Mathfolis

Summary Statistics for a Quantitative Variable

Unit 1 · Exploring One-Variable Data

What AP Stats asks here

Numerical summaries split into center (mean, median) and spread (range, IQR, SD). The decision rule is resistance: mean and SD are pulled by outliers; median and IQR are not. Almost every AP problem on this topic either asks you to compute a summary or to choose between a resistant and non-resistant pair.

Center and spread

Sample mean
xˉ=1n∑i=1nxi\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i
Sample SD
s=1n−1∑i=1n(xi−xˉ)2s = \sqrt{\dfrac{1}{n - 1}\sum_{i=1}^{n} (x_i - \bar{x})^2}
IQR
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1

1.5 × IQR outlier rule

Lower fence
Q1−1.5⋅IQRQ_1 - 1.5 \cdot \text{IQR}
Upper fence
Q3+1.5⋅IQRQ_3 + 1.5 \cdot \text{IQR}
Outlier
x<lower fence    or    x>upper fencex < \text{lower fence} \;\;\text{or}\;\; x > \text{upper fence}

Resistance

Resistant
median, IQR\text{median, IQR}
Not resistant
mean, range, SD\text{mean, range, SD}
AP Tip: Use sample SD with n − 1 for every dataset you encounter on AP. Dividing by n instead is the canonical mistake and creates a systematically smaller (biased) estimate.
Caution: For right-skewed data or data with extreme outliers, lead with median + IQR. Reporting the mean alone misrepresents the typical value because the mean tracks the tail.
Type 1

Compute mean, median, IQR, or SD

Apply the definition. Sort first for median and IQR; use n − 1 for sample SD; watch the off-by-one in the lower/upper halves when n is even vs odd.

Example 1
Quiz times (minutes) for nine students: 8, 12, 7, 15, 10, 14, 9, 11, 13. Which gives the mean and median? (A) Mean 11, median 10 (B) Mean 11, median 11 (C) Mean 12, median 11 (D) Mean 11, median 12

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Type 2

Resistance and choice of summary

When data is heavily skewed or contains extreme outliers, the resistant pair (median + IQR) better describes the typical case. The mean follows the tail.

Example 2
Household incomes in a small town (thousands of dollars): 30, 35, 38, 40, 42, 45, 48, 50, 55, 800. Which best represents the typical household, and why? (A) Mean $118.3K — it incorporates every value. (B) Median $43.5K — the single value at $800K pulls the mean far above the typical household. (C) Mean $43.5K — most households are in this range. (D) Median $118.3K — the median always equals the mean for skewed data.

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Type 3

1.5 × IQR outlier rule

Compute the IQR, then build the fences. A value strictly below the lower fence or strictly above the upper fence is an outlier by this convention.

Example 3
A dataset has Q1=40Q_1 = 40 and Q3=60Q_3 = 60. Using the 1.5 × IQR rule, classify the values 15 and 95. (A) Both outliers. (B) Neither is an outlier. (C) 15 is an outlier; 95 is not. (D) 95 is an outlier; 15 is not.

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