Hello friends! Today we shall share with you how you can calculate the standard deviation for ungrouped data. As you know, a standard deviation is an important concept used by statisticians, financial advisors, mathematicians, etc. to calculate the numerical variability of the population from the central tendencies like mean of the data.

In addition to finding out the variability of the points in a group from the mean of the data, it is also useful in proving the accuracy fo statistical conclusions. For example, in the case of analysis of polling data, the standard deviation can be used to calculate the estimated margin of error to determine how much the results of a sample population might vary from the whole population.

The standard deviation measures the variability of the statistical population, data set or a probability distribution and is the square root of its variance. So it can have various practical applications such as :

- With regard to polling data, it can be used to calculate the margin or error of the data, and calculate the exact value of sampling error, and hence determine how much closer or farther away the estimated results are from the true figures.
- in Finance, it is an important component to calculate the volatility in case of indices of assets, such as stocks, bonds, property, etc.

## Standard Deviation For Ungrouped Data

The standard deviation is represented by the symbol σ and can be calculated using the following formula :

It is expressed in the same units as the mean of the data. As you know, in statistics, data can be classified into two broad categories: grouped and ungrouped data. These data need to be handled differently mathematically and hence the procedure for calculating measures of central tendencies like standard deviation of ungrouped data is different. A data is said to be ungrouped if the observations are recorded randomly without grouping them into class intervals. For example, if we take the measurement of the height of students in a class and list them randomly, they would form an ungrouped data.

As students, you might find it confusing to understand the procedures to calculate the standard deviations for grouped and ungrouped data. You might not be able to understand some steps or might not arrive at the correct results. So to help you explain the procedure of how to calculate the standard deviation of ungrouped data, here we have provided you the step-by-step procedure of how you can find the standard deviation of any ungrouped data with frequency

For example, let us take the following data : 14,18, 12, 15,11, 19, 13, 22

Next, we shall find x-x₁ for each of the data points

x₁ | x₁- x’ | x₁= 15.5 |

11 | 11-15.5 | -4.5 |

12 | 12-15.5 | -3.5 |

13 | 13-15.5 | -2.5 |

14 | 14-15.5 | -1.5 |

15 | 15-15.5 | -0.5 |

18 | 18-15.5 | 2.5 |

19 | 19-15.5 | 3.5 |

22 | 22-15.5 | 6.5 |

Next, we shall find the squares of the values of x₁

x₁ | x₁- x’ | x₁= 15.5 | (x₁-15.5)² |

11 | 11-15.5 | -4.5 | 20.25 |

12 | 12-15.5 | -3.5 | 12.25 |

13 | 13-15.5 | -2.5 | 6.25 |

14 | 14-15.5 | -1.5 | 2.25 |

15 | 15-15.5 | -0.5 | 0.25 |

18 | 18-15.5 | 2.5 | 6.25 |

19 | 19-15.5 | 3.5 | 12.25 |

22 | 22-15.5 | 6.5 | 42.25 |

Now we shall find the mean of the squares calculated in the above step :

x₁ | x₁- x’ | x₁- 15.5 | (x₁-15.5)² |

11 | 11-15.5 | -4.5 | 20.25 |

12 | 12-15.5 | -3.5 | 12.25 |

13 | 13-15.5 | -2.5 | 6.25 |

14 | 14-15.5 | -1.5 | 2.25 |

15 | 15-15.5 | -0.5 | 0.25 |

18 | 18-15.5 | 2.5 | 6.25 |

19 | 19-15.5 | 3.5 | 12.25 |

22 | 22-15.5 | 6.5 | 42.25 |

∑(x₁- x’) = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 6.25 + 12.25 + 42.25 = 102

Var = ∑(x₁- x’)²/N

= 102/8 = 12.75

S.D. = √Var = √12.75

= 3.57

Hence using these steps, you will be able to find the standard deviation for any ungrouped data. So you will be able to find the standard deviation of a group and calculate the degree to which the observations are closely gathered or spread far away from the measures of central tendencies like mean.

To help you understand the concept of a standard deviation better and how the formula can be used in case of an ungrouped data, we have provided you a video explaining the same in easy steps :