Mathfolis

One-Variable Data: Distributions & Measures

Unit 3 · Problem-Solving & Data Analysis

What the SAT asks here

One-variable data on the SAT means computing measures of center (mean, median, mode), reasoning about how outliers shift them, and comparing standard deviations qualitatively (no formula calculation required). About one problem per test.

Measures of center

Mean — sum divided by count; sensitive to outliers
Median — middle value (or average of two middles); robust to outliers
Mode — most frequent value(s)

Spread

Range = max − min
Standard deviation — qualitative on the SAT: data clustered near the mean → smaller SD; data spread out → larger SD
SAT Tip: Mean and median are equal only for symmetric distributions. If a single outlier is added far from the mean, the mean moves more than the median.
Caution: The SAT will not ask you to compute a standard deviation. It only asks which data set has a larger or smaller SD based on visible spread.
Type 1

Compute mean / median / mode

Direct calculation from a small data set. Sort first when finding the median.

Example 1
The test scores of 5 students are 72, 85, 90, 85, 100. Find the median.

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Type 2

Effect of adding or removing data

Use total = mean × count to track how a new value shifts the mean. Reason about the median by re-sorting if the count changes parity.

Example 2
The mean of 5 numbers is 20. If a 6th number, 38, is added, what is the new mean?

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Type 3

Compare standard deviations (qualitative)

Pick the data set with values farther from the mean. No numeric computation required.

Example 3
Two data sets each have mean 50. Set A: {48, 49, 50, 51, 52}. Set B: {30, 40, 50, 60, 70}. Which has the larger standard deviation?

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