How to Calculate Standard Deviation by Hand: A Full Worked Example
Published 2026-01-11
Calculating standard deviation by hand is the best way to understand what the statistic actually does. This guide walks through every step using one small dataset, with a table showing each calculation, so you can repeat the process on any data of your own. When you are ready to check your work or handle larger datasets, the calculator on our homepage shows the same steps automatically.
The dataset
We will use six values:
4, 8, 6, 5, 3, 7
There are n = 6 values. We will compute both the population standard deviation (divide by n) and the sample standard deviation (divide by n minus 1). If you are unsure which applies to your situation, see our guide on sample vs population standard deviation.
Step 1: Add up the values
Sum all six values:
4 + 8 + 6 + 5 + 3 + 7 = 33
Step 2: Calculate the mean
Divide the sum by the number of values:
mean = 33 / 6 = 5.5
The mean, 5.5, is the reference point for every deviation that follows.
Step 3: Find each deviation from the mean
Subtract 5.5 from each value:
| Value | Deviation (value minus 5.5) |
|---|---|
| 4 | -1.5 |
| 8 | 2.5 |
| 6 | 0.5 |
| 5 | -0.5 |
| 3 | -2.5 |
| 7 | 1.5 |
Notice that the deviations add up to zero: -1.5 + 2.5 + 0.5 - 0.5 - 2.5 + 1.5 = 0. This always happens, and it is exactly why we cannot just average the deviations. We need to square them first.
Step 4: Square each deviation
Squaring does two jobs: it makes every value positive, and it gives extra weight to points far from the mean.
| Value | Deviation | Squared deviation |
|---|---|---|
| 4 | -1.5 | 2.25 |
| 8 | 2.5 | 6.25 |
| 6 | 0.5 | 0.25 |
| 5 | -0.5 | 0.25 |
| 3 | -2.5 | 6.25 |
| 7 | 1.5 | 2.25 |
Step 5: Sum the squared deviations
Add the six squared deviations:
2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 = 17.5
This total, 17.5, is called the sum of squared deviations, or sum of squares. It is the raw material for both variance formulas.
Step 6: Divide to get the variance
Now the two formulas diverge.
Population variance divides by n:
variance = 17.5 / 6 = 2.9167 (about; the exact value is 2.9166666666666665)
Sample variance divides by n minus 1, applying Bessel’s correction:
variance = 17.5 / 5 = 3.5
Variance is measured in squared units. If your data was in centimeters, the variance is in square centimeters, which is hard to interpret. That is why we take one more step.
Step 7: Take the square root
The square root returns the statistic to the original units of the data.
Population standard deviation:
square root of 2.9166666666666665 = 1.7078 (about)
Sample standard deviation:
square root of 3.5 = 1.8708 (about)
So the same dataset gives about 1.7078 as a population standard deviation and about 1.8708 as a sample standard deviation.
The full calculation table
Here is everything in one place:
| Step | Quantity | Value |
|---|---|---|
| 1 | Sum of values | 33 |
| 2 | Mean (33 / 6) | 5.5 |
| 3 | Deviations from mean | -1.5, 2.5, 0.5, -0.5, -2.5, 1.5 |
| 4 | Squared deviations | 2.25, 6.25, 0.25, 0.25, 6.25, 2.25 |
| 5 | Sum of squared deviations | 17.5 |
| 6a | Population variance (17.5 / 6) | about 2.9167 |
| 6b | Sample variance (17.5 / 5) | 3.5 |
| 7a | Population SD (square root of 2.9167) | about 1.7078 |
| 7b | Sample SD (square root of 3.5) | about 1.8708 |
Checking your answer
A few quick sanity checks catch most errors:
- The standard deviation can never be negative. If you got a negative number, you forgot the square root or mis-squared something.
- The standard deviation should be smaller than the range of the data. Our range is 8 minus 3 = 5, and both 1.7078 and 1.8708 are comfortably below that.
- The deviations must sum to zero. If yours do not, your mean is wrong.
- The sample standard deviation must be larger than the population standard deviation for the same data. Here 1.8708 is greater than 1.7078, as expected.
Common hand calculation errors
The most frequent mistake is rounding the mean too early. Keep full precision through every step and round only the final answer. Our mean here, 5.5, is exact, but with messier data premature rounding compounds through the squared deviations.
The second most frequent mistake is dividing by the wrong denominator. Decide before you start whether your data is a full population or a sample, and write the divisor down at the top of your page.
The third is forgetting the final square root and reporting the variance as the standard deviation. Remember: variance is in squared units, standard deviation is in original units.
What this number means
A standard deviation of about 1.87 tells you that the values in this dataset typically sit within roughly 1.87 units of the mean of 5.5. For a deeper explanation of interpreting spread, read what standard deviation tells you. To see where people go wrong when applying it, check common standard deviation mistakes, and use the standard deviation calculator to verify your own hand calculations step by step.