Standard Deviation Calculator

1. Paste or type your data. Copy a column from a spreadsheet or type values with commas or spaces between them. Negative numbers and decimals work.
2. Choose sample or population. Use sample (n - 1) when the values stand for a larger group. Use population (N) only when you measured every member of the group.
3. Read the result and the steps. The standard deviation shows first. The mean, variance, count and each squared deviation appear below it, so you can copy the working into homework.

The two extra modes in the first drop-down handle summary figures: a known SD with a sample size gives the standard error of the mean, and a proportion with a sample size gives the standard error of a proportion.

Standard deviation is a measure of spread: it tells you how far, on average, the values in a data set sit from their mean. A small standard deviation means the values cluster near the mean. A large one means they are scattered.

The NIST/SEMATECH e-Handbook defines it as the square root of the variance, and notes that taking the root restores the original units. Test scores in points give a standard deviation in points; heights in centimeters give a standard deviation in centimeters. That is why people quote the SD, not the variance, in reports.

Statistics uses two symbols. The lower-case letter s is the sample standard deviation, and the Greek letter sigma is the population standard deviation. The mean is the center, the variance is the average squared deviation, and the range (max minus min) is a cruder measure of spread that only looks at two values.

Standard deviation is not the same as the standard error. The SD describes single values. The standard error describes the mean of a sample, and it shrinks as the sample grows.

The formula averages the squared distances from the mean and then takes a square root. The only difference between the two versions is the divisor.

1. Add the values and divide by the count to get the mean.
2. Subtract the mean from each value. These differences are the deviations, and they always add up to 0.
3. Square each deviation so negatives do not cancel positives.
4. Add the squares. This is the sum of squared deviations.
5. Divide by n - 1 for a sample or N for a population. The result is the variance.
6. Take the square root of the variance. That is the standard deviation.

Why n - 1? A sample mean sits closer to its own values than the true population mean does, so dividing by n would understate the spread. OpenStax Introductory Statistics uses n - 1 for the sample SD and N for the population SD, the same as this tool.

A teacher samples eight scores from a large class: 72, 85, 90, 64, 78, 88, 95 and 70. What is the standard deviation?

1. Mean = 642 / 8 = 80.25.
2. Sum of squared deviations = 837.5.
3. Sample variance = 837.5 / 7 = 119.642857.
4. Sample SD = sqrt(119.642857) = 10.938 points.

If the eight scores were the whole class, the population variance would be 837.5 / 8 = 104.6875 and the population SD 10.232 points. The standard error of the mean is 10.938 / sqrt(8) = 3.867 points, and the SD is 13.6 percent of the mean, which the verdict band calls moderate spread.

The standard deviation reacts most to outliers, the sample size and the choice of divisor.

Outliers

Squaring gives far values extra weight. The NIST handbook points out that a value 10 units from the mean adds 100 to the sum, while one 2 units away adds only 4. In the example, the single score of 64 contributes 264 of the 837.5 total, almost a third. A typo such as 950 instead of 95 would swamp the result.

Sample Size

With few values, the gap between n - 1 and N is large. For 8 values the sample SD is 6.9 percent higher than the population SD. For 100 values the gap falls to about 0.5 percent.

Units and Scale

Multiplying every value by 10 multiplies the SD by 10. Adding a constant, such as 5 bonus points to every score, leaves the SD unchanged, because the mean moves by the same amount.

Grouped or Rounded Data

Rounding values to whole numbers or grouping them into classes hides small differences and can change the SD slightly. Enter raw values when you have them.

Use the sample formula unless you truly have every member of the group.

Many spreadsheets offer two functions for the same reason: one divides by n - 1 and one divides by N. The tool shows both, so you can match whichever your course or report asks for.

Homework and Exams

Students check a hand calculation step by step. The method line lists each squared deviation for up to 12 values, which matches the table most textbooks ask for.

Quality and Lab Measurements

Repeated measurements of the same part show how consistent a process is. A lower SD means tighter, more repeatable results.

Grades, Sports and Finance Data

Teachers compare how spread out two classes are. Coaches compare lap times. Analysts compare how much two return series swing. To place one value against the spread, move on to the z-score calculator, which divides the distance from the mean by the SD.

Surveys and Polls

The proportion mode gives the standard error of a percentage, the building block of a poll margin of error.

1. Using N for a Sample

Dividing a sample by N understates the spread. With 8 values the error is about 6.5 percent of the correct answer.

2. Forgetting the Square Root

The variance, 119.64 in the example, is not the SD. The SD is its square root, 10.94. Units are the quick check: variance is in squared units.

3. Pasting Thousands Separators

The value 1,200 is read as two numbers, 1 and 200. Remove thousands commas before pasting, and confirm the count n matches the number of values you meant to enter.

4. Mixing Up SD and Standard Error

The SD describes single values; the standard error describes the mean. Reporting the standard error as the SD makes data look far less variable than it is.

5. Ignoring an Outlier

One data-entry error can double the SD. Check the min and max line before you trust the result.

6. Reading the SD Without the Mean

An SD of 10 is large for values near 20 and tiny for values near 10,000. The coefficient of variation in the verdict band puts the SD in context. For a spread measure that ignores extreme values, see the percentile calculator and the interquartile range.

The arithmetic is exact to about 15 significant digits; the limits come from the data and the model, not the math.

No single value is good; it depends on the mean and the purpose. A common rule of thumb reads a coefficient of variation under 10 percent as low spread and over 30 percent as high.

No. It is a square root of a sum of squares, so it is 0 or positive. It is 0 only when every value is the same.

The two most likely causes are the divisor and rounding. Check whether the book used n - 1 or N, and whether it rounded the mean before squaring.

A sample needs at least 2 values. Estimates from fewer than about 10 values move a lot when one value changes.

The population SD is exactly 2 and the sample SD is 2.138. The mean is 5 and the sum of squared deviations is 32.

No. Shifting every value by a constant moves the mean by the same amount, so every deviation stays the same.

It is the SD divided by the square root of n. With an SD of 15 and n = 25, the standard error is 15 / 5 = 3.

Use the proportion mode. For 42 percent of 500 people, sqrt(0.42 x 0.58 / 500) = 0.0221, or 2.2 percentage points.

Yes. Copy the cells and paste them in the data box; new lines and tabs count as separators. Check that the count n matches your row count.

No. The math runs in your browser. Nothing is uploaded, and Save keeps a copy only in your own browser storage.

• Variance, standard deviation and range as measures of scale (NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6).
• Sample (n - 1) and population (N) standard deviation formulas (OpenStax Introductory Statistics 2e, section 2.7).
• Standard error of the mean, sigma / sqrt(n) (OpenStax Introductory Statistics 2e, section 7.1).
• Standard deviation of a proportion, sqrt(pq / n), and the np and nq rule (OpenStax Introductory Statistics 2e, section 8.3).