Comparing Spread and Standard Deviation

10.5: Comparing Spread and Standard Deviation. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 63 parts Practice Flashcards

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Translation and spread

Objective: Determine the effect of adding the same constant to all observations.

Prerequisite skills: mean, range, and comparing distributions. Adding a constant shifts the mean and every value equally. Each deviation from the mean stays the same, so range, IQR, and standard deviation remain unchanged. Common error: adding c to the standard deviation. Restrictions and interpretation: Standard deviation is nonnegative and measures spread around the mean; compare or transform it without requiring long manual computations.

Formulas

$SD(X+c)=SD(X)$

Worked examples

Try itA data set has standard deviation 23. Every value is increased by 28. What is the new standard deviation?Show answer

Both each observation and the mean rise by 28; their differences do not change. Therefore the standard deviation remains 23. Therefore the requested answer is 23.

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Translation and spread

Objective: Determine the effect of adding the same constant to all observations.

Formulas

$SD(X+c)=SD(X)$

Worked examples

Try itA data set has standard deviation 23. Every value is increased by 28. What is the new standard deviation?Show answer

Both each observation and the mean rise by 28; their differences do not change. Therefore the standard deviation remains 23. Therefore the requested answer is 23.

Scaling and spread

Objective: Scale standard deviation by the absolute value of a multiplier.

Multiplying every observation by k multiplies every deviation magnitude by |k|. Consequently standard deviation, range, and IQR scale by |k|. The mean scales by k, retaining its sign. Common error: squaring the multiplier for standard deviation instead of variance. Restrictions and interpretation: Standard deviation is nonnegative and measures spread around the mean; compare or transform it without requiring long manual computations.

Formulas

$SD(kX)=|k|SD(X)$

Worked examples

Try itA data set has standard deviation 24. Each value is multiplied by −3. What is the new standard deviation?Show answer

The multiplier magnitude is |−3|=3. Standard deviation cannot be negative, so the new value is 3(24)=72. Therefore the requested answer is 72.

Compare equal-center distributions

Objective: Identify greater dispersion without computing each variance.

With the same mean and sample size, moving observations farther from the mean increases the sum of squared deviations. Symmetric examples make this comparison transparent. Equal means do not imply equal spread. Common error: judging spread from the mean alone. Restrictions and interpretation: Standard deviation is nonnegative and measures spread around the mean; compare or transform it without requiring long manual computations. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

larger squared deviations imply larger standard deviation at fixed n

Worked examples

Try itWhich four-value set, all with mean 31, has the greatest standard deviation?Show answer

All sets have mean 31 and count 4. The correct set has squared-deviation sum 36+4+4+36=80. The others have 20, 8, and 0. With equal denominators, 80 gives the greatest standard deviation. Therefore the requested answer is {25, 29, 33, 37}.

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