Statistics
Standard Deviation Calculator: Find Sample And Population SD With Steps
Last updated: October 2026
Written by Jessica Hill
A standard deviation calculator measures how far the values in a data set typically are from the mean. To find it, subtract the mean from each value, square the results, average them and take the square root. Divide by n for a population or by n − 1 for a sample. A small standard deviation means the data is tightly grouped.
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Two data sets can have the same average and still look nothing alike. Standard deviation is the number that tells them apart. This guide shows how to calculate it, which formula to use, and how to read the result. Every example was checked in Python.
Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.
What Standard Deviation Tells You
Standard deviation measures spread: how far a typical value sits from the mean. A low standard deviation means the values are close together. A high one means they are scattered.
Here are two classes with the same average test score of 75:
| Class | Scores | Mean | Population SD |
|---|---|---|---|
| A | 73, 74, 75, 76, 77 | 75 | 1.41 |
| B | 55, 65, 75, 85, 95 | 75 | 14.14 |
The averages match, but class A scored almost the same, while class B ranges from 55 to 95. The standard deviation shows that difference in one number. If you only need the average itself, the mean median mode calculator covers it.
Standard deviation is written σ (sigma) for a population and s for a sample. It is always in the same units as the data. If the data is in inches, so is the standard deviation.
How To Use The Standard Deviation Calculator
The calculator above works for raw data, frequency tables and z-scores.
- Pick a tab. Data Set for a plain list, Frequency Table when values come with counts, or Z-Score to see where one value falls.
- Enter your data. Type numbers with commas or spaces, or paste a column from Excel or Google Sheets.
- Choose sample or population. The next section explains which one you need. Both results are shown either way.
The step table lets you check each part of the work, which is what most homework asks for.
Sample Vs Population Standard Deviation
Use the population formula when your data includes every member of the group you care about. Use the sample formula when your data is only part of a larger group.
| Population | Sample | |
|---|---|---|
| Symbol | σ | s |
| Divide by | N (the count) | n − 1 |
| Formula | σ = √(Σ(x − μ)² ÷ N) | s = √(Σ(x − x̄)² ÷ (n − 1)) |
| Example | Test scores of every student in one class, when you only care about that class | 50 students surveyed to estimate a whole school |
Dividing by n − 1 is called Bessel's correction. A sample tends to sit closer to its own mean than the full population does, so dividing by a slightly smaller number makes the estimate a little larger and more accurate on average.
When in doubt, use the sample formula. Most real data in science, business and school projects is a sample. The two answers get closer as the data set grows. With 1,000 values, the difference is tiny.
How To Calculate Standard Deviation By Hand
There are five steps: find the mean, subtract it from each value, square each difference, add the squares, then divide and take the square root.
Use the data set 5, 9, 8, 12, 6, 10, 6, 8.
Step 1: Find the mean. The sum is 64 and there are 8 values, so the mean is 64 ÷ 8 = 8.
Steps 2 and 3: Subtract the mean and square each difference.
| Value (x) | x − mean | (x − mean)² |
|---|---|---|
| 5 | −3 | 9 |
| 9 | 1 | 1 |
| 8 | 0 | 0 |
| 12 | 4 | 16 |
| 6 | −2 | 4 |
| 10 | 2 | 4 |
| 6 | −2 | 4 |
| 8 | 0 | 0 |
| Total | 0 | 38 |
The deviations always add up to 0. That is a quick check that your mean is right.
Step 4: Divide to get the variance.
- Population variance: 38 ÷ 8 = 4.75
- Sample variance: 38 ÷ 7 ≈ 5.429
Step 5: Take the square root.
- Population standard deviation: √4.75 ≈ 2.179
- Sample standard deviation: √5.429 ≈ 2.330
For the square root, any scientific calculator works. Round only at the very end. Rounding the mean or the squares early can throw off the final answer.
Standard Deviation Vs Variance
Variance is the average of the squared deviations. Standard deviation is its square root. They describe the same spread, but in different units.
In the example above, the population variance is 4.75 and the standard deviation is 2.179. If the data were in dollars, the variance would be in "square dollars," which is hard to picture. The standard deviation is back in dollars, so it is easier to understand and report.
Variance is still useful in math, because variances of independent things can be added. That is why many formulas work with variance first and take the square root at the end.
Why The Deviations Are Squared
The deviations are squared for two reasons. First, the raw deviations always add to zero, so their plain average would always be 0. Squaring makes every deviation positive. Second, squaring gives large deviations more weight, which makes the measure sensitive to values far from the mean.
An alternative is the mean absolute deviation (MAD), which averages the distances without squaring. For our data, the absolute deviations are 3, 1, 0, 4, 2, 2, 2, 0, which average to 14 ÷ 8 = 1.75. The MAD is easier to explain, but standard deviation has better math properties and is the standard in statistics.
The Empirical Rule (68-95-99.7)
For data that follows a normal, bell-shaped distribution, standard deviation tells you how much of the data falls near the mean. According to the NIST Engineering Statistics Handbook, about 68.27 percent of values fall within 1 standard deviation of the mean, 95.45 percent within 2, and 99.73 percent within 3.
| Range | Share of data | Test scores with mean 75, SD 5 |
|---|---|---|
| Mean ± 1 SD | About 68% | 70 to 80 |
| Mean ± 2 SD | About 95% | 65 to 85 |
| Mean ± 3 SD | About 99.7% | 60 to 90 |
This rule only holds for roughly normal data. Incomes, wait times and other skewed data do not follow it. In those cases, the standard deviation still measures spread, but the percentages do not apply.
Z-Scores: How Far Is One Value From The Mean?
A z-score tells you how many standard deviations a value is from the mean: z = (value − mean) ÷ standard deviation. A positive z-score is above the mean, and a negative one is below.
If test scores have a mean of 75 and a standard deviation of 5, a score of 85 has z = (85 − 75) ÷ 5 = 2. That score is 2 standard deviations above average, higher than about 97.7 percent of scores in a normal distribution.
Z-scores let you compare values from different scales. A z-score of 2 on a math test and 1 on a reading test means the math result was more unusual, even if the raw scores look similar. Values with z-scores beyond about ±3 are often flagged as possible outliers.
Standard Deviation From A Frequency Table
When data comes as values with counts, multiply each squared deviation by its frequency instead of writing out every value.
Values 1, 2, 3 and 4 appear 3, 5, 8 and 4 times:
| Value (x) | Frequency (f) | f × x | (x − mean)² | f × (x − mean)² |
|---|---|---|---|---|
| 1 | 3 | 3 | 2.7225 | 8.1675 |
| 2 | 5 | 10 | 0.4225 | 2.1125 |
| 3 | 8 | 24 | 0.1225 | 0.9800 |
| 4 | 4 | 16 | 1.8225 | 7.2900 |
| Total | 20 | 53 | 18.55 |
The mean is 53 ÷ 20 = 2.65. The population variance is 18.55 ÷ 20 = 0.9275, so the population SD is about 0.963. The sample SD uses 18.55 ÷ 19, giving about 0.988.
For grouped data with class ranges, use each class midpoint as x. The result is an estimate, since the exact values inside each class are unknown.
Standard Error And Relative Standard Deviation
Two related measures are built from standard deviation, and both come up often in science and business.
Standard error of the mean (SEM) shows how precise a sample mean is: SEM = s ÷ √n. For our data, 2.330 ÷ √8 ≈ 0.824. The standard deviation describes the spread of individual values. The standard error describes how much the mean itself would change from sample to sample. It shrinks as the sample gets bigger.
Coefficient of variation (CV), also called relative standard deviation, compares spread to the size of the mean: CV = (SD ÷ mean) × 100%. For our data, 2.330 ÷ 8 × 100 ≈ 29.1 percent. This lets you compare spread across data sets with very different scales, like the weights of mice and elephants. The percentage calculator helps with the final percent step.
How To Find Standard Deviation In Excel Or On A TI-84
Spreadsheets and graphing calculators have built-in standard deviation functions. The key is picking sample or population.
| Tool | Sample SD | Population SD |
|---|---|---|
| Excel | =STDEV.S(A1:A8) | =STDEV.P(A1:A8) |
| Google Sheets | =STDEV(A1:A8) | =STDEVP(A1:A8) |
| TI-84 | STAT > CALC > 1-Var Stats, shown as Sx | Same screen, shown as σx |
On a TI-84, enter the data in L1 first with STAT > EDIT. The 1-Var Stats screen shows both values, so pick Sx for a sample and σx for a population. In Excel, the older STDEV function also gives the sample SD.
Where Standard Deviation Is Used
Standard deviation shows up anywhere consistency matters.
- Manufacturing: factories track the standard deviation of part sizes. A small SD means parts are consistent and fit together.
- Finance: the standard deviation of an investment's returns is a common measure of how risky or volatile it is.
- Education: test makers use standard deviation to scale scores and to spot unusually high or low results.
- Science: lab results are often reported as mean ± standard deviation to show how much repeated measurements varied.
- Weather: the standard deviation of daily temperatures shows how much a place's weather swings.
Common Standard Deviation Mistakes
Most mistakes come from choosing the wrong formula or skipping a step.
| Mistake | Example | Fix |
|---|---|---|
| Using the wrong formula | Dividing a sample by n | Divide by n − 1 for a sample |
| Forgetting the square root | Reporting 4.75 as the SD | 4.75 is the variance. The SD is 2.179 |
| Not squaring the deviations | Averaging −3, 1, 0, 4… to get 0 | Square each deviation first |
| Rounding the mean too early | Using 8.1 instead of 8.125 in a longer data set | Keep full precision until the end |
| Applying the 68-95-99.7 rule to skewed data | Using it for incomes | Only use it for roughly normal data |
| Mixing up SD and standard error | Reporting s ÷ √n as the spread of the data | SEM is about the mean, SD is about the data |
Summary
- Standard deviation measures how spread out data is around the mean, in the same units as the data.
- Divide by N for a population and by n − 1 for a sample. Use the sample formula when in doubt.
- The five steps: find the mean, subtract, square, add, then divide and take the square root.
- Variance is the SD squared. Z-scores tell you how many SDs a value is from the mean.
- For normal data, about 68, 95 and 99.7 percent of values fall within 1, 2 and 3 SDs.
Next step: paste your data into the standard deviation calculator at the top of this page. For more tools, visit the Calc Solver homepage.
Frequently Asked Questions
Can standard deviation be negative?
No. Standard deviation is the square root of an average of squared numbers, and squares are never negative. So the smallest possible standard deviation is 0. If a calculation gives a negative number, there is a mistake, usually a sign error in the squaring step or subtracting in the wrong order somewhere.
What does a standard deviation of 0 mean?
A standard deviation of 0 means every value in the data set is exactly the same. For example, 7, 7, 7 has a mean of 7 and no spread at all, so the SD is 0. In real measurements, an SD of 0 is rare. It can also mean the measuring tool was not precise enough to show small differences.
What is a good standard deviation?
There is no single good value. It depends on the data and what you need. For parts in a factory, a tiny SD is good, because it means consistent quality. For a test meant to separate strong and weak students, a very small SD might mean the test was too easy. Compare the SD to the mean, or use the coefficient of variation, to judge it.
Is standard deviation affected by outliers?
Yes, quite strongly. Because each deviation is squared, one value far from the mean adds a lot to the total. Adding a single score of 30 to the class A scores 73 to 77 would raise the standard deviation from about 1.4 to over 16. For data with outliers, the interquartile range is often a better measure of spread.
What happens to the standard deviation if I add a constant to every value?
Nothing. Adding or subtracting the same number shifts every value and the mean by the same amount, so the distances from the mean do not change. Multiplying every value by a number does change it, though. If every value is doubled, the standard deviation doubles too, and the variance becomes four times larger.
Try it with your own numbers
The calculator at the top of this page uses the same formulas as the examples.