How to Calculate Standard Deviation by Hand: Five Test Scores, Step by Step
October 21, 2026
Standard deviation measures how spread out a set of numbers is around its mean. A small value means the numbers cluster tightly, and a large value means they range widely. Two classes can both average 82 on a test, yet one has scores between 80 and 84 while the other has scores from 60 to 100. Their means are the same, but their standard deviations tell the difference.
The calculation has five steps, and doing it by hand once makes the concept clear. We will use five test scores: 72, 85, 90, 68, and 95.
Skip the math and get your answer instantly:
Open the Standard Deviation Calculator →Step 1: Find the Mean
Add the scores and divide by how many there are: 72 + 85 + 90 + 68 + 95 = 410, and 410 ÷ 5 = 82. The mean is 82.
This is the balance point of the data. The distances of the scores from this point, some positive and some negative, always sum to zero, which is a good check.
Step 2: Find Each Deviation From the Mean
Subtract the mean from each score: 72 − 82 = −10, 85 − 82 = +3, 90 − 82 = +8, 68 − 82 = −14, 95 − 82 = +13. Check that the deviations sum to zero: −10 + 3 + 8 − 14 + 13 = 0. They do, so the mean and subtractions are right.
If they do not sum to zero, there is an arithmetic error before this point.
Step 3: Square the Deviations
Squaring removes the negative signs and gives greater weight to larger deviations: (−10)² = 100, 3² = 9, 8² = 64, (−14)² = 196, 13² = 169. The sum of squares is 100 + 9 + 64 + 196 + 169 = 538.
Using squares rather than absolute values gives the variance and standard deviation useful mathematical properties, such as additivity of variances for independent variables.
| Score | Deviation | Squared deviation |
|---|---|---|
| 72 | -10 | 100 |
| 85 | +3 | 9 |
| 90 | +8 | 64 |
| 68 | -14 | 196 |
| 95 | +13 | 169 |
| Total | 0 | 538 |
Step 4: Divide to Get the Variance
If these five scores are the entire population of interest, such as the whole class, divide by n = 5: variance = 538 ÷ 5 = 107.6. If they are a sample from a larger group, divide by n − 1 = 4: variance = 538 ÷ 4 = 134.5.
The choice matters most for small datasets. Here the sample variance is 25% larger than the population variance.
Step 5: Take the Square Root
Population standard deviation = √107.6 = 10.37. Sample standard deviation = √134.5 = 11.60. The square root returns the measure to the original units (points on the test), which makes it easy to interpret.
A standard deviation of about 10 to 12 points means a typical score lies roughly 10 to 12 points from the class average of 82.
Interpreting the Result
With a mean of 82 and a sample standard deviation of 11.6, scores between about 70.4 and 93.6 (one standard deviation either side) are typical. In this set, 72 and 90 and 85 fall inside that band, while 68 and 95 lie just outside it. A z-score converts a score to standard deviations from the mean: for 95, z = (95 − 82) ÷ 11.6 = 1.12.
Compare two classes by looking at both the mean and the standard deviation. A teacher might read a high standard deviation as a sign that the class is split between students who grasp the material and those who do not.
Frequently Asked Questions
What is standard deviation in simple terms?
It is a measure of how far typical values are from the average.
Why do we square the deviations?
To make all values positive and to emphasize larger deviations, which also gives the measure useful mathematical properties.
Should I divide by n or n − 1?
Use n if your data is the entire population and n − 1 if it is a sample.
What is a good standard deviation?
There is no universal good value. It depends on context, and you judge it relative to the mean and the scale of the data.
How can I check my arithmetic?
The deviations from the mean must sum to zero, and standard deviation can never be negative.