Standard deviation & variability · SAT Maths

Concept

Standard deviation (SD) measures how far values typically lie from the mean — a measure of spread, not centre. SAT problems mostly ask comparative questions (which dataset has greater SD?) rather than calculations.

SD = 0 ⇔ every value is identical.
Adding a constant to every value: mean shifts but SD is unchanged.
Multiplying every value by k: SD is multiplied by |k|.
Wider histograms / more spread-out box plots ⇒ greater SD.

Worked example 1

Two datasets have the same mean. Set A: {10, 10, 10, 10}; Set B: {5, 10, 10, 15}. Which has the greater standard deviation?

Solution

Set A. All values identical ⇒ no spread ⇒ SD = 0.

Set B. Values stray from the mean by 5, 0, 0, 5 ⇒ SD > 0.

Set B has the greater standard deviation.

Worked example 2

Dataset {2, 4, 6, 8} has mean 5 and SD ≈ 2.24. Add 10 to every value to get {12, 14, 16, 18}. What are the new mean and new SD?

Solution

New mean. Adding 10 shifts the mean by 10: 5 + 10 = 15.

New SD. Adding a constant doesn't change spread — SD stays at ≈ 2.24.

Mean 15, SD ≈ 2.24 (unchanged).

Practice test

8 questions on comparing spread, the effect of shifts and scaling, and reading variability from histograms.

Practice test · 8 questions Question 1 of 8 · Score 0