The standard deviation is the statistic that measures the dispersion of some dataset relative to its mean value. It is also termed as the square root of the variance. It is computed as the square root of the variance by determining the variation between each data point with respect to the mean. If the data points are different from the mean, then there is a higher deviation within the data set. Therefore, the more spread out the data, the higher the standard deviation. In this topic, we will discuss the Standard deviation formula with examples. Let us learn it!

Source: en.wikipedia.org

**Standard Deviation Formula**

**What is Standard Deviation?**

It is used under statistics to find the values of some particular data that is dispersed. In other words, the standard deviation is defined as the deviation of the values or data from their average mean. Lower standard deviation will conclude that the values are very close around their average.

On the other hand, higher values mean the values are far from the mean value. It should be noted here that the standard deviation value can never be negative.

**Formulas for Standard Deviation:**

- Population Standard Deviation Formula:

**\(\sigma= \sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}{N}}\)**

- Sample Standard Deviation Formula:

**\(\sigma= \sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}{N-1}}\)**

Where,

\(\sigma\) | Standard Deviation |

\(x_i\) | ith terms Given in the Data |

overline{x} | Mean |

N | Total number of Terms |

\(f_i\) | ith frequency |

- Standard Deviation Formula Based on Discrete Frequency Distribution:

For discrete frequency distribution of the type:

\(x= x_1, x_2, x_3,….,x_n\) and

\(f = f_1, f_2, f_3,….,f_n\)

**\(\sigma= \sqrt{\frac{1}{N}{\sum_{i=1}^{n}f_{i}\left(x_{i}-\bar{x}\right)^{2}}} \)**

Here, N is given as:

**N = \(\sum_{i=1}^{n}f_i \)**

**Standard deviation is calculated as:**

- The mean value is calculated by adding all the data points and dividing by the number of data points.
- The variance for each data point is calculated, first by subtracting the value of the data point from the mean. Each of these will result in the value, is then squared and the results finally summed. After this, the result is then divided by the number of data points less one.
- The square root of the variance—result from no. 2—is then taken to find the standard deviation.

**Standard Deviation vs. Variance:**

Variance is computed by taking the mean of the data points, then subtracting the mean from each data point individually. After this, squaring each of these results and then taking another mean of these squares. Standard deviation is the square root of the value of variance.

**Solved Examples**

**Example: Find the standard deviation of the numbers given (3, 8, 6, 10, 12, 9, 11, 10, 12, 7).**

Solution:

Step 1: First compute the mean of the 10 values given.

\(\bar X = \frac{3+8+6+10+12+9+11+10+12+7}{10}\)

= \(\frac{88}{10}\)

= 8.8

Step 2: Make a table as following with three columns, one for the X values, the second for the deviations and the third for squared deviations.

Value (X) | \(X – \bar X\) | \((X-\bar X)^2 \) |

3 | -5.8 | 33.64 |

8 | -0.8 | 0.64 |

6 | -2.8 | 7.84 |

10 | 1.2 | 1.44 |

12 | 3.2 | 10.24 |

9 | 0.2 | 0.04 |

11 | 2.2 | 4.84 |

10 | 1.2 | 1.44 |

12 | 3.2 | 10.24 |

7 | -1.8 | 3.24 |

Total | 0 | 73.6 |

Step 3:

As the data is not given as sample data, thus we use the formula for population variance.

= \(\frac{73.6} {10}\)

= 7.36

Thus standard Deviation = \(\sqrt{7.36}\)

Therefore, standard deviation = 2.71

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