Standard Deviation Calculator
Calculate standard deviation for sample or population datasets with step-by-step explanations. Enter your data below and see the complete calculation process.
How to Calculate Standard Deviation
Standard deviation is a measure of how spread out numbers are from their average (mean). It's one of the most important statistics in data analysis.
The calculation process:
- Calculate the mean (average) of all values
- Subtract the mean from each value to get deviations
- Square each deviation to make them positive
- Sum all the squared deviations
- Divide by n (population) or n-1 (sample) to get variance
- Take the square root of variance to get standard deviation
Sample vs Population: What's the Difference?
Population Standard Deviation (n)
Use when you have data for the entire population. For example, if you're calculating the standard deviation of test scores for all students in a class, and you have every student's score.
σ = √[Σ(xi - μ)² / n]
Sample Standard Deviation (n-1)
Use when you have data from a sample of the population. For example, if you're calculating standard deviation from a survey of 100 people representing a larger population. The n-1 formula (Bessel's correction) provides a better estimate.
s = √[Σ(xi - μ)² / (n-1)]
When in doubt:
Use sample standard deviation (n-1) - it's more commonly used in statistics and provides better estimates when working with real-world data.
When to Use Standard Deviation
Standard deviation is useful in many situations:
- Comparing variability: Determine which dataset has more consistent values
- Quality control: Identify if measurements are within acceptable ranges
- Financial analysis: Assess risk and volatility in investments. For personal finance calculations, check out WealthWise for wealth management tools.
- Scientific research: Report the precision of experimental measurements
- Test scores: Understand how spread out student performance is. For calculating course grades and GPA, check out GradeWise.
- Normal distribution: About 68% of values fall within 1 standard deviation of the mean
Interpreting the result:
- • Low standard deviation: Values are close to the mean (less spread)
- • High standard deviation: Values are far from the mean (more spread)
- • Zero standard deviation: All values are identical
Worked Example
Dataset: 10, 20, 30, 40, 50
Step 1: Calculate the Mean
Mean = (10 + 20 + 30 + 40 + 50) ÷ 5 = 150 ÷ 5 = 30
Step 2: Calculate Deviations from Mean
10 - 30 = -20
20 - 30 = -10
30 - 30 = 0
40 - 30 = 10
50 - 30 = 20
Step 3: Square Each Deviation
(-20)² = 400
(-10)² = 100
(0)² = 0
(10)² = 100
(20)² = 400
Step 4: Sum of Squared Deviations
400 + 100 + 0 + 100 + 400 = 1000
Step 5: Calculate Variance
Sample (n-1): 1000 ÷ (5-1) = 1000 ÷ 4 = 250
Population (n): 1000 ÷ 5 = 200
Step 6: Take Square Root
Sample Standard Deviation: √250 = 15.81
Population Standard Deviation: √200 = 14.14
💡 Try it yourself: Enter "10, 20, 30, 40, 50" in the calculator above to verify these results!
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Educational Tool Disclaimer
This calculator is provided for educational purposes only. While we strive for accuracy, please verify critical calculations independently. Always double-check your work for important assignments or professional applications.