# Standard Deviation Calculator

Standard deviation calculator with a step-by-step solution, APA format, and charts.

The tool doesn't count empty cells or non-numeric cells.

## Standard deviation calculator

Use the standard deviation calculator to compute both sample standard deviation and population standard deviation, complete with a step-by-step solution, and then present the results in APA format.

### What is standard deviation?

Standard deviation is a parameter that measures the variability of data. It represents the squared root of the average of squared differences between each value and the mean. A larger standard deviation represents more data variability.

#### Population standard deviation formula

To calculate the population standard deviation, you need the entire dataset.

${\sigma}^{}=\sqrt{\frac{\sum _{i=1}^{n}{({x}_{i}-\mathrm{x\u0304})}^{2}}{n}}$### What is sample standard deviation? (S)

When you do not have data for the entire population, you calculate the sample Standard deviation from the sampled data. Unlike population Standard deviation, when calculating the sample variance, you divide by (n - 1), making the resulting statistic less biased.

#### Sample standard deviation formula

Following the sample standard deviation formula

${S}^{}=\sqrt{\frac{\sum _{i=1}^{n}{({x}_{i}-\mathrm{x\u0304})}^{2}}{n-1}}$### What statistic should I use, population standard deviation or sample standard deviation?

Use the population standard deviation (σ) if the data contains the entire population, otherwise, use the sample standard deviation (S).

### What is the sample standard deviation

Usually, you don't have access to the entire population's data because it can be costly to gather all the data or may damage the sample. In such cases, you calculate the sample standard deviation based on a random sample of the data.

When calculating the sample standard deviation, you use a sample average (x̄) instead of the population average (µ). The sample average is a bit closer to the center of the sample than the population average. As a result, if you were to divide by n, on average, the sample average would be greater than the population standard deviation. Dividing by (n-1) will correct the biased estimation of the variance, and partially correct the biased estimation of the standard deviation (Bessel's correction). The sample standard deviation is still biased, but this correction makes it the best simple formula.

#### Simulation Example

Normal(µ=10, σ=4) distribution with a sample size of n=20, with 1,000,000 repetitions.

The averages of the statistics:

Statistic | Dividing by (n-1) | Dividing by n |
---|---|---|

Standard Deviation | 3.958 | 3.849 |

Variance | 16.008 | 15.208 |

For an unbiased statistic, we expect to get a standard deviation of 4 and a variance of 16.

You may notice that dividing by (n-1) yields better results than dividing by n. The result for the variance is not biased; it is very close to 16, while the result for the standard deviation is biased.

### Glossary

**n**-

**Sample size**, the number of values.

**Mean**-

**Average**.

**S**-

**Sample standard deviation**, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Standard Deviation.

**S**-

**Sample standard deviation**, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Standard Deviation.

**σ**-

**Population standard deviation**, if the list of values you entered is the entire population, this is the exact standard deviation.

**σ**-

**Population standard deviation**, if the list of values you entered is the entire population, this is the exact standard deviation.