9.1 Descriptive Statistics: Mean, Median, Mode, Range, SD, IQR, Quartiles, Percentiles

Descriptive Statistics: Center, Spread, and Position

Quick Answer: GRE data-analysis questions reward interpretation over raw computation. Know which measures of center are resistant to outliers (median, mode) versus sensitive (mean), how adding or multiplying a constant transforms each statistic, and how quartiles, percentiles, IQR, and standard deviation describe spread. The on-screen calculator can compute arithmetic averages, but most QC questions are decided by reasoning about properties, not by long division.

Measures of Center

Given a data set, three statistics describe its typical value:

• Mean (arithmetic average): sum of values divided by count. Sensitive to outliers — a single extreme value pulls the mean toward itself.
• Median: middle value of a sorted list. With n values, median position = (n + 1)/2. With an even count, median is the average of the two middle values. Resistant to outliers.
• Mode: most frequently occurring value. A data set can have no mode, one mode, or multiple modes. The mode is the only center measure that applies to non-numeric (categorical) data.

Worked Example — Outlier Shifts Mean, Not Median

Data: {4, 5, 6, 7, 100}. Mean = (4 + 5 + 6 + 7 + 100)/5 = 122/5 = 24.4. Median = 6 (third value of five sorted values). The outlier 100 pulls the mean to 24.4, about four times the median. A classic GRE QC exploit: Quantity A = mean of {4, 5, 6, 7, 100}; Quantity B = 7. Mean = 24.4 exceeds 7, so the answer is A — but the trap is assuming the mean equals the median and choosing C. Read carefully: outliers push the mean far above the median.

Measures of Spread

• Range = maximum − minimum. Simplest spread measure; sensitive to outliers.
• Interquartile range (IQR) = Q3 − Q1. Resistant to outliers because it ignores the bottom 25% and top 25%.
• Standard deviation (SD) measures typical distance of values from the mean. Larger SD = more spread.
• Variance = SD².

Quartiles and IQR

Sort the data, then split into four equal-sized groups. Q1 is the median of the lower half, Q2 is the overall median, Q3 is the median of the upper half. The five-number summary is {min, Q1, Q2, Q3, max}.

Worked Example — Five-Number Summary

Data: {2, 4, 6, 8, 10, 12, 14, 16}. n = 8. Median = (8 + 10)/2 = 9. Lower half {2, 4, 6, 8} has median (4 + 6)/2 = 5 = Q1. Upper half {10, 12, 14, 16} has median (12 + 14)/2 = 13 = Q3. IQR = 13 − 5 = 8. Range = 16 − 2 = 14.

Standard Deviation — Formula and Interpretation

The GRE uses the population standard deviation formula:

σ = √( (1/n) × Σ(x_i − μ)² )

where μ is the mean of the n values. The sample formula divides by (n − 1) instead of n, but the GRE does not test that distinction — focus on interpretation:

• SD ≥ 0, and SD = 0 only when every value equals the mean.
• Adding the same constant to every value does not change SD (spread unchanged).
• Multiplying every value by a positive constant c multiplies SD by c.
• Comparing two data sets: the one whose values sit farther from its mean has the larger SD.

Worked Example — Compare Spreads Without Computing

Set A: {4, 5, 6, 7, 8}. Set B: {2, 5, 6, 7, 10}. Both have mean 6. Set A's deviations from 6: {−2, −1, 0, 1, 2}. Set B's deviations: {−4, −1, 0, 1, 4}. Set B's squared deviations are larger (16 vs. 4 at the extremes), so Set B has the larger SD. No arithmetic needed — compare distances from the mean.

Percentiles

The kth percentile is the value below which k% of the data falls. To find the percentile rank of a value v: count how many values are strictly less than v, divide by n, multiply by 100. A score at the 80th percentile equals or exceeds 80% of the data — it is a position measure, not a percentage of a maximum score.

Worked Example — Percentile Trap

A student scores 72 on a test and is told she is at the 90th percentile. A common GRE trap: "She got 90% of the questions right." Wrong. She scored better than 90% of test takers; the percent correct is unrelated unless the test reports it separately.

Effect of Transformations — Memorize This Table

Worked Example — QC with Transformations

Quantity A: SD of {10, 20, 30, 40, 50}. Quantity B: SD of {20, 40, 60, 80, 100}. Set B is set A × 2, so SD(B) = 2 × SD(A). Answer: B is larger. No computation required.

Choosing the Right Measure

• Symmetric distribution: mean ≈ median; either works as "typical."
• Skewed distribution or outliers: use median.
• Categorical data: use mode.
• Spread with outliers: use IQR over range or SD.

The GRE loves to ask "which statistic is most appropriate?" — the answer hinges on whether outliers or skew are present.