Standard Deviation Calculator

Calculate the standard deviation, variance, mean, and more for any dataset. Enter your numbers separated by commas.

Quick Facts

Population vs Sample
N vs N-1
Use N-1 for sample data
68-95-99.7 Rule
Normal Distribution
68% within 1 SD of mean
Variance
SD squared
Average squared deviation
Coefficient of Variation
(SD/Mean) x 100
Relative variability measure

Your Results

Calculated
Standard Deviation (Population)
0
Using N
Standard Deviation (Sample)
0
Using N-1
Variance (Population)
0
SD squared
Mean
0
Average
Sum
0
Total
Count
0
N values

Key Takeaways

  • Standard deviation measures how spread out data is from the mean
  • A low standard deviation means data points are close to the mean
  • A high standard deviation means data points are spread out
  • Use sample SD (N-1) when analyzing a subset of a larger population
  • Use population SD (N) when you have data for the entire population

What Is Standard Deviation?

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates that data points are spread out over a wider range of values.

Standard deviation is one of the most commonly used measures of spread in statistics, used in everything from quality control in manufacturing to risk assessment in finance.

The Standard Deviation Formula

Population SD: sigma = sqrt(sum((x - mu)^2) / N)

Sample SD: s = sqrt(sum((x - x-bar)^2) / (N-1))
sigma = Population standard deviation
s = Sample standard deviation
x = Each data value
mu or x-bar = Mean
N = Number of values

How to Calculate Standard Deviation (Step-by-Step)

1

Calculate the Mean

Add all data values together and divide by the count. For example: (5+10+15+20+25)/5 = 15

2

Find Each Deviation

Subtract the mean from each data point: 5-15=-10, 10-15=-5, 15-15=0, 20-15=5, 25-15=10

3

Square Each Deviation

Square each deviation: 100, 25, 0, 25, 100

4

Calculate Variance

Find the average of squared deviations: (100+25+0+25+100)/5 = 50 (population variance)

5

Take the Square Root

Standard deviation = sqrt(50) = 7.07

Population vs Sample Standard Deviation

The key difference between population and sample standard deviation is in the denominator of the variance calculation:

  • Population SD: Divide by N (total count). Use when you have data for the entire population.
  • Sample SD: Divide by N-1 (Bessel's correction). Use when analyzing a sample from a larger population. This correction accounts for the bias in estimating population variance from a sample.

Interpreting Standard Deviation

For data that follows a normal distribution (bell curve), the standard deviation helps predict where values fall:

  • 68% of values fall within 1 standard deviation of the mean
  • 95% of values fall within 2 standard deviations of the mean
  • 99.7% of values fall within 3 standard deviations of the mean

Frequently Asked Questions

Use population standard deviation when you have data for the entire group you're interested in (e.g., test scores of all students in a class). Use sample standard deviation when you're analyzing a subset to make inferences about a larger population (e.g., surveying 100 customers to understand all customers).

A standard deviation of 0 means all values in your dataset are identical. There is no variation or spread in the data - every single data point equals the mean.

No, standard deviation cannot be negative. Since we square the deviations (which makes them all positive) and then take the square root, the result is always zero or positive. The minimum value is 0 (when all values are the same).

Variance is the square of standard deviation, or conversely, standard deviation is the square root of variance. Variance = SD^2, and SD = sqrt(Variance). Variance is useful for mathematical calculations, while standard deviation is easier to interpret because it's in the same units as the original data.

What sample size do I need for reliable results?
It depends on the desired confidence level, margin of error, and population variance. For a typical survey (95% confidence, ±5% margin), n ≈ 385 for a large population. Smaller samples are fine for exploratory analysis, but don't over-interpret the results — widen your confidence intervals to reflect the uncertainty.
How should I interpret the Standard Deviation output?
The result is a calculated estimate based on the formula and your inputs. Compare it against the reference values or benchmarks shown on this page to understand whether your result is high, low, or typical. For decisions with real consequences, use the output as one data point alongside direct measurement and professional advice.
When should I use a different approach?
Use this calculator for quick, formula-based estimates. If your situation involves multiple interacting variables, time-varying inputs, or safety-critical decisions, consider a dedicated software tool, professional consultation, or direct measurement. Calculators are most reliable within their stated assumptions — check that your scenario matches those assumptions before relying on the output.