AP Statistics Topic 1.9: Comparing Distributions and z-Scores

01 · Start here

The concept in 60 seconds

A comparison connects two or more groups using the same quantitative variable. Instead of writing separate descriptions and leaving the reader to work out the difference, say how the groups are similar or different and give evidence.

Compare whole distributions

Ask: which group has a higher typical value? Which has more variability? How do their shapes and unusual features compare?

Example: “Birch High has a higher median travel time than Cedar High: $27$ versus $19 \,\mathrm{min}$ .”

Compare individual positions

Ask: how far is this value from its own group’s mean, measured in standard deviations? A z-score answers that question.

Example: a score $2$ standard deviations above its test mean has a higher standardized position than one only $1$ standard deviation above its test mean.

Keep the two tasks distinct: a boxplot comparison describes groups. A z-score describes an individual value relative to a reference distribution.

Higher is not always preferable. A higher test score may be desirable; a shorter race time may be desirable. Explain what “better” means in the context before making a judgment.

02 · One variable, two groups

Compare school travel times

$2\text{ fictional schools}$ record students’ one-way home-to-school travel time on the Tuesday being studied, rounded to the nearest minute. We use the same $20$ Cedar High observations from earlier lessons and a new sample of $12$ Birch High observations.

See the recorded values

Cedar High, $n = 20$ : $8$ , $10$ , $12$ , $14$ , $14$ , $15$ , $16$ , $18$ , $18$ , $18$ , $20$ , $22$ , $24$ , $25$ , $26$ , $28$ , $30$ , $32$ , $35$ , $42$ .

Birch High, $n = 12$ : $18$ , $20$ , $22$ , $24$ , $24$ , $26$ , $28$ , $30$ , $30$ , $32$ , $34$ , $36$ .

$2\text{ school samples}$ on the same minute scale

$1$ dot represents $1$ student · stacks show repeated times

On a small screen, scroll the plot sideways to read the full scale.

The dotplots retain every observation. Compare their positions and patterns on the shared numerical scale rather than the total number of dots. Invented teaching data.

The Cedar observations extend farther toward the low end and have a longer upper tail. Birch’s observations are arranged symmetrically around $27 \,\mathrm{min}$ . The groups overlap: a higher typical time at Birch does not mean every Birch student travels longer than every Cedar student.

Compare the medians and the middle halves

Modified boxplots · actual non-outlier whisker endpoints · $1.5\times\operatorname{IQR}$ rule

On a small screen, scroll the plot sideways to read the full scale.

Birch’s median is higher,

27

versus

19 \,\mathrm{min}

. Cedar’s box is longer along the minute axis:

\operatorname{IQR}

12.5

versus

8 \,\mathrm{min}

. Neither sample has a flagged outlier. Sample sizes are explicitly added labels, rather than information encoded by a standard box. Invented teaching data.

Calculated summaries of these fictional samples; all times and spreads are in minutes.
Summary	Cedar High	Birch High
Sample size	$20$	$12$
Median	$19$	$27$
$Q_{1}$ to $Q_{3}$	$14.5$ to $27$	$23$ to $31$
$\operatorname{IQR}$	$12.5$	$8$
Minimum to maximum	$8$ to $42$	$18$ to $36$
Range	$34$	$18$
Mean	$21.35$	$27.00$
Sample standard deviation, $s$	$8.87$	$5.62$

Predict: which school has longer typical travel times, and which has more variability in its middle half?

Check your prediction

Birch has longer typical travel times, using its median of $27 \,\mathrm{min}$ versus Cedar’s $19$ . Cedar has more variability in the middle half, because its $\operatorname{IQR}$ is $12.5 \,\mathrm{min}$ versus Birch’s $8$ . These are two different comparisons: a higher center does not automatically mean greater spread.

Quartiles here use the median of each half of the ordered data. For odd sample sizes, exclude the overall median from the halves. Modified boxplots use the $1.5\times\operatorname{IQR}$ outlier rule. Neither school sample has a flagged observation under that rule.

03 · Read the evidence

Choose a graph and compare the important features

SOCS—shape, outliers, center and spread—is a useful reminder. Also look for clusters or gaps when the graph displays them. Let the question and available evidence determine what your answer needs; the mnemonic is not an instruction to invent missing information.

Center: which is typically higher?Name the median or mean and compare values with units. “Birch’s median is

8 \,\mathrm{min}

higher” is more informative than “Birch is higher.”

Spread: which varies more?Name the measure:

\operatorname{IQR}

, range or standard deviation. Different measures can lead to different comparisons.

Shape: how do patterns differ?Compare symmetry, skewness or peaks when visible. Identify the direction of a tail rather than the location of the tallest bar.

Unusual features: what stands out?Compare outliers, clusters and gaps that the display supports. If using a numerical outlier rule, name it.

What can each graph tell you?

Choose a display that retains the information needed for the question.
Display	Useful for	Limitation to remember
Aligned dotplots	Seeing individual values, repeated observations, clusters, gaps and overall patterns in small samples.	Large samples can become crowded. Use a common numerical scale.
Histograms	Comparing overall shape and variability, especially with larger samples.	Bins hide exact values. Match bin boundaries and consider relative frequencies when sample sizes differ.
Back-to-back stem-and-leaf plot	Comparing two small datasets while retaining every recorded value.	Read the two sides using the key; leaves on the left are usually listed in descending order.
Parallel boxplots	Quick comparisons of medians, IQRs, endpoints and flagged outliers. Asymmetry can suggest skewness.	Standard boxes hide peaks, clusters, gaps, exact means and sample sizes. Use fuller graphs for detailed shape claims.

Read a back-to-back stem-and-leaf plot

Here is a separate teaching example of travel times from $2\text{ small groups}$ . Stems are tens and leaves are ones. $\text{Every leaf represents }1\text{ student}$ .

Key: on the left, leaf $2$ beside stem $1$ means $12 \,\mathrm{min}$ for Group A; on the right, stem $1$ beside leaf $8$ means $18 \,\mathrm{min}$ for Group B.

\begin{array}{rcccl}\text{Group A }(n=10)&\mid&\text{Stem}&\mid&\text{Group B }(n=10)\\8\;8\;6\;4\;4\;2&\mid&1&\mid&8\\6\;4\;2\;0&\mid&2&\mid&0\;2\;4\;4\;6\;8\\&\mid&3&\mid&0\;0\;2\end{array}

Group A has $6$ observations in the teens, while Group B $\text{has only }1$ . Group B’s values generally sit higher on the common minute scale. Its $3$ observations in the thirties are $30$ , $30$ and $32 \,\mathrm{min}$ .

Check: what are the medians of

\text{these }2\text{ groups}

Group A’s $\text{middle }2\text{ values}$ are $18$ and $18$ , so its median is $18 \,\mathrm{min}$ . Group B’s $\text{middle }2\text{ are}$ $24$ and $26$ , so its median is $25 \,\mathrm{min}$ . Group B’s median is $7 \,\mathrm{min}$ higher. These values belong to this stem-and-leaf example, rather than to the Cedar–Birch samples.

04 · Make a fair comparison

Compare numerical summaries and graph scales fairly

Use the same feature on both sides

Compare a median with a median, an $\operatorname{IQR}$ with an $\operatorname{IQR}$ , and a standard deviation with a standard deviation. A statement such as “Cedar’s range is greater than Birch’s $\operatorname{IQR}$ ” mixes two different summaries and does not directly compare their variability using one measure.

Median and $\operatorname{IQR}$ are often informative when a distribution is skewed or contains outliers because they are resistant to extreme values. Mean and standard deviation describe the average and variation around it and are often useful for relatively symmetric distributions without strong outliers. If a question asks for a specific summary, use it and explain any relevant limitation.

The school table supports a comparison of means and standard deviations as well as medians and IQRs. These numbers were calculated from the observations; they were not read directly from the standard boxplots.

Common axes and bins keep the visual comparison honest

Graphs of the same variable should use compatible units and numerical scales. Histograms should use the same bin boundaries when comparing shapes. Check the labels before judging which distribution looks wider.

Different counts, matching proportions

Top: frequencies · bottom: percentages · matching $10$ -minute bins

On a small screen, scroll the plot sideways to read the full scale.

Group B’s count bars are

2\text{ times as tall}

. The matching percentage bars show the same proportional pattern in these bins; more observations do not by themselves mean greater spread. Invented teaching data.

In this separate fictional example, Group B has $2\text{ times as many observations}$ as Group A. Every frequency bar is $2\text{ times as tall}$ , but the percentage in each matching bin is the same: $10\%$ , $40\%$ , $40\%$ and $10\%$ . Taller count bars alone do not show greater variability or a higher center.

\text{Relative frequency}=\frac{\text{bin count}}{\text{group sample size}}

Symbol	Meaning
$x$	The individual value being standardized.
$\mu$ , pronounced “mu”	The population mean.
$\sigma$ , pronounced “sigma”	The population standard deviation.
$\bar{x}$ and $s$	The sample mean and sample standard deviation, used when appropriate population parameters are unknown.

Mistake	Why it fails	Better approach
Describe each group without comparing them.	The reader has to infer the relationship.	Use “higher than,” “less variable than” or “similar to,” with both group names.
Treat taller frequency bars as greater spread.	Bar height may reflect a larger sample rather than a wider distribution.	Check sample sizes, numerical scales, bin boundaries and relative frequencies.
Read a mean or $2$ peaks from standard boxplots.	Those details are not normally displayed.	Use supplied summaries for means, and dotplots or histograms for detailed shape features.
Call one group “more spread out” without naming a measure.	Range, $\operatorname{IQR}$ and standard deviation measure different aspects of variation.	Identify the measure and compare its values for both groups.
Use $z=\frac{x}{\sigma}$ or $z=\frac{\mu-x}{\sigma}$ .	The first omits the mean; the second reverses the intended sign.	Use $\frac{\text{value}-\text{mean}}{\text{standard deviation}}$ , then check above or below the mean.
Say $z = -2$ means a negative measurement.	The sign describes a difference from the mean, rather than the original value.	Say “ $2$ standard deviations below the reference mean.”
Convert every z-score to a percentile using a Normal table.	The distribution may not be Normal.	Interpret the standardized distance. Use a percentile model only when justified.
Claim a sample difference proves causation or a population difference.	Graphical differences alone do not establish those conclusions.	Keep the claim within the data and study design provided.

Comparisons of the Distributionsfor One Quantitative Variable

By the end of this lesson, you should be able to:

The concept in 60 seconds

Compare whole distributions

Compare individual positions

Compare school travel times

Choose a graph and compare the important features

What can each graph tell you?

Read a back-to-back stem-and-leaf plot

Compare numerical summaries and graph scales fairly

Use the same feature on both sides

Common axes and bins keep the visual comparison honest

Compare a claim with the feature it actually concerns

Understand and calculate z-scores

Calculate first, then interpret

A z-score does not automatically give a percentile

Worked examples

Example 1: compare two complete distributions

Example 2: a lower raw score can have a higher relative position

Example 3: lower values can be desirable

Example 4: recover the original value

Write a comparison and justify a claim

Improve a vague statement

Evaluate the claim the question actually makes

Common mistakes and how to fix them

Practice questions with hints and solutions

1. Compare center and variability

2. When two spread measures disagree

3. Compare unequal sample sizes

4. Calculate and interpret a negative z-score

5. Higher position or farther away?

6. Compare scores from different tests

7. Read the two sides of a stem-and-leaf plot

8. What does a z-score actually establish?

Quick revision

Quick questions students often ask

Final understanding check

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