Question type

Standard deviation on the SAT

Standard deviation measures how far values sit from the mean. On the SAT you compare it, never compute it: the data set whose values sit further from its mean has the larger standard deviation. Adding values equal to the mean makes it smaller; shifting every value by the same amount leaves it unchanged.

These questions show two data sets, often with equal means, and ask which has the greater standard deviation, or how it changes when values are added, removed or shifted.

It appeared on one of the five official practice tests we analysed. It looks like statistics but is a question about spread you can answer by looking.

On 1 of 5 official practice tests
Desmos helps

Why students miss it

Trying to calculate. Nothing in these questions needs the formula.

Confusing range with standard deviation: what matters is how many values sit far from the mean, not just the extremes.

Thinking that adding values equal to the mean changes nothing. The mean stays, but the average distance from it shrinks.

How to spot it

"Standard deviation" in the choices, usually beside "the means are equal".
Two lists of values, or two dot plots, with the same centre.
Values added at the mean, or every value shifted or scaled.

The method

Compare the spread

Find the mean, then look at how far the values sit from it.

Values bunched near the mean: smaller standard deviation.
Values spread to the ends: larger standard deviation.
Every value + 5: unchanged.
Values added at the mean: smaller.

With Desmos

If you want certainty, Desmos computes it from a list.

Type P=[20,25,30,35,40] and Q=[28,29,30,31,32], then stdev(P) and stdev(Q).

Comparing by eye is faster and is what the question is designed for.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#454Data

Data set P: 20, 25, 30, 35, 40 Data set Q: 28, 29, 30, 31, 32 Which statement correctly compares the two data sets?

Pick an answer, then check it.

Show the answer and explanation

Both sets are symmetric about

30

, so both means are

30

. Now look at spread: P's values sit

10

5

0

5

10

away from the mean, while Q's sit only

2

1

0

1

2

away. Larger typical distance from the mean means a larger standard deviation, so P's is greater.

2Question 2 of 2#421Data

A list of 10 test scores has a mean of 72 and a standard deviation of 8. Two additional scores, each equal to 72, are added to the list. Compared with the original list, the new list of 12 scores has

Pick an answer, then check it.

Show the answer and explanation

Adding values exactly equal to the mean can't shift the mean — the total rises by

144

and the count by 2, keeping it at 72. But those two new scores contribute zero to the squared deviations while the denominator grows, so the average squared deviation drops. The standard deviation gets smaller.

Questions students ask

Do you need to calculate standard deviation on the SAT?

No. SAT questions ask you to compare standard deviations or describe how they change, which you can do by looking at how spread out the values are.

What happens to the standard deviation if you add values equal to the mean?

It gets smaller. The mean stays the same, and the new values have zero distance from it, which lowers the average spread.

Practise this type until it is automatic

Five more standard-deviation questions, checked instantly, with every wrong choice explained. Free, no card.

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