Standard Deviation Formula for Ungrouped Data
For raw, ungrouped observations, subtract the mean from every value, square and add those deviations, divide by n or n − 1, and take the square root. Use the calculator to see each quantity and a graph.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 12, 2026 · How we check statistics tools
Ungrouped standard deviation calculator
Paste individual observations rather than frequency-table rows. The result includes mean, variance, SD, a number-line graph, and a square check.
How to use the standard deviation formula
- Find the mean: x̄ = Σx ÷ n.
- For each observation, calculate x − x̄ and square it.
- Add the squared deviations.
- For a population, divide by N. For a sample estimate, divide by n − 1.
- Take the positive square root of variance.
Mean, variance, and standard deviation formulas
| Measure | Formula | Unit |
|---|---|---|
| Mean | Σx ÷ n | Original unit |
| Population variance | Σ(x − μ)² ÷ N | Squared unit |
| Sample variance | Σ(x − x̄)² ÷ (n − 1) | Squared unit |
| Standard deviation | √variance | Original unit |
The calculator verifies the result by squaring the displayed SD and comparing it with variance.
How to read the standard deviation graph
The generated number line places every observation relative to the mean and marks mean − SD, mean, and mean + SD. It describes this entered data set; it does not assume the observations form a normal distribution.
Important: the familiar 68–95–99.7 rule applies to approximately normal distributions, not automatically to every data set with a standard deviation.
Ungrouped versus grouped data
Ungrouped data lists every observation. Grouped data summarizes repeated values with frequencies or combines values into intervals. For that format, use the grouped-data standard deviation calculator.
Frequently asked questions
Why does sample SD divide by n − 1?
The n − 1 divisor corrects the downward bias in the sample variance estimate when the sample mean is estimated from the same observations.
Can standard deviation be negative?
No. Variance and its principal square root are non-negative.
What happens when every value is equal?
Every deviation is zero, so variance and standard deviation are both zero.