Standard Deviation Calculator
Calculate standard deviation, variance, and coefficient of variation.
About This Calculator
Calculate standard deviation, variance, and coefficient of variation. Use the calculator above for instant results.
What Standard Deviation Measures
Standard deviation measures the typical scale of deviations from the arithmetic mean. It is small when observations cluster near the mean and larger when values are more dispersed. Because deviations can be positive or negative, they are squared before averaging so opposite signs do not cancel.
Variance is the average squared deviation, and standard deviation is its square root. Taking the square root returns the measure to the original units: if heights are measured in centimeters, standard deviation is also in centimeters, while variance is in square centimeters.
The denominator distinguishes population and sample calculations. Use N when the data constitute the entire population of interest. When a sample is being used to estimate population variability, dividing by n−1 applies Bessel’s correction and compensates for the tendency of the sample mean to make squared deviations too small on average.
Standard deviation is sensitive to outliers because deviations are squared. It also does not by itself describe distribution shape. The 68–95–99.7 rule is an approximation for normal distributions, not a universal rule for every dataset. Always identify whether the data are a population or sample and inspect the context before interpreting a numerical SD.
Worked Examples
- Mean: 30 | Pop SD: 14.14 | Sample: 15.81
Answer: SD=14.14
- All same — std dev: 0
Answer: SD=0
- Mean: 5 | Std dev: 2
Answer: SD=2
Data: 1, 2, 3. The mean is 2.
- Squared deviations: 1, 0, 1; sum = 2.
- Population variance = 2/3.
Answer: population SD ≈ 0.8165
The sample SD would be 1 because its variance divides by n−1 = 2 instead of N = 3.
Compare 10, 10, 10, 10 with 10, 10, 10, 30.
Answer: the first population SD is 0; the second is about 8.66.
The distant value contributes a large squared deviation, illustrating standard deviation’s sensitivity to outliers.
Who Uses This Calculator?
Measure data spread.
Statistical analysis.
Stats homework.
Process variation.
Common Mistakes to Avoid
❌ Population vs sample
Use population when you have ALL data. Sample when data is a subset.
❌ High SD not always bad
In quality control, low SD is better. In investments, some volatility means higher returns.
❌ Mixing population and sample formulas
Choose the denominator based on the statistical role of the data. Dividing by N and n−1 produces different answers, especially for small datasets.
❌ Applying the 68–95–99.7 rule to any dataset
That rule describes approximately normal distributions. Strongly skewed, multimodal, or heavy-tailed datasets can have very different proportions within one or two standard deviations.
Comparing sample and population results also helps confirm that the correct denominator was selected for the data context.
Frequently Asked Questions
What does SD tell you?
How spread out values are from the mean. Low: clustered. High: scattered.
68-95-99.7 rule?
68% within 1 SD, 95% within 2, 99.7% within 3 for normal distributions.
Coefficient of variation?
CV = SD/mean x 100%. Compares variation across different-scale datasets.
Can standard deviation be negative?
No. Variance is an average of squared quantities and cannot be negative, so its square root is also nonnegative. A standard deviation of zero means all observations in the dataset are identical.
Why does sample standard deviation use n minus 1?
The sample mean is estimated from the same observations, costing one degree of freedom. Dividing the squared-deviation sum by n−1 makes sample variance an unbiased estimator of population variance under standard assumptions.
What happens to standard deviation if every value is shifted by the same constant?
It does not change. Adding the same constant moves both every observation and the mean by that amount, leaving all deviations from the mean unchanged.
What happens if every value is multiplied by a constant?
Standard deviation is multiplied by the absolute value of that constant. If every measurement is converted from meters to centimeters by multiplying by 100, the standard deviation is also multiplied by 100.
Formula Explorer connections
Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.