The short answer
Standard deviation measures how spread out a data set is around its mean. The GRE almost never asks you to compute one from scratch, because the arithmetic is tedious and the exam is not testing arithmetic endurance.
What it does ask, repeatedly, is what happens to the standard deviation under a change: adding a constant to every value, multiplying every value by a constant, adding a new data point, or comparing two sets side by side.
Two rules cover most of it. Adding the same constant to every value leaves the standard deviation unchanged. Multiplying every value by a constant multiplies the standard deviation by the absolute value of that constant.
Transformations: what moves and what doesn't
Adding a constant shifts the entire data set without changing the distances between values. The mean moves by that constant; the standard deviation does not move at all. If every salary in a company rises by 3,000 dollars, the mean rises by 3,000 and the spread is identical.
Multiplying by a constant stretches or compresses the distances. Both the mean and the standard deviation are multiplied by that constant — the standard deviation by its absolute value, since spread cannot be negative. If every salary is multiplied by 1.1, both the mean and the standard deviation rise by 10 percent.
The median behaves like the mean under both operations. The range and the interquartile range behave like the standard deviation: unchanged by addition, scaled by multiplication.
| Operation on every value | Mean | Standard deviation |
|---|---|---|
| Add c | Increases by c | Unchanged |
| Subtract c | Decreases by c | Unchanged |
| Multiply by k | Multiplied by k | Multiplied by |k| |
| Add a value equal to the mean | Unchanged | Decreases |
That last row is worth pausing on. Adding a data point exactly equal to the mean leaves the mean where it is, but it adds a value with zero deviation, which pulls the average deviation down. Adding a value far from the mean does the opposite.
Comparing two sets without computing
Quantitative comparison questions often show two small data sets and ask which has the larger standard deviation. Computing both is almost always the wrong move.
Instead, look at how tightly the values cluster around their own mean. The set {10, 10, 10, 10} has a standard deviation of zero, because every value equals the mean. The set {2, 6, 14, 18} is widely dispersed and clearly has a larger one.
Two properties are worth holding. Standard deviation is never negative. And it equals zero only when every value in the set is identical — which is the basis for a fair number of quantitative comparison questions where one column is a set of repeated values.
Beware of one specific trap: a larger mean does not imply a larger standard deviation, and neither does a larger range in every case. The set {1, 2, 3} has a smaller mean than {100, 101, 102} but exactly the same spread. Spread is about distances between values, not about their magnitude.
The normal distribution and the 68–95–99.7 bands
When a GRE question specifies that a distribution is normal, it is signalling that you can use the empirical bands: approximately 68 percent of values fall within one standard deviation of the mean, approximately 95 percent within two, and approximately 99.7 percent within three.
It is usually easier to work with the segments than the cumulative bands. Moving outward from the mean, each successive band on one side holds roughly 34 percent, then 14 percent, then 2 percent of the data. Those three numbers, plus symmetry about the mean, answer most normal-distribution questions on this exam.
Take an example. Test scores are normally distributed with a mean of 500 and a standard deviation of 100. What percentage of scores fall between 500 and 700?
That is the region from the mean out to two standard deviations above it, which is 34 percent plus 14 percent, or about 48 percent. What about above 700? That is the remaining tail on that side: 50 − 48 = about 2 percent.
To find how many standard deviations a value sits from the mean, subtract the mean and divide by the standard deviation. A score of 350 in that distribution is (350 − 500)/100 = −1.5, meaning one and a half standard deviations below the mean.
Mean, median, and skew
In a symmetric distribution, including the normal distribution, the mean and median coincide. Questions that establish symmetry are often really telling you that you can substitute one for the other.
When a distribution is skewed, the mean is pulled toward the long tail while the median stays closer to the bulk of the data. A data set with a few very large values has a mean above its median; one with a few very small values has a mean below its median.
That relationship is testable in both directions. A question may state that the mean exceeds the median and ask what follows about the shape, or describe a set with one extreme outlier and ask which measure changes more. The median is resistant to outliers; the mean is not.
The same logic explains why standard deviation is sensitive to outliers. Because deviations are squared before averaging, a single distant value contributes disproportionately, which is why adding one extreme point can change the standard deviation substantially while barely moving the median.
The common mistake
The most common error is believing that adding a constant changes the spread. It feels like it should — the numbers all got bigger. But every value moved by the same amount, so the distances between them are untouched, and standard deviation is entirely a statement about those distances.
The second is computing when the question does not require it. If the question asks which of two sets has a greater standard deviation, or whether a standard deviation is zero, or what happens after a transformation, the arithmetic is a trap for your time rather than a route to the answer.
Third is misreading the bands. "Within two standard deviations" means the region between two below and two above, roughly 95 percent. "More than two standard deviations above" is one tail only, roughly 2 percent — not 5 percent, which would be both tails combined.
Fourth is applying the empirical bands to a distribution that was never stated to be normal. Those percentages are a property of the normal distribution specifically. If the problem does not say normal, or does not show a symmetric bell shape, they do not apply.
Practice this on Verbloom
Verbloom's GRE quantitative practice includes statistics questions in both problem-solving and quantitative-comparison formats, with explanations for why each distractor is attractive.
This is a topic where the errors are highly patterned, so a short review of your misses usually identifies one specific rule to relearn rather than a general weakness in statistics.
Frequently asked questions
Does adding a constant to every value change the standard deviation?
No. Adding the same constant to every value shifts the mean by that constant but leaves every distance between values unchanged, so the standard deviation is identical. Multiplying every value by a constant does change it — the standard deviation is multiplied by the absolute value of that constant.
When is a standard deviation equal to zero?
Only when every value in the set is identical, since each value then sits exactly at the mean with zero deviation. Standard deviation is never negative, so a set of repeated values gives the smallest possible value.
What are the 68–95–99.7 percentages used for?
They give the approximate proportion of a normal distribution falling within one, two, and three standard deviations of the mean. It is often easier to work with the individual bands — roughly 34 percent, 14 percent, and 2 percent moving outward on each side — and use symmetry about the mean. They apply only when the distribution is stated or shown to be normal.
How do I find how many standard deviations a value is from the mean?
Subtract the mean from the value and divide by the standard deviation. With a mean of 500 and a standard deviation of 100, a value of 650 is (650 − 500)/100 = 1.5 standard deviations above the mean.
Does the GRE ask you to calculate standard deviation directly?
Rarely, and when it does the data set is very small. Most questions test how standard deviation responds to a change, how two sets compare in spread, or how the normal distribution's bands work. Reaching for the full computation is usually a sign the question was asking something else.
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