Unit 1: Exploring One-Variable Data

Numerical Summaries

Calculating and interpreting measures of center, spread, and position (z-scores).

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Graphical Displays of Data

When the Mean and Median Disagree

Section A's median score was 74.5; Section B's was 69. Before reading on: if you computed the mean (the ordinary arithmetic average) for each section, would you expect it to land close to that section's median — for both sections equally?

Definition — Mean and Standard Deviation

The mean x̄ = (Σx)/n is the arithmetic average — unlike the median, it uses every value's actual size, not just its rank. The sample standard deviation, s = √[Σ(x−x̄)²/(n−1)], measures the typical distance of a value from the mean. Both are sensitive to every individual value, including extreme ones — the opposite of the median and IQR's resistance.

Worked Example — Computing a Mean and Standard Deviation

For Section A's 12 scores (sum = 899): x̄ = 899/12 ≈ 74.92. Computing Σ(x−x̄)² (using x̄ ≈ 74.9167) gives about 746.92; dividing by n−1 = 11 gives a variance of about 67.90, so s = √67.90 ≈ 8.24. A score of 90 has z-score (90 − 74.92)/8.24 ≈ 1.83 — about 1.83 standard deviations above the mean.

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

Worked Example — Why Mean and Median Disagree for a Skewed Distribution

Section B's scores sum to 847, giving x̄ = 847/12 ≈ 70.58 — compare that to its median, 69. Section A's mean (≈74.92) sits almost exactly on its median (74.5), since Section A is roughly symmetric. Section B's mean sits noticeably above its median, because the mean gets pulled toward a distribution's longer tail — the two large values (88, 95) that stretched Section B's shape rightward in the previous lesson pull its average upward, while the median (based only on rank) doesn't move.

Tip

Comparing a dataset's mean to its median is itself a quick skew check: mean well above the median suggests a right skew; mean well below suggests a left skew; the two landing close together suggests rough symmetry.

Common Mistakes

Using n in the denominator of s instead of n−1.
AP Statistics always uses the sample standard deviation formula with n−1 in the denominator (dividing by n instead describes a different quantity, the population standard deviation σ, appropriate only when the full population's data — not a sample — is being summarized).
Treating a z-score as automatically meaning a value is unusual, without considering the distribution's actual shape.
A z-score describes distance from the mean in standard deviation units — how unusual that actually is depends on the distribution's shape, which is why the full toolkit (shape, center, spread together) matters more than any single number alone.

Key Takeaways

Mean and standard deviation use every value in a dataset directly, unlike median and IQR, which use only the data's order.
A z-score expresses how many standard deviations a value sits from the mean.
Mean and median disagreeing — mean pulled toward a longer tail — is itself a signal of a skewed distribution.

Summary

This closes Unit 1: shape, five-number summaries, means, and standard deviations all describe a single variable. Unit 2 turns to relationships between two variables at once, starting with categorical data.

Track your progress

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