Statistics is the branch of science in which we analyze the numerical values of large quantities, collect data, and interpretation of data. Standard deviation is a prevalent and important concept in statistics. The early concept of standard deviation was introduced when in the 18^{th} century the concept of probability was in the early stage.

In this article, we’ll discuss the formula of standard deviation, its methods of finding, step-by-step guide, and applications in various fields. We will solve some examples for a better understanding of the conceptof standard deviation.

Table of Contents

## Definition of Standard Deviation

A statistical measure that quantifies the amount of variation or dispersion in a set of data values is called standard deviation. We can also say that the standard deviation is a way to assess how spread out or clustered data points are around the data set’s mean.

## Formulas of Standard deviation:

There are different formulas for standard deviation, which are below:

- Standard deviation formula for population
- Standard deviation formula for sample
- For Group data

### Standard deviation formula for population

The following formulais used when the given data is in the form of a population.

**σ = √(∑(x – μ) ^{2}) / N**

**Where:**

**σ**is denoted by the standard deviation of the population**x**indicates each data point**μ**is the population mean of the data**N**is no. of values of the population

### Standard deviation formula for sample

The following formula is used when the given data is in the form of a sample.

**s = √(∑(x –x̄****) ^{2}) / n – 1**

**Where:**

**s**is denoted by the sample standard deviation**x**indicates each data point**x̄**is the sample mean of the data**n**is no. of values of the sample

**Note:**

These formulas are used when given data is ungrouped.

### For Group data:

The following formula is used to find out the standard deviation:

**σ = √(∑f(x – μ) ^{2}) / ∑f**

**Where:**

**σ**is denoted by the population standard deviation**x**denoted the individual data point**μ**is the population mean of the data**∑f**is the total frequency

## How to Find Standard Deviation?

Below is astep-by-step guide to find out the standard deviation of the data:

**Step 1:**Collect the data given in question

**Step 2:**Find the mean of the data

**Step 3:**Find out the difference from the mean (x – x̄) or (x – μ)

**Step 4:**Take a Square of the step 3

**Step 5:**Sum of all the values of (x – x̄)^{2} or (x – μ)^{2}

**Step 6:**Divide step 5 by no. of values of data (N)

**Step 7:**Find the square root of the variance.

## Group and Ungroup Examples of Standard Deviation:

There are two types of data in the statistics group and ungroup. Here we discuss both data examples for standard deviation.

### Ungroup data:

This is an example of ungroup data for standard deviation.

**Example:**

Find the standard deviation of 2, 7, 11, 13, 17.

**Solution:**

**Step 1:**Collect data

2, 7, 11, 13, 17

**Step 2:**Find out mean

Mean = (2 + 7 + 11 + 13 + 17) / 5

Mean = 50 / 5

Mean = 10

**Step 3:**Squared sum of difference from mean

The sum of the squared difference from the mean = (2 – 10)^{2}+(7 – 10)^{2}+(11 – 10)^{2}+(13 – 10)^{2}+(17 – 10)^{2}

The sum of the squared difference from the mean = 64 + 9 + 1 + 9 + 49

The sum of the squared difference from the mean = 132

**Step 4:**Divided by no. of values

Variance = 132 / 5 – 1

Variance = 132 / 4

Variance = 33

**Step 5:**Square root of the variance

Standard deviation = √33

Standard deviation = 5.74

You can also use a standard deviation formula calculator to find the STD of the given ungrouped data values according to the formulas with steps.

### Group Data:

Following is the example of group data for standard deviation.

**Example:**

Find the standard deviation of the following data:

Class | 0 – 10 | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 |

Frequency | 2 | 6 | 11 | 16 | 9 |

**Solution:**

**Step 1:**Write data into tabular form

Class |
f |

0 – 10 | 2 |

10 – 20 | 6 |

20 – 30 | 11 |

30 – 40 | 16 |

40 – 50 | 9 |

Total |
44 |

**Step 2:**Calculate mean

Mean = 30.4

**Step 3:**Squared sum of difference from mean

Class |
f |
X |
X – X ̅ |
(X – X ̅)^{2} |
f(X – X ̅)^{2} |

0 – 10 | 2 | 5 | 5 – 30.4 | 645.16 | 1290.32 |

10 – 20 | 6 | 15 | 15 – 30.4 | 237.16 | 1422.96 |

20 – 30 | 11 | 25 | 25 – 30.4 | 29.16 | 320.76 |

30 – 40 | 16 | 35 | 35 – 30.4 | 21.16 | 338.56 |

40 – 50 | 9 | 45 | 45 – 30.4 | 213.16 | 1984.44 |

Total |
44 |
5357.04 |

**Step 4:**Write the formula of standard deviation for group data

σ = √(∑f(x – μ)^{2}) / ∑f

**Step 5:**Put values in the formula

σ = √5357.04 / 44

σ = 73.19 / 44

σ = 1.66

## Wrap up:

In the conclusion, we explored the history, definition, formulas, and applications of standard deviation in statistics. Introduced in the 18th century during the early stages of probability theory, Carl Friedrich Gauss and later Karl Pearson played key roles in shaping the modern concept.

The standard deviation is a crucial measure of data variability, with formulas for both population and sample data. We provided step-by-step procedures for calculating standard deviation and illustrated examples for ungrouped and grouped data. Understanding standard deviation is essential in various fields.