What Is Standard Deviation?
Standard deviation tells you how much the values in a dataset differ from the average. When the standard deviation is small, most values sit close to the mean. When it is large, the values are more spread out.
Imagine you and your classmates scored between 78 and 82 on a test. The standard deviation would be low because everyone scored near the average. Now imagine scores ranged from 40 to 100 — the standard deviation would be much higher because results varied widely.
The Formulas
Population Standard Deviation (σ)
σ = √( Σ(xᵢ - μ)² / N )
Use when your data includes every member of the group you are studying.
Sample Standard Deviation (s)
s = √( Σ(xᵢ - x̄)² / (n - 1) )
Use when your data is a subset (sample) of a larger population. Dividing by (n - 1) instead of n is called Bessel's correction.
How to Calculate Standard Deviation by Hand
- Find the mean (average) of the data: add all values and divide by the count.
- Subtract the mean from each value to get the deviation from the mean.
- Square each deviation (this removes negative signs and gives more weight to larger deviations).
- Find the average of the squared deviations. Divide by N for population, or by (n - 1) for a sample. This gives you the variance.
- Take the square root of the variance. The result is the standard deviation.
Our calculator above does all of this automatically and shows you each step. Click "Show step-by-step solution" after entering your data.
Population vs. Sample: Which One Do I Use?
If your data covers the entire group you care about, use population standard deviation (σ). For example, test scores of every student in your class, or heights of every player on a team.
If your data is a sample drawn from a larger group, use sample standard deviation (s). For example, a survey of 500 people representing a city of 100,000, or 30 measurements from a manufacturing line that produces thousands.
Not sure? Most homework and research problems use sample standard deviation. When in doubt, go with (s).
Common Mistakes
- Forgetting Bessel's correction: dividing by n instead of (n - 1) for a sample will underestimate the true standard deviation.
- Confusing SD with variance: variance is the square of standard deviation. They measure the same thing but in different units.
- Comparing SDs across different scales: a standard deviation of 5 means something very different for test scores (0-100) than for heights in meters (1.5-2.0). Use the coefficient of variation (CV) to compare.
- Assuming normality: standard deviation works for any distribution, but the "68-95-99.7 rule" only applies to normally distributed data.
Frequently Asked Questions
What is standard deviation?
Standard deviation measures how spread out numbers are from their average (mean). A low standard deviation means data points cluster near the mean; a high one means they are spread over a wider range.
What is the difference between population and sample standard deviation?
Population SD (σ) divides by N and is used for complete datasets. Sample SD (s) divides by (n-1) to correct for bias when working with a subset of data. Most academic work uses sample SD.
When should I use standard deviation vs variance?
Both measure spread. SD is in the same units as your data, making it easier to interpret. Variance (SD squared) is used more in formulas. For reporting, SD is preferred.
What is a "good" standard deviation?
There is no universal good or bad value. Compare the SD to the mean using the coefficient of variation (CV = SD/mean × 100%). A CV under 15-20% is generally considered low variability.
Can I calculate standard deviation in Excel?
Yes. Use =STDEV.S() for sample SD and =STDEV.P() for population SD. For example: =STDEV.S(A1:A10).
What does the 68-95-99.7 rule mean?
For normally distributed data, about 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This rule does not apply to non-normal distributions.