Standard Deviation, Quartiles & Percentiles • Topic 1 of 4

Spread & Standard Deviation

Standard deviation measures the typical distance of the values from their mean: the more the data clusters near the mean, the smaller the SD; the more it spreads out, the larger. You will rarely compute one by hand on the GRE — instead you compare. Two facts unlock most questions. First, adding the same constant to every value does not change the SD, because sliding the whole set left or right does not change how spread out it is; the mean shifts but the distances to it are identical. Second, multiplying every value by k multiplies the SD by |k|, because every distance scales. To compare two sets by eye, look at how tightly the values cluster: {19, 20, 21} has a far smaller SD than {5, 20, 35}, even though both average 20. A set where every value is identical has an SD of exactly zero. Keep the concept front of mind and skip the arithmetic the test does not want.

Two dot plots sharing the same mean of 20: one tightly clustered, one widely spreadSame mean, different spreadSmall spread (small SD)Large spread (large SD)mean = 20 (both sets)

✅ Solved examples

1. Which has the larger standard deviation: A = {10, 10, 10, 10} or B = {7, 9, 11, 13}?
Set A has every value equal to its mean, so SD = 0. Set B spreads around its mean of 10, so its SD is positive. B has the larger SD.
2. A data set has standard deviation 4. Every value is increased by 20. What is the new standard deviation?
Adding a constant shifts the whole set without changing its spread, so the SD stays 4.
3. A data set has SD 3. Every value is multiplied by 5. New SD?
Multiplying every value by 5 multiplies the SD by 5: new SD = 15. (The mean also scales, but the point is that distances stretch by the same factor.)
4. Quantitative Comparison. Set S = {4, 6, 8, 10, 12}. Quantity A: the SD of S. Quantity B: the SD of {104, 106, 108, 110, 112}. Which is greater?
The second set is S with 100 added to every value. Adding a constant leaves the SD unchanged, so the two standard deviations are identical. Answer: the two are equal.

✏️ Practice — try these, take hints as needed

1. Which set has SD zero: {3, 3, 3, 3} or {2, 3, 4}?
SD is zero only when every value equals the mean.
Check whether all values are identical.
Only one set is constant.
{3, 3, 3, 3}
2. A set has SD 6. Each value is decreased by 15. New SD?
Subtracting a constant shifts the set.
Spread does not depend on location.
SD is unchanged.
6
3. A set has SD 2.5. Each value is doubled. New SD?
Multiplying by k scales SD by |k|.
k = 2.
2.5 × 2.
5
4. Order by SD, smallest first: P = {50, 50, 50}, Q = {48, 50, 52}, R = {40, 50, 60}.
Look at how far values sit from the mean of 50.
P is constant; R is the most spread.
Rank by cluster tightness.
P < Q < R
5. True or false: two data sets with the same range must have the same standard deviation.
Range uses only the extremes.
SD uses every value.
Consider {0, 5, 10} vs {0, 0, 10}.
False — same range can hide different SDs

📝 Topic test — 8 questions

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