Standard Deviation Calculator - Sample & Population

Paste your numbers and instantly get the sample and population standard deviation, variance, mean, and a step-by-step breakdown. Free and fully browser-based, your data never leaves your device.

Sample standard deviation (s)

13.4907

Sample variance: 182

Population standard deviation (σ)

12.3153

Population variance: 151.6667

Count (n)

6

Mean

18

Sum

108

Sum of squares (SS)

910

Min

4

Max

42

Step-by-step breakdown

  1. Find the mean: add all 6 values and divide by 6.

    mean = 108 / 6 = 18

  2. Square each deviation from the mean: for every value x, compute (x - mean)².
    x = 4(x - 18)² = 196
    x = 8(x - 18)² = 100
    x = 15(x - 18)² = 9
    x = 16(x - 18)² = 4
    x = 23(x - 18)² = 25
    x = 42(x - 18)² = 576
  3. Sum the squared deviations (SS):

    SS = 910

  4. Divide to get variance: by n - 1 = 5 for a sample, or by n = 6 for a full population.

    s² = 910 / 5 = 182 · σ² = 910 / 6 = 151.6667

  5. Take the square root:

    s = 13.4907 · σ = 12.3153

Sample vs Population Standard Deviation Explained

Standard deviation measures how spread out your numbers are around the mean. A small standard deviation means the values cluster tightly (test scores of 78, 80, 82 have a sample standard deviation of just 2), while a large one means they are scattered (scores of 40, 80, 120 have a sample standard deviation of 40). It is the most widely used measure of variability in statistics because it is expressed in the same units as the data itself, unlike variance which is in squared units.

The sample formula divides the sum of squared deviations by n - 1 instead of n. This is called Bessel's correction: a sample tends to underestimate the true spread of the population it came from, and dividing by n - 1 corrects that bias. Use the sample standard deviation (s) when your data is a subset, such as 30 survey responses out of thousands of customers. Use the population standard deviation (σ) only when you truly have every data point, such as the exam scores of all 25 students in one class.

A practical rule of thumb for roughly bell-shaped data is the 68-95-99.7 rule: about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. If a factory produces bolts with a mean length of 50 mm and a standard deviation of 0.2 mm, you would expect about 95% of bolts to measure between 49.6 mm and 50.4 mm. That is exactly how quality control limits, lab reference ranges, and investment volatility bands are constructed.

This calculator accepts numbers separated by commas, spaces, semicolons, or line breaks, so you can paste a column straight from Excel or Google Sheets. It shows every intermediate step (mean, squared deviations, sum of squares, variance) so you can check homework or verify a spreadsheet formula. For the average alone, try the Average Calculator, and for quick percent math the Percentage Calculator is one click away. Both are free and run entirely in your browser.

How the Standard Deviation Calculator Works

  1. 01Paste or type your numbers, separated by commas, spaces, or new lines - straight from Excel works too.
  2. 02The calculator finds the mean, then squares each value's deviation from that mean.
  3. 03It divides the sum of squares by n - 1 for the sample result and by n for the population result.
  4. 04Read both standard deviations, both variances, and the full step-by-step breakdown, then copy the results.

Sample or population: which one do you need?

Use the sample standard deviation (s, divided by n - 1) when your data is a subset of a larger group, such as 50 survey answers out of thousands of customers. Use the population standard deviation (sigma, divided by n) only when you have every single data point, like all exam scores of one class. When in doubt, the sample version is almost always the right choice - it is what Excel's STDEV.S and most statistics courses use by default.

When to Use This Tool

Use it for statistics homework where you must show every step, for checking a spreadsheet formula, for quality control (how consistent are your measurements?), or for understanding the spread in any data set: test scores, response times, sales per day, or lab results. Because everything runs in your browser, it is also safe for data you would rather not upload anywhere.

Common Use Cases

  • Show every calculation step for statistics homework → Average Calculator
  • Check the consistency of manufacturing or lab measurements
  • Compare the volatility of two data series, like daily sales or returns

Tips

  • When in doubt, use the sample standard deviation - it matches Excel's STDEV.S and most textbooks.
  • Outliers have a big effect because deviations are squared; one extreme value can double your result.
  • Standard deviation only tells the full story for roughly symmetric data; for skewed data also look at min and max.

Frequently Asked Questions

What is the difference between sample and population standard deviation?→
The sample version divides the sum of squared deviations by n - 1, the population version by n. For the data set 2, 4, 6 the population standard deviation is 1.63 while the sample standard deviation is 2.0. The n - 1 correction (Bessel's correction) compensates for the fact that a sample underestimates the true spread of the population it was drawn from.
What does a standard deviation of 0 mean?→
It means every value in your data set is identical, so there is no spread at all. The data set 5, 5, 5, 5 has a mean of 5 and a standard deviation of 0. Any variation between values immediately makes the standard deviation positive.
How is variance related to standard deviation?→
Variance is simply the standard deviation squared. If the standard deviation is 4, the variance is 16. Standard deviation is usually preferred for interpretation because it is in the same units as your data - a standard deviation of 4 cm means something directly, while a variance of 16 square centimeters does not.
What is the 68-95-99.7 rule?→
For roughly bell-shaped (normal) data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. With a mean of 100 and a standard deviation of 15 (like IQ scores), about 95% of values fall between 70 and 130.
Can I paste data straight from Excel or Google Sheets?→
Yes. Copy a column or row of cells and paste it into the input box. The calculator splits on commas, spaces, tabs, semicolons, and line breaks, and ignores anything that is not a number, so header text is skipped automatically.

Related tools

12 suggested

From the blog

Further reading