Standard Deviation, Explained
Standard deviation measures spread: how far the values in a data set typically sit from their mean. Two data sets can share a mean and look nothing alike, and this is the number that tells them apart.
The five steps
- Find the mean of the data.
- Subtract the mean from each value to get the deviations.
- Square every deviation, which removes the negative signs.
- Average the squared deviations. This is the variance.
- Take the square root of the variance to get back to the original units.
Sample or population
If your data is the entire group you care about, divide by n. If it is a sample being used to estimate a wider population, divide by n - 1. Dividing by n - 1 makes the estimate slightly larger, which corrects a known bias in small samples. Reading the question to see which one you have is worth more marks than the arithmetic.
What the number tells you
A standard deviation near zero means the values cluster tightly around the mean. A large one means they are spread out. Because it is in the same units as the data, a standard deviation of 2 cm is directly comparable with the measurements themselves, which is why it is preferred over the variance in reporting.
Worked examples
Example 1
Question
Find the population standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9
Answer
2
Steps
- 1Mean: the values sum to 40, and 40 / 8 = 5.
- 2Deviations: -3, -1, -1, -1, 0, 0, 2, 4.
- 3Squares: 9, 1, 1, 1, 0, 0, 4, 16, summing to 32.
- 4Population variance: divide by n = 8, giving 4.
- 5Standard deviation: the square root of 4 is 2.
Example 2
Question
Find the sample standard deviation of: 4, 8, 6, 5, 3
Answer
About 1.92
Steps
- 1Mean: the values sum to 26, and 26 / 5 = 5.2.
- 2Deviations: -1.2, 2.8, 0.8, -0.2, -2.2.
- 3Squares: 1.44, 7.84, 0.64, 0.04, 4.84, summing to 14.8.
- 4Sample variance: divide by n - 1 = 4, giving 3.7.
- 5Standard deviation: the square root of 3.7 is about 1.92.
Dividing by 5 instead of 4 here would give 1.72. The difference matters most in small samples, which is exactly when exams ask.
Example 3
Question
Two classes both average 70. Class A has a standard deviation of 3, class B has 15. What does that tell you?
Answer
Class A is consistent; class B has both much stronger and much weaker results.
Steps
- 1The mean says nothing about spread, so identical means can hide very different distributions.
- 2A standard deviation of 3 means most of class A scored roughly between 67 and 73.
- 3A standard deviation of 15 means class B typically ranges from about 55 to 85.
- 4So class B contains students who need very different support, despite the identical average.
- 5This is why reporting a mean without a measure of spread is misleading.
Common mistakes
Dividing by n when the data is described as a sample.
Samples divide by n - 1. Look for the words sample, survey, or selected in the question.
Forgetting the final square root and reporting the variance.
Variance is in squared units. The standard deviation is its square root, in the original units.
Adding the deviations without squaring them.
They always sum to zero, which is why squaring comes first.
Rounding the mean early and carrying the error through every deviation.
Keep the mean exact until the final answer, then round once.
Practice it
Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.
1. Find the population standard deviation of 10, 12, 14.
Show answer
About 1.63. The mean is 12, the squared deviations sum to 8, and 8 / 3 = 2.67.
2. Find the sample standard deviation of the same data: 10, 12, 14.
Show answer
2. The squared deviations still sum to 8, but dividing by n - 1 = 2 gives a variance of 4.
3. A data set has variance 49. What is its standard deviation?
Show answer
7. Take the square root of the variance.
Questions people ask
- Why square the deviations at all?
- Because they sum to zero otherwise. Squaring removes the signs, and it also weights larger departures more heavily.
- Why divide by n - 1 for a sample?
- Because a sample's own mean sits closer to its data than the true population mean does, which understates the spread. Dividing by n - 1 corrects that bias.
- Is a high standard deviation bad?
- It is neither good nor bad on its own. It is bad for a manufacturing tolerance and neutral for exam results, so context decides.
- How does it relate to the normal distribution?
- For normally distributed data, about 68% of values lie within one standard deviation of the mean and about 95% within two.
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