 # Difference Between Variance and Standard Deviation

Understanding the main difference between Variance and Standard deviation is important to know. These mathematical terms are usually used in normal mathematical equations to solve problems. Primarily variance and standard deviation are used as metrics to solve statistical problems. The standard deviation formula is used to measure the standard deviation of the given data values. It is important to understand the difference between variance, standard deviation, as they are both commonly used terms in the probability theory and statistics.  These two terms are used to determine the spread of the data set. Both the standard deviation and the variance are numerical measures, which calculates the spread of data from the mean value.

In short, the mean is the average of the range of given data values, a variance is used to measure how far the data values are dispersed from the mean, and the standard deviation is the used to calculate the amount of dispersion of the given data set values. We know that the measures of dispersion can be categorised into two different types, namely absolute measure of dispersion and the relative measure of dispersion. When we consider the variance and standard deviation, both fall under the absolute measure of dispersion. Before discussing the key difference between the variance and the standard deviation let’s discuss the definition of variance and the standard deviation here.

## Definition of Variance and Standard Deviation

Variance: Variance can simply be defined as a measure of variability to represent members of a group. The variance measures the closeness of data points corresponding to a greater value of variance.

Standard Deviation: Standard deviation, on the other hand, observes the quantifiable amount of dispersion of observations when approached with data. We must understand that variance and standard deviation differ from each other. Variances describe the variability of the observed observations while standard deviation measures the dispersion of observations within a set.

## What is the Difference Between Variance and Standard Deviation

Here, the list of comparative differences between the variance and the standard deviation is given below in detail:

Difference between Variance and Standard Deviation

Variance

Standard Deviation

It can simply be defined as the numerical value, which describes how variable the observations are. It can simply be defined as the observations that get measured are measured through dispersion within a data set.
Variance is nothing but the average taken out of the squared deviations. Standard Deviation is defined as the root of the mean square deviation
Variance is expressed in Squared units. Standard deviation is expressed in the same units of the data available.
It is mathematically denoted as (σ2) It is mathematically denoted as (σ)
Variance is a perfect indicator of the individuals spread out in a group. Standard deviation is the perfect indicator of the observations in a data set.

### Variance and Standard Deviation Problem

Example:

Find the mean, standard deviation and variance for the following data: 6, 7,10, 12, 13, 4, 8, 12.

Solution:

Given data: 6, 7,10, 12, 13, 4, 8, 12

Finding Mean:

We know that mean is the ratio of the sum of observations to the total number of observations.

(i.e) Mean = Sum of observations / Total number of observations.

Mean = (6+7+10+12+13+4+8+12)/8

Mean = 72/8

Mean = 9.

Finding Variance:

Variance =

$$\begin{array}{l}\frac{\sum (x_{i}-\bar{x})^{2}}{n}\end{array}$$

Here, n=8

$$\begin{array}{l}\sum (x_{i}-\bar{x})^{2}\end{array}$$
can be calculated as follows:

 $$\begin{array}{l}x_{i}\end{array}$$ $$\begin{array}{l}x_{i}-\bar{x}\end{array}$$ $$\begin{array}{l}(x_{i}-\bar{x})^{2}\end{array}$$ 6 6 – 9 = -3 9 7 7 – 9 = -2 4 10 10 – 9 = 1 1 12 12 – 9 = 3 9 13 13 – 9 = 4 16 4 4 – 9 = -5 25 8 8 – 9 = -1 1 12 12 – 9 = 3 9 $$\begin{array}{l}\sum(x_{i}-\bar{x})^{2}\end{array}$$ 74

Hence, Variance =

$$\begin{array}{l}\frac{\sum (x_{i}-\bar{x})^{2}}{n}\end{array}$$
= 74/8

Variance = 9.25

Finding Standard Deviation:

We know that variance is the square of standard deviation. Hence, the standard deviation can be found by taking the square root of variance.

Therefore, standard deviation = √variance

Standard deviation = √(9.25) = 3.041.

Hence, the mean, variance and standard deviation of the given data are 9, 9.25, 3.041 respectively.

Thus, these are the key differences between variance and standard deviation. To know more about Maths-related articles, register with BYJU’S – The Learning App today.

## Frequently Asked Questions on Difference Between Variance and Standard Deviation

Q1

### What does the variance and the standard deviation tell us?

In probability theory and statistics, both the variance and standard deviation tell us how far the data values are spread out/dispersed from the mean of the given data set

Q2

### How to derive the variance from the standard deviation?

The variance can be easily derived from the standard deviation by taking the square of the standard deviation.

Q3

### Mention the use of variance in statistics.

In statistics, the variance is used to determine the measure of dispersion and the uncertainty in the given data set values.

Q4

### What exactly is SD in statistics?

The standard deviation, SD is the number which gives information about the spread of data values from the mean value. If SD is small, the data values are close to the mean value. If SD is high, the data values are widely spread out from the mean value.

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