# GRE Standard Deviation and Normal Distributions | topin

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## Standard deviation measures spread around the mean

[ETS](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html) includes standard deviation and normal distributions in GRE Quant’s data-analysis scope. Its [Mathematical Conventions](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-conventions.pdf) defines the standard deviation of a finite list using the mean of squared deviations, followed by the nonnegative square root. This is the population form, with division by the number of values. Do not silently replace that denominator with n − 1 from a separate sample-statistics course.

The mean alone does not determine spread. Lists 8, 10, 12 and 2, 10, 18 both have mean 10, but the second list places its outside values much farther from that mean. Standard deviation incorporates every value, unlike the range, which uses only the extremes. A problem about spread therefore needs either the relevant data or enough structural information to compare their deviations.

This concept sits within the wider [GRE Quant syllabus](/gre/quant). The practical distinction is between a numerical standard deviation and a relative comparison. If the question only asks which list is more spread out, calculating square roots may be unnecessary. Our method is to look first for identical means, translations, scaling or a simple pattern of distances.

## Use transformations before arithmetic

If every value receives the same added constant, the mean receives that constant too, leaving each deviation unchanged. If every value is multiplied by c, the mean is multiplied by c and each deviation is multiplied by c. Squaring then taking the square root gives a standard deviation multiplied by |c|. A negative multiplier reflects the list but cannot create negative spread.

These are mathematical deductions from the definition, illustrated by our original examples below. They are especially useful when two lists are written as expressions rather than explicit values. Check that the transformation applies to every observation and the weights remain the same. Changing one value or duplicating only one observation is not a uniform transformation.

__How common changes affect standard deviation__
| Change to every value          | Effect on mean               | Effect on standard deviation            |
| ------------------------------ | ---------------------------- | --------------------------------------- |
| Add 7                          | Mean increases by 7          | Unchanged                               |
| Multiply by 3                  | Mean triples                 | Triples                                 |
| Multiply by −2                 | Mean multiplies by −2        | Doubles                                 |
| Make every value identical     | Mean equals the common value | Becomes zero                            |
| Repeat the entire list equally | Mean unchanged               | Population standard deviation unchanged |

## Original problem: compute a small standard deviation

Original question: find the standard deviation of the values 4, 6, 8 and 10, using the GRE convention for a finite list. The mean is 28/4 = 7\. The deviations are −3, −1, 1 and 3\. Squaring gives 9, 1, 1 and 9, whose sum is 20\. Divide by four to obtain variance 5, then take the nonnegative square root.

The standard deviation is √5, approximately 2.236\. Leave it as √5 when the question needs an exact expression or only a comparison. An average of the unsquared deviations would be zero and lose the spread information. An average of absolute deviations would be two, a different statistic. The order of operations in the definition matters, even though all three calculations start with the same mean.

Now add 20 to every value, making 24, 26, 28 and 30\. The mean becomes 27 but the deviations remain −3, −1, 1 and 3, so the standard deviation stays √5\. Multiply the original values by four instead, making 16, 24, 32 and 40: the standard deviation becomes 4√5\. These checks demonstrate why larger numbers do not automatically imply greater spread.

If the list 4, 6, 8, 10 is repeated in full, its population mean and standard deviation remain seven and √5\. Both the squared-deviation sum and the number of observations double, leaving their ratio unchanged. This frequency reasoning connects with [weighted averages](/articles/gre-weighted-averages-mixtures). Repeating only the value ten would be a different transformation and needs a fresh calculation.

## Original problem: equal ranges do not fix spread

Original question: compare the standard deviations of A = 0, 0, 10, 10 and B = 0, 5, 5, 10\. Both means are five, and both ranges are ten. For A, every deviation has magnitude five, so the variance is 25 and standard deviation is five. For B, squared deviations are 25, 0, 0 and 25, averaging to 12.5.

B therefore has standard deviation √12.5, less than five. Two central values reduce its average squared distance without changing the endpoints. A range comparison alone cannot settle this pair. For larger data sets, an outlying observation can affect both the mean and spread, so compare the complete specified transformation rather than reasoning from only one endpoint.

For a [Quantitative Comparison](/gre/quant/quantitative-comparison), a variable may leave the relationship open. If a list is transformed by multiplication with positive k, its nonzero standard deviation is multiplied by k. When the only condition is k > 0, values below one reduce spread and values above one increase it. Do not assume a positive multiplier must enlarge the quantity.

## Original problem: compare distance in standard deviations

Original question: distribution A has mean 50 and standard deviation five; distribution B has mean 80 and standard deviation ten. A value of 60 in A is (60 − 50)/5 = 2 standard deviations above its mean. A value of 95 in B is (95 − 80)/10 = 1.5 standard deviations above its mean. The first value is farther above its own centre on this standardised scale.

The calculation z = (value − mean)/standard deviation measures relative position when the standard deviation is positive. A negative z denotes a value below the mean. A z of zero means the value equals the mean. If standard deviation is zero, all values are equal and division by it is undefined; do not manufacture a standardised distance for that case.

Standardisation by itself does not identify a percentile unless the distributional information supports that interpretation. For normal distributions, the same z corresponds to the same relative area. For arbitrary differently shaped distributions, the same z need not imply the same percentile. Keep “two standard deviations above the mean” distinct from a claimed universal percentage of observations below the value.

## Original problem: reason with symmetry and supplied areas

ETS’s Math Review describes a normal distribution as symmetric about its mean, with mean, median and mode coinciding. It describes about two-thirds of the area within one standard deviation and almost all within two. These are approximate statements. Read any supplied diagram or percentage first rather than replacing given information with a memorised approximation or claiming an exact value from a rough sketch.

Original question: a normal distribution has mean 100 and standard deviation 15\. A supplied diagram states that 68% of its area lies between 85 and 115\. What proportion lies above 115? The remaining area is 32%, split equally between the two tails by symmetry, so the upper tail is 16%. This answer uses the problem’s stated 68%, not an assertion that the exact normal probability is precisely 68%.

Under the same supplied approximation, the area from 100 to 115 is half of 68%, or 34%. The area above the mean is exactly 50% for a normal distribution. If the question asks for an arbitrary threshold such as 109 without more information, do not interpolate linearly along the horizontal axis: equal-width intervals need not contain equal areas because the curve height varies.

## Separate exact rules from approximate curve readings

Our recommended practice sequence uses a small-list calculation, a translation, a scaling change, equal-range lists and a normal-area question. For each answer, write “exact” or “uses supplied approximation”. That label prevents a common transfer error: carrying an approximate percentage into a problem that requests an exact comparison. Review the reasoning even when the final choice happens to be correct.

Use [topin’s free GRE mock](/gre/practice-test), marked on the official scale, after practising the concept. Review whether you spent time calculating a value that a transformation already settled. For table or graph contexts, the [Data Interpretation guide](/gre/quant/data-interpretation) covers display-reading habits. The statistical rule stays the same, but frequencies and axis labels determine which observations are actually represented.

When using a calculator, compute the mean first, retain exact deviations where possible and take the square root last. ETS notes that its on-screen calculator respects operation order, so use parentheses for totals divided by counts. A plausible decimal should not override your structural check: standard deviation cannot be negative, constant lists have zero spread, and uniform shifts leave spread unchanged.

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## FAQs

Do I use n or n minus one for GRE standard deviation?

ETS’s finite-list convention uses the mean of squared deviations, so divide by n. Sample standard deviation is a related statistic with n − 1; do not substitute it unless the problem explicitly defines that different measure.

Does adding a constant increase standard deviation?

Adding the same constant to every observation leaves standard deviation unchanged. Both the values and their mean shift together, preserving all deviations.

Can standard deviation be negative?

No. It is the nonnegative square root of a mean of squares. Multiplying every value by a negative number scales standard deviation by the absolute value of that multiplier.

Does the same range mean the same standard deviation?

No. Range uses only the minimum and maximum; standard deviation depends on every observation. Lists with the same endpoints can put different numbers of values near or far from their mean.

Can I use normal-distribution percentages for every data set?

No. Normal-area reasoning requires a normal or appropriately specified approximately normal distribution. Standard deviation alone does not establish a normal shape, and supplied approximations should remain labelled approximate.

## Sources (checked 5 October 2026)

* [ETS: Quantitative Reasoning overview and calculator guidance (checked 5 October 2026)](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html)
* [ETS: GRE Math Review (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-review.pdf)
* [ETS: GRE Mathematical Conventions (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-conventions.pdf)
* [ETS: Guidelines specific to the on-screen calculator (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/on-screen-calculator-guidelines.pdf)

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