Standard Deviation Calculator
Sample and population deviation, variance, and mean for number lists.
What it does
Standard deviation reports how far values typically sit from their mean. A small value means the group clusters tightly, while a large one means the values scatter widely, even when two groups share the same average.
The tool above takes a typed list and a mode switch. Sample mode divides by n-1 and fits data drawn from a larger group, while population mode divides by n and fits a complete set. It returns mean, deviation, and variance together.
Variance is the average squared distance from the mean, and deviation is its square root. Squaring first gives extra weight to far values, then rooting returns the spread to original units like points, seconds, or dollars.
The formulas follow the NIST engineering statistics handbook definitions for variance and deviation, with the same n-1 sample form taught in Penn State and Khan Academy statistics units. Pick the mode that matches what the list represents.
How to use
- Type the numbers separated by commas or spaces.
- Pick sample for a subset drawn from more, population for the full set.
- Read the mean first to anchor the center.
- Read the deviation for the typical distance from that center.
- Compare variance when you need the squared form for further math.
Formula
- mean
- Center that distances are measured from.
- n - 1
- Sample divisor correcting small group bias.
- n
- Population divisor for complete sets.
- root
- Square root returning spread to original units.
Single value lists are rejected because spread needs at least two points. Sample mode on two values divides by one, which keeps the math defined but very sensitive to the pair.
Worked example
Example: 4, 8, 15, 16, 23, 42, sample mode
Six values, sample divisor n-1 = 5.
- Mean: (4 + 8 + 15 + 16 + 23 + 42) / 6 = 108 / 6 = 18.
- Squared gaps: 196 + 100 + 9 + 4 + 25 + 576 = 910.
- Sample variance: 910 / 5 = 182.
- Deviation: root of 182 = 13.49.
- Population check: 910 / 6 = 151.67, root 12.32. Tool shows 18, 13.49, 182.
Reference table
| Mode | Variance | Deviation |
|---|---|---|
| Sample (n-1) | 182 | 13.49 |
| Population (n) | 151.67 | 12.32 |
Limits
- Outliers inflate the result because gaps are squared first.
- Needs numbers only and at least two values.
- Skewed groups may read better with median and quartiles beside it.
- Units follow the data; mixing units in one list corrupts the answer.
Common mistakes
- Using population mode on a small sample, which understates spread.
- Forgetting the final square root and reporting variance as deviation.
- Rounding each gap before squaring instead of once at the end.
- Comparing deviations across different units without converting.
- Reading a single number as if two close means always agree.
Try it yourself
Quick checks. Answers sit tucked under each question.
Sample deviation of 2, 4, 4, 4, 5, 5, 7, 9?
Mean 5, squared sum 32, variance 32/7 = 4.57, deviation 2.14.
Population deviation of 2, 4?
Mean 3, squared sum 2, variance 1, deviation 1.
Variance when deviation is 5?
Square it: 25 in the same squared units.
Questions
Sample or population?
Sample for a subset (divide by n-1), population for the full group (divide by n). Labs and polls use sample.
What does 13.49 mean here?
Values sit typically about 13.49 away from the mean 18 in original units.
Why square and then root?
Squaring weights far values and removes signs, rooting returns the spread to plain units.
Can deviation be zero?
Yes, when every value matches. Any spread at all pushes it above zero.
Why n-1 for samples?
One degree of freedom goes to estimating the mean, so n-1 keeps small sample spreads unbiased.
Variance vs deviation?
Variance is the squared spread (182 here), deviation its root (13.49). Most reports quote the deviation.