The variance and the standard deviation are two different terms in mathematics and statistics that are used in various fields such as finance, accounting, economics, and investing. Both terms are the measure of the various groups of data values from the expected value.

Standard deviation and variance are helpful for determining the volatility and distribution of the data values. In this blog post, we are going to explore the key difference between the variance and the standard deviation along with the solved examples.

## What is the variance?

In statistical analysis, the variance is the measure of the squared differences from the mean. The evaluation of the variance is done by evaluating the difference between each data point of the given step of data and the expected value.

After finding it, take a square of the differences and take the average from the total data values. For assistance, if the group of the number isthe first ten even numbers then the averageis 11 after calculating the differences from the mean and finding the square the result is 330.

After that divide, the square of the sum by N – 1, and the result will be 36.67. Hence the variance of the first ten even numbers is 36.67.

## What is the standard deviation?

The term standard deviation is a statistical measurement that is helpful in determining how far a group of data values from the average. In simple words, the standard deviation is the measure of the data set from the mean to know how far apart numbers are in the given data set.

In other words, the standard deviation is the square root of the variance. Once you evaluate the variance of the given data set, you can easily find the standard deviation by taking the square root of the result of the variance.

After calculating the standard deviation, you can check the closeness and farness of the data values from the mean and also check the higher or lower deviation. If the result of the standard deviation is far away from the mean, then there is a higher deviation within the given data set.

If the result is closer to the mean, then there is a lower deviation within the given data set.

## What is the difference between the variance and the standard deviation?

The difference between variance and the standard deviation is:

Variance | Standard Deviation |

It is the measure of the data values from the mean to check how much data values vary from the average. | It measures how far the numbers are apart from the given set of data. |

It is measured in squared units. | It is not measured in squared units as it is the square root of the variance. |

It is less than 1. | It can be greater than the variance. When the variance is less than 1. |

It could be greater than 1.2 | It would be less than the variance. When the variance is greater than 1.2 |

For sample variance, its formula is: s^{2} = [∑ (z_{i} – z̄)^{2}/(N – 1)] | For sample standard deviation, its formula is: s = √ [∑ (x_{i} – x̄)^{2 }/ (N – 1)] |

For population variance, its formula is: σ^{2} = [∑ (z_{i} – μ)^{2}/N] | For population standard deviation, its formula is: σ = √ [∑ (x_{i} – µ)^{2}/ N] |

## Examples of variance and standard deviation

In this section, we’ll discuss examples of the variance and the standard deviation.

**Example 1: For Variance**

Evaluate the sample variance of the given sample data.

Sample data = 1, 9, 11, 18, 20, 22, 25, 28, 30, 36, 42

**Solution**

**Step 1:**First of all, evaluate the sample mean of the given sample data by adding the data values and dividing the sum by the total number of sample values.

Sample data = 1, 9, 11, 18, 20, 22, 25, 28, 30, 36, 42

Sum of sample data = 1 + 9 + 11 + 18 + 20 + 22 + 25 + 28 + 30 + 36 + 42

Sum = 242

Total number of observation = N = 11

Sample Mean = 242/11

Sample Mean = 22

**Step 2:**Now evaluate the difference of each data point from the above calculate average and take the square of each difference.

Data values | z_{i} – z̄ | (z_{i} – z̄)^{2} |

1 | 1 – 22= -21 | (-21)^{2} = 441 |

9 | 9 – 22 = -13 | (-13)^{2} = 169 |

11 | 11 – 22 = -11 | (-11)^{2} = 121 |

18 | 18 – 22 = -4 | (-4)^{2} = 16 |

20 | 20 – 22 = -2 | (-2)^{2} = 4 |

22 | 22 – 22 = 0 | (0)^{2} = 0 |

25 | 25 – 22 = 3 | (3)^{2} = 9 |

28 | 28 – 22 = 6 | (6)^{2} = 36 |

30 | 30 – 22 = 8 | (8)^{2} = 64 |

36 | 36 – 22 = 14 | (14)^{2} = 196 |

42 | 42 – 22 = 20 | (20)^{2} = 400 |

**Step 3:**Now evaluate the sum of the squared differences.

∑ (z_{i} – z̄)^{2} = 441 + 169 + 121 + 16 + 4 + 0 + 9 + 36 + 64 + 196 + 400

∑ (z_{i} – z̄)^{2} = 1456

**Step 4:**Now divide the above result by N – 1 to get the result of the sample variance.

∑ (z_{i} – z̄)^{2} / (n – 1) = 1456 / 11 – 1

∑ (z_{i} – z̄)^{2} / (n – 1) = 1456 / 10

∑ (z_{i} – z̄)^{2} / (n – 1) = 145.6

Hence the sample variance of the given sample data is 145.6

The variance of the given step of data values can also be evaluated with the help of an online variance calculators to get results in no time. Here is the above example solved by using the variance calculator by AllMath.

**Example 2: for STD**

Evaluate the sample standard deviation of the given sample data.

Sample data = 5,6, 7, 12,13, 14, 20, 21, 22, 30

**Solution**

**Step 1:**First of all, evaluate the sample mean of the given sample data by adding the data values and dividing the sum by the total number of sample values.

Sample data = 5, 6, 7, 12,13, 14, 20, 21, 22, 30

Sum of sample data = 5 + 6 + 7 + 12 + 13 + 14 + 20 + 21 + 22 + 30

Sum = 150

Total number of observation = N = 10

Sample Mean = 150/10

Sample Mean = 15

**Step 2:**Now evaluate the difference of each data point from the above calculate average and take the square of each difference.

Data values | z_{i} – z̄ | (z_{i} – z̄)^{2} |

5 | 5 – 15 = -10 | (-10)^{2} = 100 |

6 | 6 – 15 = -9 | (-9)^{2} = 81 |

7 | 7 – 15 = -8 | (-8)^{2} = 64 |

12 | 12 – 15 = -3 | (-3)^{2} = 9 |

13 | 13 – 15 = -2 | (-2)^{2} = 4 |

14 | 14 – 15 = -1 | (-1)^{2} = 1 |

20 | 20 – 15 = 5 | (5)^{2} = 25 |

21 | 21 – 15 = 6 | (6)^{2} = 36 |

22 | 22 – 15 = 7 | (7)^{2} = 49 |

30 | 30 – 15 = 15 | (15)^{2} = 225 |

**Step 3:**Now evaluate the sum of the squared differences.

∑ (z_{i} – z̄)^{2} = 100 + 81 + 64 + 9 + 4 + 1 + 25 + 36 + 49 + 225

∑ (z_{i} – z̄)^{2} = 594

**Step 4:**Now divide the above result by N – 1 to get the result of the sample variance.

∑ (z_{i} – z̄)^{2} / (n – 1) = 594 / 10 – 1

∑ (z_{i} – z̄)^{2} / (n – 1) = 594 / 9

∑ (z_{i} – z̄)^{2} / (n – 1) = 66

**Step 5:**Now evaluate the variance by taking the square root.

√ [∑ (z_{i} – z̄)^{2} / (n – 1)] = √66

√ [∑ (z_{i} – z̄)^{2} / (n – 1)] = 8.124

There is another way to evaluate the standard deviation that is by using online calculators. The online calculators will help you to get the result on just a single click. Let us solve the above example of STD with the help of standarddeviationcalculator.io

## Conclusion

Now you can get all the basic information about the variance and the standard deviation from this post. We have discussed all the basics of the variance and the standard deviation along with definitions, formulas, differences, and solved examples.