Which data set has a larger standard deviation?
The data set whose values sit further from their own mean has the larger standard deviation. You can usually tell by looking: {11, 12, 13} has a smaller standard deviation than {10, 15, 20}, because its values are 1 away from the middle while the other set's are 5 away. The exact sample SDs are 1 and 5.
━ Set A: {11, 12, 13}, s = 1 ━ Set B: {10, 15, 20}, s = 5
Is the SD of {11, 12, 13} greater than, less than or equal to {10, 15, 20}?
Less than. Work it out with deviations from each set's own mean:
- {11, 12, 13}: mean 12, deviations −1, 0, +1. Squares add to 2; s = √(2 ÷ 2) = 1.
- {10, 15, 20}: mean 15, deviations −5, 0, +5. Squares add to 50; s = √(50 ÷ 2) = 5.
The second set is exactly five times as spread out, so its SD is five times as large. With the population formula (divide by n) the SDs are 0.82 and 4.08: different numbers, same comparison. Which formula you use never changes which set is larger when the sets are the same size.
How to compare standard deviations without calculating
- Find the middle of each set. Roughly where is the mean?
- Look at the typical distance from it. Ignore where the values are on the number line; only their distance from their own centre counts.
- Watch the extremes. Squaring makes values far from the mean count much more. A set with values piled at both ends beats a set with most values in the middle.
- Check for shortcuts. If one set is another plus a constant, the SDs are equal. If it is another times a constant, the SD scales by that constant.
If two sets look close, calculate. The standard deviation calculator shows both results with the steps.
Does adding a constant change the standard deviation?
No. {111, 112, 113} has s = 1, the same as {11, 12, 13}. The mean moves up by 100, every value moves with it, and the distances between values and mean stay the same. A test where everyone gets 10 bonus points has a higher average but the same spread.
Does multiplying by a constant change it?
Yes, in proportion. {10, 15, 20} is {2, 3, 4} × 5, so its SD is 5 × 1 = 5. That is also why {10, 15, 20} has five times the SD of {11, 12, 13}: subtract 9 from the second set and you get {2, 3, 4}, which has the same spread. Converting metres to centimetres multiplies the SD by 100 and the variance by 10,000.
What if the data are flat?
If every value is the same, such as {7, 7, 7, 7}, the standard deviation is 0. There is no spread to measure. That is the smallest an SD can be; it is never negative. On a standard deviation graph, a flat data set is a single spike at the mean.
Same range, different standard deviation
The range only looks at the two extremes, so it can hide a difference the SD picks up. All three of these sets run from 1 to 9:
| Data | Range | Sample SD |
|---|---|---|
| {1, 5, 5, 5, 9} | 8 | 2.83 |
| {1, 3, 5, 7, 9} | 8 | 3.16 |
| {1, 1, 5, 9, 9} | 8 | 4 |
The more values sit at the extremes, the larger the SD. When a question gives two sets with the same range, look at where the middle values fall.
Related calculators
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Standard deviation calculator
Check any comparison with the exact SDs and steps.
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Standard deviation graph
What a small and a large SD look like.
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Range calculator
The quickest measure of spread, and its limits.
Common questions
Is the standard deviation of {11, 12, 13} greater than, less than or equal to that of {10, 15, 20}?
Less than. The means are 12 and 15, but the values in {11, 12, 13} are 1 away from their mean while those in {10, 15, 20} are 5 away. The sample SDs are 1 and 5 (population SDs 0.82 and 4.08).
What is the standard deviation if all the values are the same?
Zero. If the data are flat, such as {7, 7, 7, 7}, every value equals the mean, every deviation is 0, and so is the standard deviation. It is the only way to get an SD of 0.
Does adding the same number to every value change the standard deviation?
No. Adding or subtracting a constant shifts the mean by the same amount and leaves every deviation unchanged, so {111, 112, 113} has the same SD as {11, 12, 13}.
Does multiplying every value change the standard deviation?
Yes, by the same factor (its absolute value). Multiplying by 5 multiplies the SD by 5 and the variance by 25.
Can two data sets with the same range have different standard deviations?
Yes. {1, 1, 5, 9, 9} and {1, 5, 5, 5, 9} both have a range of 8, but the first has s = 4 and the second s = 2.83, because the first piles its values at the ends.
