Topic 1.9 Notes – Comparisons of the Distributions for One Quantitative Variable

Comparing distributions of one quantitative variable

Here, you are comparing the same quantitative variable in different groups. That could be test scores for two classes, commute times for two routes, or heights before and after a program. The variable has to be the same, with the same units.

A real comparison does more than describe each graph separately. You need to say how they differ.

A full comparison usually checks these four features together:

• Shape
  Are the distributions symmetric or skewed? Unimodal or bimodal?
• Center
  Which group has the higher typical value?
• Variability
  Which group is more spread out, and by what measure?
• Unusual features
  Are there outliers, gaps, or clusters?

And always say it in context. For example, say “Route A commute times have a lower median by about 3 minutes than Route B commute times,” not just “A is lower.”

A quick graph reminder helps because the graph controls what you can honestly claim:

• Histograms, dotplots, back-to-back stem-and-leaf plots show shape well. You can see clusters, gaps, and possible outliers.
• Side-by-side boxplots are great for medians, IQRs, spread, and possible outliers. They do not show clusters or modality.

What to compare in the distributions

Shape

Skew comes from the longer tail.

• Right-skewed means the tail stretches to larger values.
• Left-skewed means the tail stretches to smaller values.
• If the graph shows two clear peaks, it is bimodal.

Boxplots can hint at skewness if one whisker is longer or the median is off-center, but they do not show detailed shape.

Center

You usually compare mean or median.

• Use median with IQR for skewed data or data with outliers.
• Use mean with standard deviation for roughly symmetric data without strong outliers.

Use comparison language. Say “Group A has a median about 3 units higher than Group B,” not just list both medians.

Also remember:

• mean $\approx$ median for symmetric distributions
• mean $>$ median for right-skewed distributions
• mean $<$ median for left-skewed distributions

Variability

Spread needs a named measure.

• Range = max minus min. It uses only the extremes, so it is nonresistant.
• IQR = $Q_3 - Q_1$. It shows the spread of the middle 50% and is resistant.
• Standard deviation measures typical distance from the mean and is nonresistant.

One group can have a smaller IQR but a larger range or standard deviation. That shows why “more variable” needs evidence.

Unusual features

Outliers, gaps, and clusters can matter a lot.

A modified boxplot uses the outlier rule

$$\text{below } Q_1 - 1.5(\text{IQR}) \quad \text{or above } Q_3 + 1.5(\text{IQR})$$

An outlier is unusual relative to its own distribution, not because it looks far from another group.

How to read comparative graphs

Comparative histograms

Use the same scale and same bins. Different bins can make the shape look different.

If sample sizes are very different, relative-frequency histograms are better than raw counts.

The image below contrasts counts with relative frequency for the same distribution. For comparisons between groups of different sizes, focus on the relative histogram rather than the cumulative plots.

Counts vs. relative-frequency histograms

Aligned dotplots and back-to-back stem-and-leaf plots

These keep individual values, so they are excellent for small data sets. You can see overlap clearly, which matters because a higher center does not mean every value is higher.

Stem-and-leaf plots need a key, and both groups must use the same stem units.

Side-by-side boxplots

These let you compare medians, IQRs, nonoutlying spread, and possible outliers quickly.

A very common trap is thinking a longer box means more data. It does not. The box length shows IQR, and each quartile still contains about 25% of the data.

Writing and justifying a comparison

A strong AP Stats comparison usually sounds like this:

1. Identify the variable, groups, and graph.
2. Compare center with evidence.
3. Compare variability with a named measure.
4. Compare shape if the graph supports it.
5. Mention outliers, gaps, or clusters.
6. Tie the evidence to the claim.

If a histogram only gives rough values, use approximate language like “about,” “roughly,” or “appears to.”

Common weak answers:

• “A is higher”
• listing two medians with no actual comparison
• saying “more spread out” without naming how
• claiming exact values from a histogram

Z-scores as relative position

A z-score tells how many standard deviations a value is from the mean.

With population values,

$$z=\frac{x-\mu}{\sigma}$$

With sample statistics,

$$z=\frac{x-\bar{x}}{s}$$

Interpretation:

• positive $z$ means above the mean
• negative $z$ means below the mean
• larger magnitude means farther from the mean
• z-scores have no units

Example:

If a score is 84 on a test with mean 76 and SD 4,

$$z=\frac{84-76}{4}=2$$

That score is 2 standard deviations above the mean.

Raw score alone does not settle which performance is more impressive across different distributions. The larger signed z-score is the higher relative position. The larger absolute z-score is the more extreme value.

For times, where lower is better, a more negative z-score can mean better performance.

This normal curve example shows how z-scores place two values on the same standardized scale, even when the original measurements come from different contexts.

Comparing relative position with z-scores