Difference Between Variance and Standard Deviation (original) (raw)
Last Updated : 4 Nov, 2025
Variance and Standard deviation are both widely used in mathematics to solve statistical problems. They provide various ways to extract information from the group of data.
They are also used in probability theory and other branches of mathematics.
**Standard Deviation
Standard deviation is an important method of measuring statistical deviation.
- It is the measure of the extent to which the numbers in a statistical series are spread from their arithmetic mean.
- It is always a non-negative value, and its unit is the same as that of the items in the given series.
- The same unit among both makes comparison and interpretation easier and more detailed.
- It is a common tool used to measure central tendency.
Formula of Standard Deviation
There are two formulas for Standard Deviation:
**Variance
Variance is the squared deviation of items/values in a statistical series from its arithmetic mean.
- This numerical value quantifies the average magnitude to which the data set is dispersed around its mean.
- It is simply the square of the standard deviation and is denoted as σ2.
- Its unit is different from that of the individual items in the statistical series and, hence, it is less suitable for direct comparison as a measure of dispersion.
Formula for Variance
There are two formula for Variance:
**Variance vs Standard Deviation
The basic difference between Variance and Standard Deviation is discussed in the table below,
| Standard Deviation | Variance |
|---|---|
| It is the measure of the dispersion of values in a given data set relative to their mean. | It is the statistical measure of how far the numbers are spread in a data set from their average. |
| It measures the absolute variability of the dispersion. | It helps determine the size of the data spread. |
| It is calculated by taking the square root of the variance. | It is calculated by taking the average of the squared deviations of each value from the average. |
| It is primarily used as a measure of market and security volatility in finance. | It is one of the key aspects of asset allocation in investing portfolios. |
**Also Check:
**Solved Examples on Variance and Standard Deviation
**Example 1: Find the variance and standard deviation of the following data
| **5 | **8 | **3 | **6 | **7 | **12 | **5 | **2 |
|---|
**Solution:
**X **x = X - X̄ **x 2 5 -1 1 8 2 4 3 -3 9 6 0 0 7 1 1 12 6 36 5 -1 1 2 -4 16 **Σx = 48 **Σx 2 = 68 Arithmetic Mean = Σx/N = 48/8 = 6
Population Variance = \Sigma\dfrac{(X_i-\bar{X})^2}{N}\\=\frac{68}{8}
**Population Variance = 8.5
**Standard Deviation = √8.5 = 2.91
**Example 2: Find the standard deviation of first n natural numbers.
**Solution:
Mean of first n natural numbers = Sum of first n natural numbers/n
= [n(n+1)/2/]n
= (n+1)/2
Sum of squares of first n natural numbers = \sum x^2=\frac{n(n+1)(2n+1)}{6}
Now, standard deviation = σ=\sqrt{\frac{\sum x^2} N-(\frac{\sum x} N)^2}\\=\sqrt{\frac{n(n+1)(2n+1)}{6}-(\frac{n+1}2)^2}\\=\sqrt{\frac{n^2-1}{12}}
Thus the standard deviation of first n natural numbers is \sqrt{\frac{n^2-1}{12}} .
**Example 3: Find the standard deviation of the following data:
**Solution:
**X **f **fX **x = X - X̄ **x 2 **fx 2 5 6 30 -7 49 294 10 7 70 -2 4 28 15 3 45 3 9 27 20 2 40 8 64 128 25 1 25 13 169 169 30 1 30 18 324 324 **N = 20 **ΣfX = 240 **Σfx 2 = 970 Arithmetic Mean = ΣfX/N = 240/20 = 12
Population Variance = \Sigma\dfrac{(X_i-\bar{X})^2}{N} = 970/20
Population Variance = 48.5
**Thus, Standard Deviation = √48.5 = 6.96
**Example 4: Find the variance of the variable Z, where Z represents the sum of all observations upon rolling a pair of dice.
**Solution:
There are total 36 observations when a pair of dice is rolled. The probabilities of getting different sums upon rolling a pair of dice are:
P(Z=2) = 1/36
P(Z=3) = 2/36 = 1/18
P(Z=4) = 3/36 = 1/12
P(Z=5) = 4/36 = 1/9
P(Z=6) = 5/36
P(Z=7) = 6/36 = 1/6
P(Z=8) = 5/36
P(Z=9) = 1/9
P(Z=10) = 1/12
P(Z=11) = 1/18
P(Z=12) = 1/36
Since, E(Z)=∑Zi . P(Zi) = (1/18) + (1/6) + (1/3) + (5/9) + (5/6) + (7/6) + (10/9) + 1 + (5/6) + (11/18) + (1/3)
Thus, E(Z) = 7
And, E(Z2) = 54.833
Thus,
Var(Z) = E(Z2) - E(Z)2
= 54.833 – 49
**Var(Z) = 5.833
**Example 5: Calculate variance for the following grouped data
| **Intervals | **20 - 25 | **25 - 30 | **30 - 35 | **35 - 40 | **40 - 45 | **45 - 50 |
|---|---|---|---|---|---|---|
| **Frequency | **170 | **110 | **80 | **45 | **40 | **35 |
**Solution:
**X **f **M **d **d 2 **fd **fd 2 20 - 25 170 22.5 -2 4 -340 680 25 - 30 110 27.5 -1 1 -110 110 30 - 35 80 32.5 0 0 0 0 35 - 40 45 37.5 1 1 45 45 40 - 45 40 42.5 2 4 80 160 45 - 50 35 47.5 3 9 105 315 **N = 480 **Σfd = -220 **Σfd 2 = 1340 Variance = \Sigma\dfrac{f(M-\bar{X})^2}{N}
**Population Variance = 62.98
**Example 6: Find the variance of the first 69 natural numbers.
Solution:
Standard deviation of first n natural numbers is \sqrt{\frac{n^2-1}{12}} .
Thus, Variance = {n2-1}/12
Here, n = 69
Thus,
Var = {692-1}/12
**Var = 396.67
Practice Problems
**Question 1 : Find the variance and standard deviation of all the possibilities of rolling a die.
**Question 2 : Find the variance and standard deviation of all the even numbers less than 10.
**Question 3 : Find the standard deviation for the data: 42, 38, 35, 26, 45, 52, 48.
**Answer 1 : Mean, x̅ = 3.5
Variance (σ²) = 2.917
Standard Deviation (σ) = √(2.917) = 1.708
**Answer 2 : Mean, x̅ = 4
Variance (σ²) = 8
Standard Deviation (σ) = √(8) = 2.828
**Answer 3 : Mean, x̅ = 40.85
Standard Deviation (σ) = 8.07