standard deviation is tedious to calculate by hand for large data sets. Most graphing calculators have a 1-Var Stats function that gives $\bar{x}$, $s$, and other summaries directly. Spreadsheets use =STDEV() for sample standard deviation and =STDEVP() for population standard deviation. Whether you use technology or calculate by hand, you must be able to interpret the result in context.
- Find the range: 22, 27, 19, 31, 24, 28, 35
- Find Q1, Q3, IQR: 4.2, 3.8, 5.1, 4.7, 3.9, 5.5, 4.1, 4.8, 3.6
- Five-number summary: 6, 7, 8, 8, 9, 7, 6, 10, 5, 9, 8, 7, 6, 4
- Apply 1.5 IQR rule to: Q1 = 20, Q3 = 35, values 8 and 52
- $s$ = 0 for which data set? 5, 5, 5, 5, 5 or 3, 5, 7, 5, 5?
- Two classes: same mean = 80. $s_A = 3$, $s_B = 12$. Describe the difference.
Sections 5.12 – 5.15
Relative Position and Choosing Summaries
Percentiles · z-Scores · Choosing Mean/SD or Median/IQR · Common Mistakes
Section 5.12
Percentiles
A percentile tells you the relative position of a particular value within a distribution. The $k$th percentile is the value below which approximately $k$ percent of the data fall. The median is the 50th percentile; Q1 is the 25th percentile; Q3 is the 75th percentile.
Example 5.12.1
Finding a Percentile Rank
- Problem20 students took an exam. Scores (ordered): 54, 58, 62, 64, 65, 67, 68, 70, 72, 74, 75, 77, 78, 79, 81, 83, 85, 88, 91, 96. Maria scored 79. What is her percentile rank?
- CalculateValues below 79: 54, 58, 62, 64, 65, 67, 68, 70, 72, 74, 75, 77, 78 → 13 values$$\text{Percentile rank} = \frac{13}{20} \times 100 = 65$$
- InterpretMaria's score is at approximately the 65th percentile. She scored higher than about 65% of students in the group.
Answer: 65th percentile
Example 5.12.2
Finding a Value at a Given Percentile
- ProblemUsing the same 20 exam scores, find the score at the 25th percentile (Q1).
- CalculateLower half of 20 values: 54, 58, 62, 64, 65, 67, 68, 70, 72, 74$$Q1 = \frac{65 + 67}{2} = 66$$
- InterpretThe 25th percentile score is 66. About 25% of students scored at or below 66.
Answer: Q1 = 66 (25th percentile)
Context Note — Percentiles in Real Life
Percentile ranks are widely used in standardized testing, growth charts in medicine, athletic performance rankings, and income comparisons. When a doctor says a child's height is at the 60th percentile, it means the child is taller than about 60% of children the same age. Percentile ranks always describe position within a specific group — they lose meaning without knowing the comparison group.
Section 5.13
z-Scores
A z-score (also called a standardized score) measures how many standard deviations a particular value lies above or below the mean. A positive z-score means the value is above the mean; a negative z-score means it is below the mean; a z-score of 0 means the value is exactly at the mean.
Example 5.13.1
Calculating and Interpreting a z-Score
- ProblemClass exam scores: $\bar{x} = 74$, $s = 8$. Alex scored 90. Calculate Alex's z-score.
- Calculate$$z = \frac{90 - 74}{8} = \frac{16}{8} = 2.0$$
- InterpretAlex's score is 2.0 standard deviations above the class mean. A z-score of 2.0 is high — very few students in a typical class score that far above the mean.
Answer: $z = 2.0$ (2 standard deviations above the mean)
Example 5.13.2
Comparing Scores from Different Distributions
- ProblemMath scores: $\bar{x} = 72$, $s = 6$. Reading scores: $\bar{x} = 68$, $s = 10$. Carlos scored 78 in math and 82 in reading. On which test did he perform better relative to his class?
- Calculate
Math z-score: z = (78 − 72) / 6 = 6/6 = 1.0
Reading z-score: z = (82 − 68) / 10 = 14/10 = 1.4
- InterpretAlthough Carlos's raw reading score (82) is higher than his raw math score (78), his reading performance was relatively stronger — 1.4 standard deviations above his class mean vs. only 1.0 in math. The z-score is the fair comparison because it accounts for the different scales and spreads.
Answer: Reading ($z = 1.4$) was the relatively stronger performance
Example 5.13.3
Identifying an Unusual Value
- ProblemCollege students' sleep: $\bar{x} = 6.8$ hours, $s = 1.1$ hours. One student reports 3.5 hours. How unusual is this?
- Calculate$$z = \frac{3.5 - 6.8}{1.1} = \frac{-3.3}{1.1} = -3.0$$
- InterpretThis student's sleep time is 3.0 standard deviations below the mean — very unusual. In most symmetric distributions, fewer than 1% of values fall more than 3 standard deviations from the mean.
Answer: $z = -3.0$ — very unusual (3 standard deviations below the mean)
Convention for Unusual Values
Values with $|z| > 2$ are sometimes considered unusual; values with $|z| > 3$ are considered very unusual. These are guidelines, not strict rules — context always matters. A z-score does not by itself tell you the probability of a value; that connection to probability requires a distribution model (introduced in Chapter 11).
Section 5.14
Choosing Mean/Standard Deviation or Median/IQR
Numerical summaries come in pairs. The choice of pair depends on the shape of the distribution and the presence of outliers.
| Pair | Center | Spread | Use When |
|---|
| Pair 1 | Mean $\bar{x}$ | Standard Deviation $s$ | Distribution is approximately symmetric; no strong outliers |
| Pair 2 | Median | IQR | Distribution is skewed, has outliers, or values are open-ended |
Example 5.14.1
Choosing Summaries Based on Shape
- Situation AA sample of 50 adult women's heights (cm) forms a roughly symmetric, bell-shaped distribution with no outliers. → Report mean and standard deviation: "Mean height = 163.5 cm, $s$ = 6.2 cm."
- Situation BA sample of 50 household incomes is strongly right-skewed with a few very high earners. → Report median and IQR: "Median income = $52,400, IQR = $18,600." The median resists the pull of the high earners.
- Situation CA sample of 20 exam scores is left-skewed, with several very low scores pulling the mean down. → Report median and IQR. The median gives a better sense of typical performance than the deflated mean.
Rule: Match the summary pair to the shape of the distribution — always graph first.
Shape, Center, and Spread — The Core Principle
Always look at a graph of the data before choosing or reporting numerical summaries. The shape of the distribution determines which summaries are most appropriate. Two data sets can have identical means and standard deviations while looking completely different in shape.
Section 5.15
Common Summary Statistic Mistakes
Mistake 1 — Reporting the Mean for a Skewed Distribution
Mean household income = $84,000 → "Most families earn around $84,000"
If a distribution is strongly skewed or contains outliers, the mean may not represent what is typical. Check the distribution shape first. Report the median and IQR for skewed data.
Mistake 2 — Confusing Population Parameters with Sample Statistics
Using μ and σ when describing a sample of 30 students
$\mu$ and $\sigma$ describe populations; $\bar{x}$ and $s$ describe samples. Identify whether you have a sample or a population and use the correct notation.
Mistake 3 — Ignoring the Distribution Before Summarizing
Jumping directly to calculating the mean without graphing the data
Two distributions with identical means and standard deviations can have completely different shapes. Always create a dot plot, histogram, or box plot before summarizing numerically.
Mistake 4 — Treating Outliers as Automatic Errors
Deleting a flagged value from the data set without investigation
The 1.5 IQR rule flags potential outliers. Investigate every outlier in context. Correct genuine data entry errors; report real values honestly. Some outliers represent the most important data points.
Mistake 5 — Confusing Percentile and Percentage
"A percentile rank of 80 means the student got 80% of questions correct."
A percentile rank of 80 means the student scored higher than 80% of other students. It says nothing about how many questions were answered correctly.
Mistake 6 — Interpreting a z-Score as a Probability
"z = 2.0 means there is a 2% probability of that value."
z-scores measure position in standard deviation units, not probability. The connection to probability requires a specific distribution model, developed in Chapter 11.
Mistake 7 — Omitting Context from Any Summary
"The mean is 74."
"The mean exam score in this class of 28 students is 74 out of 100 points" is a complete summary. Every number in statistics needs context: what is being measured, in what units, for what group.
- 25 ordered scores: 41, 48, 52, 55, 58, 60, 61, 63, 65, 66, 68, 69, 70, 71, 72, 74, 75, 77, 78, 80, 83, 85, 87, 90, 95. What is the percentile rank of 70?
- Exam: $\bar{x} = 78$, $s = 6$. A student scores 88. Calculate and interpret the z-score.
- Birth weights: $\bar{x} = 7.4$ lb, $s = 0.9$ lb. Find the z-scores for babies weighing 9.2 lb and 5.6 lb. Which is more unusual?
- Long jump: $\bar{x} = 5.2$ m, $s = 0.4$ m. High jump: $\bar{x} = 1.7$ m, $s = 0.12$ m. Marcus jumped 5.8 m and 1.92 m. In which event did he perform better relative to the competition?
- A distribution is strongly right-skewed. Should you report mean/SD or median/IQR? Explain.
Chapter 5 — Back Matter
Chapter Summary
- Sec 5.1Summary statistics compress distributions to compact numbers. This is useful but costly — always examine a graph before choosing or reporting numerical summaries.
- Sec 5.2The mean $\bar{x}$ is the arithmetic average. It is the balancing point of the distribution and is sensitive to every value, including outliers. Population mean is $\mu$.
- Sec 5.3The median is the middle value of an ordered data set. It is resistant to outliers and skewness. When distributions are skewed, the median is usually the better measure of center.
- Sec 5.4The mode is the most frequently occurring value. It is most useful for categorical data or identifying the most common response in a discrete distribution.
- Sec 5.5Range = Maximum − Minimum. Simple but sensitive to outliers. Use range as a quick first look, not as the primary spread measure.
- Sec 5.6Quartiles divide the ordered distribution into four roughly equal parts: Q1 (25th percentile), Q2 (median, 50th percentile), Q3 (75th percentile).
- Sec 5.7IQR = Q3 − Q1. Measures the spread of the middle 50% of the data. Resistant to outliers; paired naturally with the median.
- Sec 5.8The five-number summary (Min, Q1, Median, Q3, Max) compactly describes a full distribution and is the basis for box plots. Shape can be read from the relative positions of these five values.
- Sec 5.9The 1.5 IQR rule: potential outliers fall below $Q1 - 1.5 \times \text{IQR}$ or above $Q3 + 1.5 \times \text{IQR}$. Outliers should be investigated in context, not automatically deleted.
- Sec 5.10Sample variance $s^2 = \sum(x_i - \bar{x})^2 / (n-1)$ measures average squared deviation from the mean. Squared deviations are used because raw deviations sum to zero.
- Sec 5.11Standard deviation $s = \sqrt{s^2}$ measures typical distance from the mean in original units. Not resistant. Always reported alongside the mean. Population SD is $\sigma$.
- Sec 5.12Percentile rank = (values below $v$) ÷ $n$ × 100. Describes relative position within a specific group. The median = 50th percentile; Q1 = 25th; Q3 = 75th.
- Sec 5.13z-score = $(x - \bar{x})/s$. Measures how many standard deviations a value is from the mean. Allows fair comparison across distributions with different scales. Not a probability.
- Sec 5.14Use mean/SD for symmetric distributions without strong outliers. Use median/IQR for skewed distributions or data with outliers. Shape determines the appropriate summary pair.
- Sec 5.15Seven recurring mistakes: mean for skewed data; wrong notation; ignoring shape; deleting outliers without investigation; confusing percentile with percentage; treating z-scores as probabilities; omitting context.
Key Terms Review
mean $\bar{x}$
The arithmetic average of all values. The balancing point of the distribution. Sensitive to outliers. Sample mean = $\bar{x}$; population mean = $\mu$.
median
The middle value of an ordered data set. Resistant to outliers and skewness. Preferred for skewed distributions.
mode
The most frequently occurring value. May be unimodal, bimodal, or absent. Most useful for categorical data.
range
Maximum − Minimum. Simplest measure of spread; sensitive to extreme values.
quartile
Values that divide the ordered distribution into four roughly equal parts: Q1 (25th percentile), Q2 (median), Q3 (75th percentile).
IQR
Interquartile range = Q3 − Q1. Measures the spread of the middle 50% of data. Resistant to outliers.
five-number summary
Min, Q1, Median, Q3, Max. Compact description of a full distribution; basis for the box plot.
outlier
A value unusually far from the bulk of the data. The 1.5 IQR rule provides a consistent identification method.
1.5 IQR rule
A value is a potential outlier if it falls below $Q1 - 1.5 \times \text{IQR}$ or above $Q3 + 1.5 \times \text{IQR}$.
variance $s^2$
The average squared deviation from the mean. Measured in squared units. Sample variance uses $n-1$ in the denominator.
standard deviation $s$
The square root of variance. Measures typical distance from the mean in original units. Not resistant. Sample SD = $s$; population SD = $\sigma$.
percentile rank
The percent of values in a data set that fall below a particular value. Describes relative position within a specific group.
z-score
$(x - \bar{x})/s$. The number of standard deviations a value lies above (positive) or below (negative) the mean.
resistant measure
A statistic not strongly affected by outliers. The median and IQR are resistant; the mean and standard deviation are not.
deviation from the mean
The difference $x_i - \bar{x}$ for each value. Deviations always sum to zero. Squared deviations are used to compute variance.
skewed distribution
A distribution in which one tail is longer than the other. Right-skewed: long right tail; mean > median. Left-skewed: long left tail; mean < median.
symmetric distribution
A distribution in which the left and right sides are mirror images. The mean and median are approximately equal.
Concept Review Questions
- Explain in your own words why the mean of a data set is called the "balancing point" of the distribution.
- Under what circumstances would the median be a better measure of center than the mean? Give an example.
- A data set has a five-number summary: 12, 25, 34, 47, 88. Without calculating, what is the IQR and what does it measure?
- Is standard deviation a resistant or non-resistant measure? Explain why.
- What does a z-score of −1.5 tell you about a value?
- What are the two summary pairs? When is each appropriate?
Skill Practice
Calculate the mean and median for each data set, then identify which measure you would report as the center and why.
- 5, 8, 6, 7, 9, 8, 10, 7, 6, 8
- 14, 17, 15, 14, 16, 98, 15, 13, 14, 16
- 100, 110, 120, 130, 140, 150, 160
- 37, 38, 39, 40, 41, 42, 44, 210
Find the five-number summary and IQR. Apply the 1.5 IQR rule to identify any potential outliers.
- 22, 27, 19, 31, 24, 28, 20, 35, 23, 26
- 6, 7, 8, 8, 9, 7, 6, 10, 5, 9, 8, 7, 6, 4
- 0, 1, 2, 2, 3, 3, 3, 4, 5, 6, 8, 22
- 18, 22, 24, 25, 26, 27, 28, 30, 54
Calculate the mean and standard deviation using a deviation table. Interpret the standard deviation in context.
- 10, 14, 12, 16, 13
- 3, 3, 3, 3, 3
- 12, 15, 11, 14, 13, 16
- 20, 25, 22, 28, 25
- Score of 70 in a group of 20 (13 below it). Percentile rank?
- Score = 90, $\bar{x} = 74$, $s = 8$. Find and interpret the z-score.
- $\bar{x} = 7.4$ lb, $s = 0.9$ lb. z-score for a 9.2 lb baby?
- Sleep: $\bar{x} = 6.8$ hr, $s = 1.1$ hr. z-score for 3.5 hours?
- Two events: long jump $\bar{x} = 5.2$, $s = 0.4$; high jump $\bar{x} = 1.7$, $s = 0.12$. Marcus: 5.8 m and 1.92 m. In which did he perform better relative to competitors?
Real-World Applications
- College readiness: A college reports that the median SAT score of admitted students is 1,210 and the mean is 1,250. What do the relative positions of the mean and median suggest about the shape of the score distribution?
- Income data: A report states: "The average household income is $84,000." A community organizer argues this figure is misleading. What concept explains why the organizer might be correct? What summary would be more honest?
- Quality control: Bolts must have diameter between 9.9 mm and 10.1 mm. A sample has $\bar{x} = 10.0$ mm and $s = 0.05$ mm. A bolt of 10.15 mm is found. (a) Calculate its z-score. (b) Is it within spec? Is it unusual?
- Healthcare: A doctor says a child is at the 35th percentile for height. (a) What does this mean? (b) Is the child unusually short? What percentile would suggest clinical concern?
- Sports: A basketball team's points over 15 games: 88, 92, 79, 104, 87, 95, 91, 86, 110, 84, 93, 90, 85, 97, 88. (a) Find mean and median. Which better represents typical scoring? (b) Find the five-number summary. (c) Apply the 1.5 IQR rule. (d) Calculate the standard deviation and interpret it.
- Finance: Monthly portfolio returns (%) over 8 months: 2.1, −0.4, 3.2, 1.8, −1.1, 2.7, 0.9, 1.5. (a) Find the mean and standard deviation. (b) In which month was the return most unusual? Calculate a z-score.
Error Analysis
Each problem shows a student's work or claim. Identify the error and write the correct answer or explanation.
Problem 1 — Finding the Median
Data: 51, 43, 37, 62, 29, 45, 38 — "Median = 43 because it is the middle number in the list."
What mistake did the student make? Find the correct median.
Problem 2 — Misleading Average
Salaries: $38K, $41K, $40K, $39K, $42K, $38K, $600K — reporter says "average salary is $119,714; workers are doing well."
Is the reporter's claim accurate? Calculate the median and explain which summary is more representative.
Problem 3 — Interpreting the IQR
"The IQR of this data set is 12, so all values must be within 12 units of each other."
Is this true? Explain what the IQR actually measures.
Problem 4 — Interpreting a z-Score
A student gets z = −0.6 and says: "This student scored in the bottom 60 percent."
What is the correct interpretation of z = −0.6?
Problem 5 — Standard Deviation of a Constant
"The standard deviation of 8, 8, 8, 8, 8 is 8."
Without calculating, explain why this cannot be correct. What is the standard deviation of a constant data set?
Chapter 5 Checkpoint
Chapter 5 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 5
Part A — Mean, Median, and Shape
- Data set: 14, 18, 12, 21, 15, 19, 13, 17, 20, 16.
(a) Calculate the mean.
(b) Find the median.
(c) Are the mean and median close? What does this suggest about the shape?
Part B — Five-Number Summary and Outliers
- Using the same 10 values above:
(a) Find Q1, Q3, and the IQR.
(b) Write the five-number summary.
(c) Apply the 1.5 IQR rule. Are there any potential outliers?
Part C — Standard Deviation
- Calculate the standard deviation of: 20, 25, 22, 28, 25. Show your deviation table and interpret the result.
- Two distributions are described: Distribution X — symmetric, bell-shaped, $\bar{x} = 50$, $s = 5$. Distribution Y — strongly right-skewed, $\bar{x} = 50$, $s = 5$, Median = 43.
(a) For Distribution X, which summary pair would you report?
(b) For Distribution Y, the mean and median differ by 7 points. What does this suggest?
(c) Why would reporting only the mean for Distribution Y be misleading?
Part D — Percentiles and z-Scores
- A student scored 72 on an exam. The class mean is 65 and the standard deviation is 9.
(a) Calculate the student's z-score.
(b) Is this score above or below average? By how much in standard deviation terms?
- 30 values are arranged in order. The 24th value is 88. What is this value's approximate percentile rank? What does it mean in context?
Part E — Application
- A school district records days absent per student for 12 students: 0, 1, 2, 2, 3, 3, 3, 4, 5, 6, 8, 22.
(a) Calculate the mean and median.
(b) Which is more representative of a typical student's absences?
(c) Apply the 1.5 IQR rule. Is 22 a potential outlier?
(d) Should the value 22 be removed? Explain your reasoning.
(e) Calculate the z-score for 22 days using the full data set's mean and SD. Is this value unusual?
Answers may be placed in the answer key.
Looking Ahead
Chapter 6: Scatter Plots, Correlation, and Linear Models
Chapter 5 developed tools for summarizing a single quantitative variable. Chapter 6 will ask: what if we have two quantitative variables measured on the same individuals? A scatter plot will display the relationship; correlation will measure its strength and direction; and regression will build a line that models the trend. The mean of $x$ and the mean of $y$ play a central role in fitting a regression line — the line of best fit always passes through the point $(\bar{x}, \bar{y})$. The z-scores you learned in Section 5.13 have a direct connection to the correlation formula. Every numerical summary skill from Chapter 5 will be used in Chapters 6 through 14.
MGU Library Connection
- Appendix F: Summary Statistics Formula Sheet — all formulas from this chapter in one reference
- Appendix O: Calculator and Spreadsheet Skills — computing mean, median, SD, and five-number summaries with technology
- Chapter 4 of this book: Displaying Quantitative Data — box plots, histograms, and shapes that guide summary choice
- Chapter 6: Scatter Plots and Linear Models — mean and SD used in correlation and regression
- Chapter 11: Normal Distributions — z-scores connect to probability under the normal curve
- Chapter 12: Sampling Distributions — standard error is built from standard deviation and sample size
- MGU Dictionary entries: mean, median, standard deviation, IQR, z-score, percentile, five-number summary