What is standard deviation?
Standard deviation measures how spread out values are around their mean. A small value means data points cluster close to the average; a large one means they're widely scattered. Two classes might both average 70, but with standard deviations of 5 and 20 — the first class performs evenly, while the second mixes top performers and students who are struggling.
Formulas
- Mean:
x̄ = Σx ÷ n - Sample variance:
s² = Σ(x − x̄)² ÷ (n − 1) - Population variance:
σ² = Σ(x − μ)² ÷ n - Standard deviation: the square root of the variance.
Sample or population?
Use the population formula when your data includes every member of the group you're studying — for example, the scores of every student in one class. Use the sample formula (dividing by n − 1) when your data is a subset drawn from a larger group and you want to estimate its spread, which is the usual case in research. Dividing by n − 1 (Bessel's correction) offsets a sample's tendency to underestimate true variability.
Other statistics included
- Median: the middle value once data is sorted — less affected by outliers than the mean.
- Mode: the most frequent value; there may be several or none.
- Range: the difference between the largest and smallest values.
- Quartiles Q1, Q3 and IQR: useful for spotting outliers — values more than 1.5 × IQR beyond the quartiles are usually considered outliers.
- Coefficient of variation: standard deviation as a percentage of the mean, for comparing spread across different units.
To visualize your data, use the Chart Maker.
Frequently Asked Questions
What's the difference between variance and standard deviation?
Standard deviation is the square root of variance and is expressed in the same units as the data, making it easier to interpret.
How do I enter data?
Separate numbers with commas, spaces or new lines — you can paste a column straight from Excel.
Are negative numbers and decimals supported?
Yes, both are supported.
Why does sample SD differ from population SD?
The sample formula divides by n − 1 instead of n to correct bias, so it's slightly larger.
What does “no mode” mean?
Every value appears only once, so no value is more frequent than another.
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