ToolsWebBeta

Standard Deviation Calculator

Use our Standard Deviation Calculator to find standard deviation, variance, mean, and median from sample or population data with clear results for quick analysis.
Standard Deviation
15.81
Count

5

Mean

30

Variance

250

Median

30

Recent
Nothing here yetYour recent activity will appear here

What is the Standard Deviation?

Standard deviation measures how far data values spread from the mean. A small value shows little variation, while a large value shows greater variation.

Population Standard Deviation

Population standard deviation measures the spread of every value in a complete population. Use it when your dataset includes the full group you want to study.

Sample Standard Deviation

Sample standard deviation estimates spread when your data represents part of a larger population. It uses n−1n-1 to reduce bias in the estimate.

Standard Deviation Calculator showing a student analyzing statistical data with distribution charts, data points, and a calculator on a laptop.

How can you use our Standard Deviation Calculator?

Our standard deviation calculator quickly calculates standard deviation, variance, mean, and median. Enter your data, choose its type, and view clear statistics.

Step 1

Select Your Data Type

Choose sample or population data.

Step 2

Enter Your Numbers

Enter the numbers you want to calculate.

Step 3

View Your Results

Get the standard deviation, variance, mean, and median.

Step 4

Copy Your Result

Copy the calculated statistics for future use.

What Results Does the Calculator Provide?

The calculator provides key statistics that help you understand the center, spread, and variation within your dataset.

Standard Deviation

Standard deviation shows how much values differ from the mean. A result near zero means the data points stay close to their average.

Variance

Variance measures spread using squared differences from the mean. Its units are squared, while standard deviation uses the original data units.

Mean

Mean is the arithmetic average of your dataset. Add every data value, then divide the total by the number of values.

Median

Median is the middle value after sorting the dataset. It helps show the data center and stays less affected by extreme values.

How to calculate Standard Deviation manually?

You can calculate standard deviation by finding the mean, measuring each deviation, squaring those values, and using the correct formula. Choose the population or sample formula based on your data.

Formula

For population standard deviation:

σ=√[Σ(x−μ)2/N]σ = √[Σ(x − μ)² / N]

For sample standard deviation:

s=√[Σ(x−xˉ)2/(n−1)]s = √[Σ(x − x̄)² / (n − 1)]

Here, xx means each data value. The symbol μμ means population mean, while xˉx̄ means sample mean. NN represents the population size, and nn represents the sample size.

Step by Step Guide

  1. Find the mean of your dataset.
  2. Subtract the mean from each data value.
  3. Square each difference.
  4. Add all squared differences.
  5. Divide the total by NN for population data.
  6. Divide by n−1n − 1 for sample data.
  7. Find the square root of the result.

Practical Example

Use the dataset 2,4,6,82, 4, 6, 8.

  1. Mean = (2+4+6+8)÷4=5(2 + 4 + 6 + 8) ÷ 4 = 5.
  2. Differences = −3,−1,1,3−3, −1, 1, 3.
  3. Squared differences = 9,1,1,99, 1, 1, 9.
  4. Sum of squares = 2020.
  5. Population variance = 20÷4=520 ÷ 4 = 5.
  6. Population standard deviation = √5≈2.24√5 ≈ 2.24.

The result, about 2.24, shows how far the values typically spread from their mean of 55.

Explore Statistics Calculators

Explore statistics calculators for averages, probability, distributions, percentages, and data analysis.

Frequently Asked Questions

These answers explain sample and population standard deviation, n−1, z-scores, data entry, and how to interpret your results.