Standard Deviation Calculator
Calculate standard deviation, variance, and mean for your data set
Standard Deviation Formulas
What is Standard Deviation?
Standard deviation (SD) is a measure of how spread out numbers are in a data set. It tells you, on average, how far each value lies from the mean. A low standard deviation means values cluster close to the mean, while a high standard deviation indicates values are spread over a wider range.
Standard deviation is the square root of variance. While variance gives us squared units, standard deviation returns to the original units of measurement, making it more interpretable. For example, if you're measuring height in centimeters, the standard deviation is also in centimeters.
In a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3. This 68-95-99.7 rule is fundamental in statistics and quality control.
Population vs Sample Standard Deviation
Population SD (σ)
Use when you have data for the entire population. Divide by N (total count).
Sample SD (s)
Use when data is a sample from a larger population. Divide by n-1 (Bessel's correction).
Variance (σ² or s²)
Standard deviation squared. Useful for statistical calculations.
Coefficient of Variation
SD/Mean × 100%. Compares variability between datasets with different scales.
Standard Deviation Interpretation
What different standard deviation values mean for your data:
| SD Range | Interpretation | Distribution | Action |
|---|---|---|---|
| Very Low | Values tightly clustered | Narrow bell curve | High consistency |
| Low | Moderate clustering | Normal distribution | Typical variation |
| Medium | Moderate spread | Wider distribution | Review outliers |
| High | Values widely spread | Flat distribution | Investigate causes |
| Very High | Extreme variation | Possible bimodal | Check data quality |
Step-by-Step Calculation
Calculate the Mean
Add all values and divide by count. Mean = Σx / n. This is your center point for measuring deviation.
Find Deviations
Subtract the mean from each value. Some deviations will be positive, some negative. They sum to zero.
Square the Deviations
Square each deviation to make all values positive. This also gives more weight to larger deviations.
Calculate Variance & SD
Average the squared deviations (use n for population, n-1 for sample), then take the square root for SD.
Frequently Asked Questions
When do I use population vs sample standard deviation?
Use population SD when your data includes every member of the group you're studying (e.g., all employees). Use sample SD when you've taken a subset from a larger group (e.g., surveyed 100 of 10,000 customers).
Why divide by n-1 for sample standard deviation?
This is Bessel's correction. It corrects for the bias that occurs because we're estimating the population mean from the sample. Dividing by n-1 gives an unbiased estimate of population variance.
Can standard deviation be negative?
No, standard deviation is always non-negative. It's the square root of variance (which is always positive because we square deviations). SD = 0 only when all values are identical.
What's the difference between standard deviation and standard error?
Standard deviation measures spread in your data. Standard error (SE = SD/√n) measures how precisely your sample mean estimates the population mean. SE decreases as sample size increases.
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