The Central Idea
The correct display depends first on the type of variable. For categorical data, we usually compare categories. For quantitative data, we usually study how observations are distributed along a numerical scale.
The first decision is therefore:
\[ \boxed{\text{What kind of variable am I displaying?}} \]Why Visual Displays Matter
A graph can make patterns visible almost immediately.
- which categories are most common;
- how groups compare;
- where numerical values are concentrated;
- whether a distribution is symmetric or skewed;
- whether gaps or clusters appear;
- whether unusual observations may be present.
Categorical and Quantitative Displays Serve Different Purposes
Categorical displays emphasize counts or proportions by group. Quantitative displays preserve or approximate position on a numerical scale.
A bar chart for departments and a histogram for salaries may both use rectangles, but the rectangles have different meanings.
Common Displays for Categorical Data
- bar charts;
- relative-frequency bar charts;
- pie charts;
- segmented or stacked bar charts;
- side-by-side bar charts.
Bar Charts
A bar chart represents each category with a separate bar. The height or length of the bar corresponds to a frequency, relative frequency, or percentage.
| Payment Method | Frequency |
|---|---|
| Cash | 12 |
| Credit | 28 |
| Debit | 20 |
| Digital Wallet | 10 |
A bar chart would place the four categories along one axis and use bar heights \(12\), \(28\), \(20\), and \(10\).
Bars Are Separated
In a bar chart, bars are usually separated by visible gaps. The gaps reinforce that the categories are distinct groups rather than adjoining numerical intervals.
Category Order in a Bar Chart
Nominal categories may be arranged:
- alphabetically;
- chronologically when time matters;
- from largest frequency to smallest;
- according to a natural organizational order.
Ordinal categories should usually preserve their meaningful ranking.
Relative-Frequency Bar Charts
If a category has frequency \(f\) in a dataset of size \(n\), then
\[ \boxed{\text{relative frequency}=\frac{f}{n}}. \]Relative-frequency bar charts are especially useful when comparing groups with different sample sizes.
Example: Counts Versus Proportions
Branch A has \(50\) purchases among \(200\) customers, while Branch B has \(100\) purchases among \(500\) customers.
\[ \frac{50}{200}=0.25 \] \[ \frac{100}{500}=0.20 \]Branch B has the larger count, but Branch A has the larger proportion.
Pie Charts
A pie chart represents a whole as a circle divided into sectors. Each sector represents a category's share of the total.
If a category has relative frequency \(r\), then its sector angle is
\[ \boxed{\theta=360^\circ r}. \]Example: Pie-Chart Angle
If \(25\%\) of observations belong to one category, then
\[ \theta=360^\circ(0.25)=\boxed{90^\circ}. \]When Pie Charts Work Well
- the categories represent parts of one meaningful whole;
- there are relatively few categories;
- the category shares differ enough to compare visually;
- the total corresponds naturally to \(100\%\).
Limitations of Pie Charts
- many categories are difficult to compare;
- similar sector sizes are hard to distinguish;
- precise comparisons are less direct than with bars;
- overlapping categories cannot represent exclusive parts of one whole.
Segmented Bar Charts
A segmented bar chart represents a whole as one bar divided into category segments. A \(100\%\) segmented bar shows the proportion assigned to each category.
Side-by-Side Bar Charts
A side-by-side bar chart places bars for different groups beside one another within each category. Relative frequencies are often preferable when the group sizes differ.
Common Displays for Quantitative Data
- dot plots;
- stem-and-leaf plots;
- histograms;
- frequency polygons;
- cumulative-frequency displays.
Dot Plots
A dot plot places one mark for each observation above its position on a numerical scale. Repeated observations stack above the same value.
For the data
\[ 1,\ 2,\ 2,\ 2,\ 3,\ 4,\ 4,\ 5, \]the value \(2\) would have three dots and \(4\) would have two.
Why Dot Plots Are Useful
- exact observed values remain visible;
- repeated values are clear;
- gaps and clusters can be seen;
- possible unusual values are easy to inspect.
Stem-and-Leaf Plots
A stem-and-leaf plot separates each numerical value into a leading part called the stem and a trailing digit called the leaf.
For
\[ 21,\ 24,\ 27,\ 31,\ 33,\ 38, \]2 | 1 4 7 3 | 1 3 8
A key should explain the scale:
\[ 2|1=21. \]Advantages of Stem-and-Leaf Plots
Stem-and-leaf plots show the overall distribution while preserving the original observations. They work best for small or moderate datasets with a convenient decimal structure.
Ordered Leaves
4 | 1 3 3 6 9
represents
\[ 41,\ 43,\ 43,\ 46,\ 49. \]Histograms
A histogram divides a quantitative scale into intervals called bins or classes. Bar height represents frequency, relative frequency, or an appropriate density.
Histogram Bars Usually Touch
Histogram bars usually touch because the intervals form adjacent parts of one numerical scale.
\[ 20\le x<30 \]is immediately followed by
\[ 30\le x<40. \]Bar Chart Versus Histogram
| Feature | Bar Chart | Histogram |
|---|---|---|
| Data type | Categorical | Quantitative |
| Horizontal positions | Categories | Numerical intervals |
| Bars | Usually separated | Usually touching |
| Order | May be rearranged for nominal data | Fixed numerically |
| Bar width | Mostly visual | Represents interval width |
Example: Histogram Classes
| Time (minutes) | Frequency |
|---|---|
| 10–19 | 3 |
| 20–29 | 7 |
| 30–39 | 11 |
| 40–49 | 6 |
| 50–59 | 3 |
A histogram places these intervals along the numerical axis and uses corresponding heights \(3,7,11,6,3\).
Bin Width Affects Appearance
Very wide bins can hide structure. Very narrow bins can produce a noisy, fragmented appearance. A useful histogram reveals structure without implying more precision than the data support.
Histogram Shape
Important features include:
- number of major peaks;
- symmetry or skewness;
- gaps;
- clusters;
- possible outliers;
- overall spread.
Symmetric Distributions
A distribution is approximately symmetric when the patterns on the left and right sides of its center are roughly balanced.
Right-Skewed Distributions
A distribution is right-skewed when its longer tail extends toward larger numerical values.
Income, waiting times, and transaction sizes can sometimes exhibit this pattern.
Left-Skewed Distributions
A distribution is left-skewed when its longer tail extends toward smaller numerical values. The direction of skewness is named for the tail.
Unimodal and Multimodal Distributions
A distribution with one major peak is often called unimodal. One with two major peaks is bimodal. Multiple peaks can suggest distinct underlying groups or processes.
Gaps and Clusters
A gap is a region with few or no observations. A cluster is a region where many observations are concentrated.
Possible Outliers
An outlier is an observation unusually far from the main body of the data. A graph can identify possible outliers visually, but a formal rule may be needed before classifying them.
Frequency Polygons
A frequency polygon plots a point at each class midpoint with vertical position determined by the class frequency. Adjacent points are connected by line segments.
Cumulative-Frequency Displays
A cumulative-frequency graph shows how many observations are at or below successive values or class boundaries. Such displays are useful for locating percentiles and comparing accumulation across distributions.
Choosing a Display for Categorical Data
Use a bar chart when:
- precise category comparison matters;
- many categories are present;
- frequencies or proportions are being compared.
Use a pie chart when:
- categories form one clear whole;
- there are only a few categories;
- part-to-whole comparison is the main message.
Choosing a Display for Quantitative Data
Use a dot plot when exact observations should remain visible and the dataset is relatively small.
Use a stem-and-leaf plot when exact values should remain recoverable and the numerical scale supports a clean stem structure.
Use a histogram when the dataset is larger and overall distribution shape matters more than individual observations.
Axes and Scales
A numerical axis should clearly indicate:
- the variable;
- the units;
- the scale increments;
- the range displayed.
Truncated Axes
A truncated axis begins at a value other than the natural baseline. This is not automatically wrong, but it can visually magnify small differences. For bar charts, a nonzero baseline can be especially misleading if it is not clearly indicated.
Area and Volume Can Mislead
Doubling both height and width of a pictorial symbol multiplies its area by
\[ 2\cdot2=4. \]A value that merely doubled can therefore appear four times as large.
Labels and Legends
A useful display should make clear:
- what variable is shown;
- what each category or symbol means;
- whether values are counts, proportions, percentages, or densities;
- what population or sample the data describe.
Example: Choosing the Correct Display
| Variable | Type | Possible Display |
|---|---|---|
| Department | Categorical nominal | Bar chart |
| Annual salary | Quantitative | Histogram |
| Performance category | Categorical ordinal | Ordered bar chart |
| Completed projects | Quantitative discrete | Dot plot or histogram |
Example: Reading a Categorical Display
| Category | Frequency |
|---|---|
| A | 15 |
| B | 27 |
| C | 18 |
- Category B is most common.
- Category A is least common.
- The total is \(15+27+18=60\).
- Category B represents \(\frac{27}{60}=45\%\).
Example: Reading a Quantitative Distribution
Suppose a histogram has most observations between \(20\) and \(40\), with a few observations extending toward \(80\).
- the data are concentrated at lower-to-middle values;
- the distribution has a longer right tail;
- the distribution may be right-skewed;
- the largest values may deserve inspection.
Common Error: Using a Histogram for Categories
Departments are categories, not numerical intervals. A bar chart is appropriate. A histogram would incorrectly suggest a continuous numerical scale connecting the categories.
Common Error: Interpreting Category Position Numerically
In a nominal bar chart, horizontal distance between categories does not represent numerical difference.
Common Error: Unequal Histogram Bin Widths Without Adjustment
If histogram bins have unequal widths, raw bar height alone may not fairly represent frequency. Frequency density may be needed so that bar area corresponds to frequency. Equal-width bins avoid this complication in introductory work.
Common Error: Changing Bin Width to Force a Story
Histogram appearance depends on binning. Selectively choosing bins to exaggerate or conceal structure can mislead. A defensible display uses a reasonable binning rule.
Common Error: Using Counts When Proportions Are Needed
If two groups have different sizes, raw counts can reflect group size rather than the phenomenon being compared. Relative frequencies or percentages are often more informative.
Common Error: Omitting the Sample Size
The value \(50\%\) could mean \(1\) of \(2\) observations or \(5{,}000\) of \(10{,}000\). The percentage is identical, but the amount of information supporting it differs greatly.
A Reliable Procedure for Choosing a Display
- Identify the variable.
- Classify it as categorical or quantitative.
- Determine whether the goal is comparison, composition, exact-value inspection, or distribution shape.
- Choose a display appropriate for that goal.
- Label categories, axes, units, and scales clearly.
- Use frequencies, relative frequencies, or percentages consistently.
- Check that visual proportions accurately reflect numerical values.
- Inspect the display for patterns, gaps, clusters, skewness, and unusual observations.
- State conclusions in the language of the variable and population.
Practice 1: Choose the Display
- favorite streaming service;
- customer waiting time in minutes;
- shirt size: small, medium, large;
- number of calls received per hour.
Practice 2: Bar Chart or Histogram?
- department name;
- annual income;
- vehicle color;
- exam score.
Practice 3: Relative Frequency
A category contains \(24\) of \(80\) observations. What relative-frequency bar height should be used?
Practice 4: Pie-Chart Angle
A category represents \(35\%\) of a dataset. Find the corresponding pie-chart angle.
Practice 5: Stem-and-Leaf Interpretation
3 | 2 5 8 4 | 1 1 7
Using the key \(3|2=32\), list the original values.
Practice 6: Identify the Shape
A distribution has most observations between \(10\) and \(30\), with a small number extending as high as \(90\). Which direction is it likely skewed?
Practice 7: Identify the Error
A graph displays three departments with bars touching one another and calls the result a histogram. Explain the problem.
Practice 8: Compare Unequal Groups
Group A has \(30\) successes among \(100\) cases. Group B has \(45\) successes among \(300\) cases. Which group should have the taller bar in a relative-frequency comparison?
Practice 9: Histogram Interpretation
A histogram shows two clear peaks separated by a low-frequency interval. What feature of the distribution does this suggest?
Practice 10: Misleading Scale
Two bar heights represent values \(98\) and \(100\), but the vertical axis begins at \(97\). Why might the visual difference appear misleadingly large?
Practice Answers
Practice 1
- favorite streaming service: bar chart;
- customer waiting time: histogram, dot plot, or stem-and-leaf plot;
- shirt size: ordered bar chart;
- calls per hour: dot plot or histogram.
Practice 2
- department name: bar chart;
- annual income: histogram;
- vehicle color: bar chart;
- exam score: histogram.
Practice 3
\[ \frac{24}{80}=\boxed{0.30}. \]The bar height is \(0.30\), or \(30\%\).
Practice 4
\[ 360^\circ(0.35)=\boxed{126^\circ}. \]Practice 5
\[ \boxed{32,\ 35,\ 38,\ 41,\ 41,\ 47}. \]Practice 6
The distribution is likely right-skewed.
Practice 7
Departments are categorical rather than numerical intervals. The proper display is a bar chart, whose bars should ordinarily be separated.
Practice 8
\[ \frac{30}{100}=0.30,\qquad \frac{45}{300}=0.15. \]Group A should have the taller relative-frequency bar.
Practice 9
Two distinct peaks suggest a bimodal distribution and may indicate two underlying subgroups or processes.
Practice 10
Beginning the axis at \(97\) removes most of the baseline and visually magnifies a difference of only
\[ 100-98=\boxed{2}. \]What to Remember
- Categorical displays compare categories using frequencies, proportions, or percentages.
- Bar charts are a standard choice for categorical data.
- Pie charts emphasize part-to-whole relationships.
- Quantitative displays place observations or intervals along a numerical scale.
- Dot plots and stem-and-leaf plots preserve individual values.
- Histograms group quantitative observations into adjacent numerical intervals.
- Bar-chart categories may often be reordered; histogram intervals cannot be arbitrarily rearranged.
- Distribution shape includes symmetry, skewness, peaks, gaps, clusters, and possible outliers.
- Relative-frequency displays are often preferable for groups of unequal size.
- A statistical display should clarify the data without distorting magnitude or structure.
Where We Go Next
Visual displays reveal the overall structure of a dataset, but we also need numerical summaries that describe where observations are concentrated.
The next lesson, Mean, Median, Mode, and Quantiles, develops measures of center, position, and ordered location for quantitative data.