Variance and Standard Deviation (original) (raw)

Last Updated : 5 Dec, 2025

Variance and Standard Deviation are the important measures used in Mathematics and statistics to find meaning from a large set of data. The different formulas for Variance and Standard Deviation are highly used in mathematics to determine the trends of various values in mathematics. Variance is the measure of how the data points vary according to the mean, while standard deviation is the measure of the central tendency of the distribution of the data.

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Variance and Standard Deviation

The major difference between variance and standard deviation is in their units of measurement. Standard deviation is measured in a unit similar to the units of the mean of data, whereas the variance is measured in squared units.

Variance

Variance is defined as "The measure of how far the set of data is dispersed from their mean value". Variance is represented with the symbol σ2. In other words, we can also say that the variance is the average of the squared difference from the mean.

Properties of Variance

Various properties of the Variance of the group of data are,

Variance Formula

There are two formulas for Variance, that are:

Formula for Population Variance

The mathematical formula to find the variance of the given data is,

\bold{\sigma^2 = \sum_{i=1}^n{\frac{(x_i - \bar{x})^2}{n}}}

Where,

This formula is also called the Population variance formula as it is used for finding the variation in the population data.

Formula for Sample Variance

Also, The other formula for finding the variance is the sample variance formula, which is discussed in the image.

Sample-Variance

Formula for Sample Variance

Standard Deviation

How far our given set of data varies from. the mean of the data is measured in standard deviation. Thus, we define standard deviation as the "spread of the statistical data from the mean or average position". We denote the standard deviation of the data using the symbol σ.

We can also define the standard deviation as the square root of the variance.

Properties of Standard Deviation

Various properties of the Variance of the group of data are,

Standard Deviation Formula

There are two formulas for the standard deviation listed as follows:

Formula for Population Standard Deviation

The mathematical formula to find the standard deviation of the given data is,

s = \sqrt{\frac{\sum_{i=1}^n (x_i - x̄)^2}{n}}

Where,

This formula is also called the Population standard deviation formula as it is used for finding the standard deviation in the population data.

Formula for Sample Standard Deviation

Sample variance formula is discussed in the image below,

Sample-Standard-Deviation

Formula for Sample Standard Deviation

**Learn more about, **Standard Deviation Formula

Relation between Standard Deviation and Variance

Variance and Standard deviation are the most common measure of the given set of data. They are used to find the deviation of the values from their mean value or the spread of all the values of the data set.

The relationship between Variance and Standard Deviation is discussed below.

Variance = (Standard Deviation)2
OR
√(Variance) = Standard Deviation

Variance vs Standard deviation

The differences between Standard Deviation and Variance are discussed in the table below.

Standard Deviation Variance
Standard Deviation is defined as the square root of the variance. Variance is defined as the average of the squared differences from the mean.
Standard deviation provides a measure of the typical distance between data points and the mean. Variance provides a measure of the average squared distance between data points and the mean.
It is represented by the Greek symbol σ. It is represented by a square of the Greek symbol sigma. i.e. σ2.
It has the same unit as the data set. Its unit is the square of the unit of the data set.
It represents the volatility in the market or given data set. It represents the degree to which the average return varies according to the long-term change in the market.

Solved Questions on Variance and Standard Deviation

**Question 1: Find the variance and standard deviation of all the possibilities of rolling a die.

**Solution:

All possible outcomes of rolling a die are {1; 2; 3; 4; 5; 6}.

This data set has six values (n) = 6

Before finding the variance, we need to find the mean of the data set.

Mean, x̅ = (1+2+3+4+5+6)/6 = 3.5

We can put the value of data and mean in the formula to get;

σ2 = Σ (xi – x̅)2/n
⇒ σ2 = [(1-3.5)2 + (2-3.5)2 + (3-3.5)2 + (4-3.5)2 + (5-3.5)2 + (6-3.5)2]/6
⇒ σ2 = (6.25+2.25+0.25+0.25+2.25+6.25)/6

Variance (σ2) = 2.917

Now,

Standard Deviation (σ) = √ (σ2)
⇒ Standard Deviation (σ) = √(2.917)
⇒ Standard Deviation (σ) = 1.708

**Question 2: Find the variance and standard deviation of all the even numbers less than 10. , Find the variance and standard deviation of all the even numbers less than 10.

Even Numbers less than 10 are {2, 4, 6, 8}

This data set has five values (n) = 4

Before finding the variance, we need to find the mean of the data set.

Mean, x̅ = (2+4+6+8)/4= 5

We can put the value of data and mean in the formula to get;

σ2 = Σ (xi – x̅)2/n
⇒ σ2 = [(2-5)2 + (4-5)2 + (6-5)2 + (8-5)2]/4
⇒ σ2 = (9 + 1 + 1 + 9 )/5 = 20/4

Variance (σ2) = 5

Now, Standard Deviation (σ) = √ (σ2)
⇒ Standard Deviation (σ) = √(5)
⇒ Standard Deviation (σ) = 2.238

Practice Problems - Variance and Standard Deviation

Try solving the given question to grab the idea of what we just learned :

**Question 1: Find the variance and standard deviation of the following data

5 8 3 4 7 12 5 2

**Question 2: Find the standard deviation of the following data:

X 5 10 15 20 25 30
f(x) 6 7 3 2 1 1

**Question 3: Find the variance of the first 69 natural numbers.

**Answer 1 : Mean x̄ = 6

**Answer 2 : Mean x̄ = 12

**Answer 3 : Variance = 396.67

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