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All courses Math III · S-ID.4 53 of 55
Use mean and standard deviation to fit normal distributions and estimate population percentages with technology.

Read normal mean and positive standard deviation separately

Problem
Heights follow X ~ N(68 inches, 3 inches), with the second parameter representing standard deviation. State the mean, standard deviation, and variance with units.
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Hint

Identify center language versus spread/typical-distance language.

Mean mu is the center; standard deviation sigma>0 is the typical spread in the original units; variance is sigma².

Solution walkthrough

01

Read the normal-model parameters

\[X~N(68~\text{inches},3~\text{inches})\]

The prompt states that the first parameter is the mean and the second is the standard deviation.

02

Name mean and standard deviation

\[\text{mean}=68~\text{inches};~\text{standard}~\text{deviation}=3~\text{inches}\]

Both location and spread retain the original height unit, inches.

03

Compute variance

\[\text{variance}=(\text{standard}~\text{deviation})^2=(3~\text{inches})^2=9~\text{square}~\text{inches}\]

Variance is the square of standard deviation, so its unit is squared.

04

State all three quantities

\[\text{The}~\text{mean}~\text{is}~68~\text{inches},~\text{the}~\text{standard}~\text{deviation}~\text{is}~3~\text{inches},~\text{and}~\text{the}~\text{variance}~\text{is}~9~\text{square}~\text{inches}.\]

The parameter order, squaring relationship, and units are all preserved.

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Another way

  1. Use variance=3*3=9 and attach inches times inches, which gives square inches.

!

Common mistake

Do not report variance as 3 square inches; square both the numerical standard deviation and its unit.