Standard Deviation Calculator
Calculate sample or population standard deviation from a list of numbers. Get the variance, mean, and a step-by-step breakdown of the calculation.
Standard deviation calculator inputs
What Is Standard Deviation?
Standard deviation measures how far typical values sit from the mean. A low value means the numbers cluster near the average. A higher value means more spread. The number is only meaningful in the units of the data: 2 points on a 100-point test is small; 2 inches on a 3-inch part is large.
How to Use This Standard Deviation Calculator
- Paste or type your numbers. Commas, spaces, tabs, and new lines all work.
- Choose sample (n − 1) or population (N). Sample is the default because most homework and lab lists are subsets.
- Read the selected standard deviation and variance. The other formula stays visible beside it.
- Open the steps to see the count, mean, deviations, and square-root finish.
Standard Deviation Formulas
Population: σ = √[Σ(xᵢ − μ)² / N]. Sample: s = √[Σ(xᵢ − x̄)² / (n − 1)].
- σ is the population standard deviation; s is the sample standard deviation.
- xᵢ is one observation. μ is the population mean; x̄ is the sample mean.
- N or n is the count. Σ means add the squared deviations.
Bessel's correction (n − 1) makes the sample variance unbiased under ordinary independent-sampling assumptions. The sample standard deviation, the square root of that variance, is not in general an exactly unbiased estimator of σ.
How to Calculate Standard Deviation Step by Step
Dataset: 2, 4, 4, 4, 5, 5, 7, 9. Count = 8. Sum = 40. Mean = 5.
| Value (xᵢ) | xᵢ − mean | (xᵢ − mean)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
Population variance = 32 / 8 = 4. Population SD = 2. Sample variance = 32 / 7 = 4.5714285714. Sample SD = 2.1380899353.
Sample vs. Population Standard Deviation
| Sample | Population | |
|---|---|---|
| Symbol | s | σ |
| Denominator | n − 1 | N |
| When to use | The list is a subset of a larger group | The list is the entire group of interest |
| Excel / Sheets | =STDEV.S(A1:A8) | =STDEV.P(A1:A8) |
Standard Deviation Examples
- Simple list 2, 4, 6, 8: mean 5, population SD 2.2360679775.
- Test scores 72, 85, 90, 90, 94: sample SD 8.5557.
- Lab decimals 0.1, 0.2, 0.3: sample SD 0.1000.
- Negatives −2, −1, 0, 1, 2: mean 0, sample SD 1.5811388301. CV is undefined because the mean is 0.
- Identical values 10, 10, 10, 10: SD 0.
- Outlier 10, 11, 12, 13, 50: sample SD 17.2540, versus 1.2910 without the 50.
Standard Deviation vs. Variance
Variance is the average squared deviation, so its units are squared (points², inches²). Standard deviation is the positive square root, back in the original units. If variance = 25, standard deviation = 5. They answer related questions; they are not interchangeable on a report.
How to Calculate Standard Deviation in Excel and Google Sheets
- Put one number in each cell of a column, for example A1:A8.
- Use =STDEV.S(A1:A8) for a sample and =STDEV.P(A1:A8) for a population.
- Copy the column and paste it into this calculator to check the same split of sample versus population.
Older STDEV and STDEVP names still work in many workbooks. They match S and P. Do not type thousands separators into the paste box here; 1250 stays 1250.
Standard Deviation From a Frequency Table
Exact-value frequency mode treats 10 | 3 and 20 | 2 as the list 10, 10, 10, 20, 20 without building that list in memory. Count = 5, mean = 14, population SD = 4.8989794856, sample SD = 5.4772255751.
That is not a grouped class-interval calculation. Midpoint approximations for bins such as 10–20 belong on a different tool.
How to Interpret Standard Deviation
- High SD: values wander far from the mean.
- Low SD: values hug the mean.
- Zero SD: every value equals the mean.
- Always report the mean with the SD. 76 ± 4 is readable; 4 alone is not.
- One outlier can inflate SD because deviations are squared.
- The 68–95–99.7 empirical rule applies when the shape is approximately normal. This calculator does not assume that shape.
Common Standard Deviation Mistakes
- Using N on a sample, or n − 1 on a complete census.
- Reporting variance when the reader expects original units.
- Stopping at the variance and forgetting the square root.
- Ignoring an obvious outlier before comparing two groups.
- Pasting 1,250 and expecting one number — commas separate values here.
- Applying the empirical rule to skewed data.
- Asking for sample SD from a single observation.
- Comparing SDs from incompatible units without a relative measure such as CV.
Frequently Asked Questions
- What is a standard deviation calculator?
- A tool that takes a list of numbers and returns sample or population standard deviation, variance, and the mean.
- How do I calculate standard deviation?
- Find the mean, square each deviation, average those squares with N or n − 1, then take the square root.
- What is the difference between sample and population standard deviation?
- Sample uses n − 1. Population uses N. Sample is for a subset; population is for the whole group.
- What is the formula for standard deviation?
- σ = √[Σ(xᵢ − μ)² / N] or s = √[Σ(xᵢ − x̄)² / (n − 1)].
- What does a high standard deviation mean?
- Values are more spread out around the mean, in the data's own units.
- What does a low standard deviation mean?
- Values sit close to the mean.
- Can standard deviation be zero?
- Yes, when every observation equals the mean.
- Can standard deviation be negative?
- No. It is a square root of a sum of squares.
- Is standard deviation the same as variance?
- No. Standard deviation is the square root of variance.
- How do I calculate standard deviation in Excel?
- =STDEV.S(range) for a sample, =STDEV.P(range) for a population.
- Can I calculate standard deviation with decimals and negative numbers?
- Yes. The same formulas apply. Scientific notation such as 1.5e3 is accepted.
- What happens when my dataset contains an outlier?
- Squared deviations magnify that point. Compare the Outlier and Textbook examples above.
Methodology and Limits
Totals use a Welford / West running algorithm so a large offset with a small spread does not cancel in floating point. Displayed decimals are rounded to the selected places; internals stay unrounded. Frequency mode uses weights instead of repeating rows. Datasets stay in this browser. There is no share URL that copies your numbers into the address bar.