# Population Standard Deviation Calculator

Calculate population standard deviation from population variance using the n denominator

> Canonical page: https://elysiatools.com/en/tools/population-standard-deviation-calculator

- **Category:** Math & Numbers

- **Keywords:** population standard deviation, standard deviation, population variance, statistics, spread, sigma

## Overview

Calculate population standard deviation for a complete numeric dataset.

Population standard deviation is the square root of population variance:

sigma = sqrt(sum((x - mean)^2) / n)

Use it when your dataset contains every item in the population you want to describe, such as a full production batch, all class scores, or a complete measurement series.

## Inputs

- **Dataset** (textarea): e.g. 2, 4, 4, 4, 5, 5, 7, 9
- **Decimal Places** (number)
- **Include Sample Comparison** (checkbox)

## When to use

- When analyzing a complete dataset that represents an entire population rather than a subset or sample.
- When you need to calculate the exact dispersion of scores, such as final grades for a whole class.
- When evaluating quality control metrics for a complete, limited production run where every item was measured.

## How it works

- Enter your complete numeric dataset into the text area, separating values with commas, spaces, or newlines.
- Specify the number of decimal places you want for the final calculated result.
- Optionally, enable the sample comparison checkbox to see the difference between population and sample standard deviations.
- The tool calculates the mean, computes the squared differences, divides by the total population size (n), and returns the square root.

## Use cases

- Teachers calculating the exact grading curve and score dispersion for an entire class.
- Quality assurance engineers analyzing the exact variance of a complete, finite batch of manufactured parts.
- Researchers measuring the precise spread of a small, closed population where every individual was surveyed.

## Frequently asked questions

### What is the difference between population and sample standard deviation?

Population standard deviation divides by n (the total number of items) and is used when you have all the data. Sample standard deviation divides by n-1 to estimate the spread of a larger population from a smaller subset.

### How should I format my dataset?

You can paste numbers separated by commas, spaces, or newlines. The tool will automatically parse the numeric values from your input.

### What does the 'Include Sample Comparison' option do?

It calculates and displays the sample standard deviation alongside the population standard deviation, allowing you to easily compare the two values for the exact same dataset.

### Can I adjust the precision of the result?

Yes, you can use the 'Decimal Places' setting to round the output to your preferred number of decimal digits, up to 10 places.

### Is there a limit to how many numbers I can input?

The tool can handle large datasets pasted into the text area, limited only by standard browser memory and text input constraints.

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