Standard Deviation Calculator
Calculate statistical measures including mean, median, mode, standard deviation, and probability.
Statistical Analysis
Enter your data points separated by commas, spaces, or line breaks
Use when your data represents a sample from a larger population (divides by n-1)
For confidence interval calculation
Key Concepts
- • Standard Deviation: Measures data spread from the mean
- • Variance: Average of squared differences from mean
- • Sample vs Population: Different denominators (n-1 vs n)
- • Outliers: Values beyond 1.5 × IQR from quartiles
Statistical Results
Standard Deviation
Sample (s)
Variance
s²
Mean
Average value
Descriptive Statistics
Five Number Summary
95% Confidence Interval
Sample vs Population
Interpretation Tips
- • Lower standard deviation = data points closer to mean
- • Higher standard deviation = more spread out data
- • ~68% of data falls within 1 standard deviation of mean
- • ~95% of data falls within 2 standard deviations of mean
- • Use sample std dev when data is a sample from larger population
How it works
Standard deviation measures how spread out a data set is around its mean. You find each value's distance from the mean, square those distances (so positives and negatives don't cancel), average them to get the variance, then take the square root to return to the original units. A small σ means values cluster near the mean; a large one means they're scattered.
Standard deviation
σ = √[ Σ(xᵢ − μ)² ÷ N ] (sample: divide by N − 1 instead of N)
- xᵢ
- each data value
- μ
- the mean of the data
- N
- number of values
Worked example
- Data: 2, 4, 6 (population)
- Mean μ = 4
- Squared deviations: (−2)², 0², 2² = 4, 0, 4
- Variance = (4+0+4) ÷ 3 = 2.67
σ = √2.67 ≈ 1.63.
Good to know
- Use the population formula (÷ N) when you have every data point; use the sample formula (÷ N − 1) when your data is a sample of a larger group.
- Variance is just σ² — same information, but in squared units, which is why standard deviation (back in original units) is easier to interpret.
- In a normal distribution, ~68% of values fall within 1 σ of the mean and ~95% within 2 σ.
Related Calculators
Frequently Asked Questions
What does standard deviation actually measure?
It measures how spread out values are around the mean, in the same units as the data. A small standard deviation means values cluster tightly near the average; a large one means they vary widely.
Should I use sample or population standard deviation?
Use population SD (divide by n) only when you have every member of the group. When your data is a sample of a larger population — the usual case — divide by n - 1 (Bessel's correction) to avoid underestimating the spread.
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value and square the results, average those squared deviations (using n or n - 1), and take the square root. The calculator shows each of these steps.
What is the 68-95-99.7 rule?
For roughly normal data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. It is a quick way to judge whether a particular value is unusual.
What is the difference between variance and standard deviation?
Variance is the average of squared deviations; standard deviation is its square root. Because SD is in the original units (dollars, points, cm), it's far easier to interpret than variance's squared units.