Standard Deviation Examples With Steps
Follow three standard deviation examples covering a population, a sample, and data sets with equal means but different spread.
How should you use these Standard Deviation Calculator examples?
These examples turn the Standard Deviation Calculator inputs into complete scenarios with stated values, calculation steps, and interpreted results. Use them to check how the calculator behaves, compare your own situation with realistic cases, and identify which inputs have the greatest effect before relying on an estimate.
Example 1: Population Standard Deviation
Suppose the complete population is 2, 4, 4, 4, 5, 5, 7, 9. The sum is 40 and N = 8.
| x | x - μ | (x - μ)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
| Total | 0 | 32 |
The population variance is 4 and the population standard deviation is 2. The standard deviation uses the same unit as the entered observations.
Example 2: Sample Standard Deviation
Now treat the same eight values as a sample used to estimate a larger population. The mean and sum of squared deviations remain 5 and 32, but the denominator changes from 8 to 7.
At three displayed decimal places, sample variance is 4.571 and sample standard deviation is 2.138. This is larger than the population result because the sample formula divides by n - 1.
Example 3: Same Mean, Different Spread
Data set A is 4, 5, 5, 6. Data set B is 1, 5, 5, 9. Both populations have a mean of 5, but their distances from 5 differ.
| Population | Mean | Sum of squared deviations | Variance | Standard deviation |
|---|---|---|---|---|
| A: 4, 5, 5, 6 | 5 | 2 | 0.500 | 0.707 |
| B: 1, 5, 5, 9 | 5 | 32 | 8.000 | 2.828 |
Data set B has the same center but four times the standard deviation of data set A. This illustrates why the mean alone cannot describe consistency or dispersion.
What These Examples Show
For “What These Examples Show,” the Standard Deviation Calculator examples organizes the relevant inputs, applies the documented method, and shows a result that can be compared across scenarios. The points below identify the specific considerations and assumptions users should review before relying on the estimate.
- Population and sample modes can produce different spread estimates from identical observations.
- Standard deviation uses every observation, not only the minimum and maximum.
- Two data sets can have the same mean and very different variability.
- Rounding belongs at the display stage, after the variance and square root calculations.
Enter any scenario in the Standard Deviation Calculator or review the formula definitions and calculation order.
Corrections and Review
Report an issue or error in the calculator using the Contact Us page.