How to Find Standard Deviation- Step-by-Step Guide
What is Standard Deviation?
Standard deviation measures how spread out numbers are in a dataset. That's it. A low standard deviation means numbers cluster near the mean. A high one means they're all over the place.
You need this when comparing variability between datasets, checking consistency, or doing any real statistical work. Scientists, analysts, and researchers use it constantly.
The Formula
There are two versions depending on whether you're working with an entire population or just a sample:
- Population standard deviation: σ = √[Σ(xi - μ)² / N]
- Sample standard deviation: s = √[Σ(xi - x̄)² / (n-1)]
The difference? Population uses N (total items). Sample uses n-1 (Bessel's correction). Most of the time you're calculating sample standard deviation.
Step-by-Step Calculation
Example dataset: Test scores for 6 students
Scores: 72, 85, 90, 78, 88, 95
Step 1: Find the mean (average)
Add all values and divide by how many there are.
(72 + 85 + 90 + 78 + 88 + 95) ÷ 6 = 508 ÷ 6 = 84.67
Step 2: Subtract the mean from each value
- 72 - 84.67 = -12.67
- 85 - 84.67 = 0.33
- 90 - 84.67 = 5.33
- 78 - 84.67 = -6.67
- 88 - 84.67 = 3.33
- 95 - 84.67 = 10.33
Step 3: Square each result
- (-12.67)² = 160.45
- (0.33)² = 0.11
- (5.33)² = 28.41
- (-6.67)² = 44.49
- (3.33)² = 11.09
- (10.33)² = 106.71
Step 4: Add all squared values
160.45 + 0.11 + 28.41 + 44.49 + 11.09 + 106.71 = 351.26
Step 5: Divide by (n-1) for sample
Since this looks like a sample (6 students from a larger class), use n-1:
351.26 ÷ (6-1) = 351.26 ÷ 5 = 70.25
Step 6: Take the square root
√70.25 = 8.38
The sample standard deviation is 8.38 points. This tells you test scores deviate from the average by about 8 points.
Population vs Sample Standard Deviation
Use population standard deviation when you have every single data point in your group. Every employee, every transaction, every measurement.
Use sample standard deviation when you're working with a subset. Survey respondents, test samples, randomly selected items. Always use n-1 when calculating from sample data.
Quick Reference Table
| Type | Formula | When to Use |
|---|---|---|
| Population | σ = √[Σ(x-μ)² / N] | All data available |
| Sample | s = √[Σ(x-x̄)² / (n-1)] | Partial data, inferring about population |
Common Mistakes to Avoid
- Using N instead of n-1 for sample data. Your result will be slightly smaller than it should be.
- Forgetting to square the deviations. Negative and positive values will cancel out and give you zero.
- Confusing variance with standard deviation. Variance is the squared result before taking the square root.
- Using population formula on sample data. Makes your estimate unreliable.
Using a Calculator or Spreadsheet
You don't need to do this by hand every time. Most tools have built-in functions:
- Excel/Google Sheets: =STDEV.S(data) for sample, =STDEV.P(data) for population
- TI calculators: Use the 1-Var-Stats function
- Online calculators: Plug in values, get results instantly
But knowing the steps helps you understand what the number actually means. That's worth more than just getting an answer.
What Your Standard Deviation Tells You
With our test score example (mean 84.67, SD 8.38):
- About 68% of scores fall between 76.29 and 93.05 (one SD above and below mean)
- About 95% fall between 67.91 and 101.43 (two SDs)
- About 99.7% fall between 59.53 and 109.81 (three SDs)
This is the empirical rule. It works well for normally distributed data. Skewed distributions don't follow this pattern.