Measure how spread out your data really is.
Paste any set of numbers and get the full picture: mean, variance and standard deviation for both a population and a sample, plus a chart that shows where every point sits around the average.
Your data
Separate values with commas, spaces or new lines. For example: 4, 8, 15, 16, 23, 42
The headline uses your choice. Both results are always shown below so you can compare.
What your numbers are telling you
Where every point falls
Each value, the mean and the +/-1 SD band
Key measures
Deviation from the mean, point by point
Standard deviation starts here: each value minus the mean, then squared so nothing cancels out. The squared column sums to the total that gets divided by N or by n-1.
| # | Value | Deviation (x - mean) | Squared deviation |
|---|
Standard deviation, explained
What Is Standard Deviation?
Standard deviation measures how spread out numbers are from the mean. A low standard deviation means values cluster close to the average, while a high standard deviation indicates values are spread over a wider range. It is one of the most important concepts in statistics and data analysis.
Population vs Sample Standard Deviation
Population standard deviation (σ) divides by N, while sample standard deviation (s) divides by N-1 (Bessel's correction). Use population SD when you have data for the entire group. Use sample SD when working with a subset of a larger population, which is more common in practice.
How to Interpret Standard Deviation
In a normal distribution, about 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This is known as the 68-95-99.7 rule (empirical rule). It helps identify outliers and understand the reliability of your data.
Standard Deviation in Real Life
Standard deviation is used in finance to measure investment risk (volatility), in manufacturing for quality control, in weather forecasting for temperature variability, and in test scoring to establish grade curves. A higher SD in investments means more risk but potentially higher returns.
Common questions
What is the difference between variance and standard deviation?
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. SD is preferred for interpretation because it uses the same units as the original data.
Can standard deviation be zero?
Yes, a standard deviation of zero means all values in the dataset are identical, there is no spread at all.
Is a higher or lower standard deviation better?
It depends on context. In manufacturing, low SD means consistent quality. In investing, low SD means less risk. In research, context determines whether variability is desirable or not.
Why do we square the deviations?
Squaring ensures all deviations are positive (so negatives don't cancel out positives) and gives more weight to larger deviations, making the measure more sensitive to outliers.
How many data points do I need?
Technically you need at least 2, but more data points give a more reliable standard deviation. For meaningful analysis, aim for at least 5-10 data points.
For learning and planning only. Results depend on the numbers you enter and on choosing the right formula for your data. Double-check whether you have a full population or a sample before you report a standard deviation.