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Interpreting Standard Deviation

SAT Math · Problem-Solving & Data Analysis · Updated October 2026

What standard deviation tells you

Standard deviation measures how far the values in a data set are from the mean. The more scattered the values, the larger it is. It tells you nothing about whether the average itself is high or low.

On the SAT you will almost never have to calculate it. You compare the spread of two data sets, or you decide what happens to the standard deviation when the data changes. I teach this every year in AP Statistics, and the students who do well are the ones who picture the spread instead of reaching for a formula.

Standard deviation questions on the SAT come in three kinds: comparing two data sets, predicting what a change does to the spread, and reading the spread from a dot plot or a list. In every one of them, the question you are really answering is how far the values are from the middle.

Spread, and what changes it

Tightly clustered data has a small standard deviation. Widely scattered data has a large one. Two data sets can have exactly the same mean and very different standard deviations. If every value is identical, the standard deviation is $0$.

Most SAT questions about standard deviation are really about change. Once you know the rules in this table, you can answer them without any arithmetic.

What happens to the dataMeanStandard deviation
Add the same number $c$ to every valueGoes up by $c$No change
Subtract the same number from every valueGoes down by that numberNo change
Multiply every value by $k$Multiplied by $k$Multiplied by $|k|$
Remove a value far from the mean (an outlier)Moves toward the rest of the dataDecreases
Add a value far from the meanMoves toward the new valueIncreases
Add a value equal to the meanNo changeDecreases slightly

Three examples

Example 1: two sets with the same mean

Set A is $48, 49, 50, 51, 52$. Set B is $30, 40, 50, 60, 70$. Which has the larger standard deviation?

  1. Find the middle of each set. Both have a mean of $50$.
  2. Look at the distances from $50$. Set A: $2, 1, 0, 1, 2$. Set B: $20, 10, 0, 10, 20$.
  3. Set B's values are ten times farther from the mean, so Set B has the larger standard deviation. You don't need the formula for this.

Example 2: removing an outlier

A data set is $12, 14, 15, 16, 43$. What happens to the mean and the standard deviation if $43$ is removed?

  1. The mean is $\dfrac{12 + 14 + 15 + 16 + 43}{5} = \dfrac{100}{5} = 20$.
  2. Without $43$, the mean is $\dfrac{57}{4} = 14.25$. The mean decreases.
  3. $43$ was the value farthest from the mean. The four values left are close together, so the standard deviation decreases too.

Example 3: adding the same amount to every value

Quiz scores have a mean of $62$ and a standard deviation of $4$. The teacher adds $7$ points to every score.

  1. Every score moves up by $7$, so the mean becomes $62 + 7 = 69$.
  2. Every score moves the same amount, so each score's distance from the mean is unchanged.
  3. The standard deviation stays $4$. Moving the whole data set up does not spread it out.

The empirical rule

When data is roughly bell-shaped (symmetric, with most values near the middle), the standard deviation tells you where the values fall:

−3 SD−2 SD−1 SDmean+1 SD+2 SD+3 SD 68% within 1 SD 13.5% 13.5%

The darker band holds about $68\%$ of the data. Add the two lighter bands and the whole shaded area, within $2$ standard deviations, holds about $68\% + 13.5\% + 13.5\% = 95\%$.

Example 4: using the empirical rule

Heights of a large group of students are roughly bell-shaped with a mean of $64$ inches and a standard deviation of $3$ inches.

  1. One standard deviation on each side: $64 - 3 = 61$ and $64 + 3 = 67$. About $68\%$ of heights are between $61$ and $67$ inches.
  2. Two standard deviations on each side: $64 - 6 = 58$ and $64 + 6 = 70$. About $95\%$ are between $58$ and $70$ inches.
  3. The other $5\%$ is split evenly between the two ends, so about $2.5\%$ of students are taller than $70$ inches.

The SAT itself rarely asks about the empirical rule by name. It is still worth knowing: it turns "one standard deviation" into a picture you can use, and it is standard material in AP Statistics and on many state tests. It only works for data that is roughly bell-shaped, not for skewed data.

Mistakes I see often

Linking standard deviation to the mean. A higher standard deviation means more spread, not a higher average.

Confusing standard deviation with range. Range uses only the two extreme values. Standard deviation reflects every value.

Thinking a shift changes the spread. Adding or subtracting the same number from every value moves the mean but leaves the standard deviation unchanged.

Ignoring outliers. One value far from the rest raises the standard deviation a lot. Removing it lowers the standard deviation.

Using the empirical rule on skewed data. The $68\%$, $95\%$, $99.7\%$ pattern only holds when the data is roughly bell-shaped.

Practice questions

Most of these can be done without calculating anything. Picture the spread first, then open the solution to check your reasoning.

Easy
Dataset X has values $10, 10, 10, 10, 10$. Dataset Y has values $2, 6, 10, 14, 18$. Which dataset has the larger standard deviation?
  • ADataset X
  • BDataset Y
  • CThey have the same standard deviation
  • DIt cannot be determined without calculating
Show solution
Answer: B. Both datasets have a mean of $10$. Every value in Dataset X equals the mean, so its standard deviation is $0$. Dataset Y's values sit as far as $8$ away from the mean, so it has the larger standard deviation.
Easy
A teacher reports that every student in a class scored exactly $78$ on a quiz. What is the standard deviation of the scores?
  • A$1$
  • B$78$
  • C$0$
  • DIt cannot be determined from the information given
Show solution
Answer: C. The mean is $78$ and every score equals the mean. No score sits any distance from the mean, so there is no spread and the standard deviation is $0$.
Medium
Two groups of students recorded their daily exercise time. Group 1 has a mean of $45$ minutes and a standard deviation of $20$ minutes. Group 2 has a mean of $30$ minutes and a standard deviation of $5$ minutes. Which group's data is more spread out?
  • AGroup 2
  • BThey are equally spread out
  • CIt cannot be determined without the data values
  • DGroup 1
Show solution
Answer: D. Standard deviation measures spread. Group 1's standard deviation of $20$ is four times Group 2's standard deviation of $5$. The means do not matter for this comparison.
Medium
A set of test scores has a mean of $62$ and a standard deviation of $4$. The teacher adds $7$ points to every score. What are the mean and standard deviation of the new scores?
  • AMean $62$, standard deviation $11$
  • BMean $69$, standard deviation $4$
  • CMean $69$, standard deviation $11$
  • DMean $62$, standard deviation $4$
Show solution
Answer: B. Every score goes up by $7$, so the mean goes up by $7$ to $69$. Every score moves the same amount, so the distances from the mean do not change and the standard deviation stays $4$.
Medium
A data set consists of the values $12, 14, 15, 16, 43$. If the value $43$ is removed, what happens to the mean and the standard deviation?
  • AThe mean decreases and the standard deviation increases.
  • BThe mean stays the same and the standard deviation decreases.
  • CBoth increase.
  • DBoth decrease.
Show solution
Answer: D. The mean drops from $\dfrac{100}{5} = 20$ to $\dfrac{57}{4} = 14.25$. The value $43$ was far from all the others, so removing it leaves values that are close together, and the standard deviation decreases.
Hard
A data set has a standard deviation of $3$. Each value in the data set is multiplied by $2$. What is the standard deviation of the new data set?
  • A$3$
  • B$5$
  • C$6$
  • D$9$
Show solution
Answer: C. Multiplying every value by $2$ also multiplies the mean by $2$, so every distance from the mean doubles. The standard deviation doubles from $3$ to $6$. Choice B comes from adding $2$ instead of multiplying.
Hard
List A is $1, 5, 9, 13, 17$. List B is $6, 7, 9, 11, 12$. Which statement is true?
  • AThe means are equal, and List B has the larger standard deviation.
  • BList A has the larger mean and the larger standard deviation.
  • CThe means are equal, and List A has the larger standard deviation.
  • DThe lists have the same mean and the same standard deviation.
Show solution
Answer: C. Both lists add to $45$, so both means are $\dfrac{45}{5} = 9$. List A's values sit $8, 4, 0, 4, 8$ from the mean. List B's sit $3, 2, 0, 2, 3$ from the mean. List A is more spread out, so it has the larger standard deviation.

Empirical rule practice

For each question, the data is roughly bell-shaped, so the $68\%$, $95\%$, $99.7\%$ pattern applies.

Medium
Scores on a practice test are roughly bell-shaped with a mean of $500$ and a standard deviation of $100$. About what percent of scores fall between $400$ and $600$?
  • A$34\%$
  • B$68\%$
  • C$95\%$
  • D$99.7\%$
Show solution
Answer: B. $400 = 500 - 100$ and $600 = 500 + 100$, so the interval is $1$ standard deviation on each side of the mean. About $68\%$ of scores fall there. Choice A is only the half from $500$ to $600$.
Medium
Heights of a large group of students are roughly bell-shaped with a mean of $64$ inches and a standard deviation of $3$ inches. About $95\%$ of the heights fall between which two values?
  • A$61$ and $67$ inches
  • B$58$ and $70$ inches
  • C$55$ and $73$ inches
  • D$62$ and $66$ inches
Show solution
Answer: B. About $95\%$ of values lie within $2$ standard deviations of the mean. $64 - 2(3) = 58$ and $64 + 2(3) = 70$. Choice A is $1$ standard deviation ($68\%$) and choice C is $3$ standard deviations ($99.7\%$).
Hard
The battery life of a phone model is roughly bell-shaped with a mean of $40$ hours and a standard deviation of $5$ hours. About what percent of these batteries last longer than $50$ hours?
  • A$2.5\%$
  • B$5\%$
  • C$16\%$
  • D$32\%$
Show solution
Answer: A. $50 = 40 + 2(5)$, so $50$ hours is $2$ standard deviations above the mean. About $95\%$ of batteries fall within $2$ standard deviations, which leaves $5\%$ outside. That $5\%$ splits evenly between the two ends, so about $2.5\%$ last longer than $50$ hours.
Hard
Scores on a test are roughly bell-shaped with a mean of $70$ and a standard deviation of $8$. About $16\%$ of students scored above what value?
  • A$62$
  • B$70$
  • C$78$
  • D$86$
Show solution
Answer: C. About $68\%$ of scores lie within $1$ standard deviation of the mean, so $100\% - 68\% = 32\%$ lie outside it. Half of that, $16\%$, is above the mean. One standard deviation above the mean is $70 + 8 = 78$.

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