Difference Between Variance and Standard Deviation

Variance and Standard deviation both formulas are widely used in mathematics to solve statistics 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. So it is important to distinguish between them. Let’s learn about Standard Deviation, Variance, and their difference in detail in this article.

What is 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 and is denoted as σ and is given as:

S.D.(x) = σ =[Tex]\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(X_{i}-\mu\right)^{2}}[/Tex]

What is 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 extent the data set is dispersed around itself. It is simply the square of the standard deviation and is denoted as σ². Since it is the squared deviation, its unit is different from that of the individual items in the statistical series and is, hence, not a reliable measure of dispersion because of inaccurate and vague comparison. Variance is calculated as

Var(x) = [Tex]\sum \frac{f\left ( M_{i}-\overline{X} \right )^{2}}{N}[/Tex]

Difference Between Variance and 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.

Do Check,

• Standard Deviation Formula
• Variance and Standard Deviation

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	x2
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		Σx2 = 68

Arithmetic Mean = Σx/N = 48/8 = 6

Population Variance = [Tex]\Sigma\dfrac{(X_i-\bar{X})^2}{N}\\=\frac{68}{8}    [/Tex]

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 = [Tex]\sum x^2=\frac{n(n+1)(2n+1)}{6}[/Tex]

Now, standard deviation = σ=[Tex]\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}}[/Tex]

Thus the standard deviation of first n natural numbers is [Tex]\sqrt{\frac{n^2-1}{12}}   [/Tex].

Example 3: Find the standard deviation of the following data:

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

Solution:

X	f	fX	x = X – X̄	x2	fx2
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			Σfx2 = 970

Arithmetic Mean = ΣfX/N = 240/20 = 12

Population Variance = [Tex]\Sigma\dfrac{(X_i-\bar{X})^2}{N}    [/Tex] = 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)=∑Z_i . P(Z_i)  = (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	d2	fd	fd2
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	Σfd2 = 1340

Variance = [Tex]\Sigma\dfrac{f(M-\bar{X})^2}{N}[/Tex]

Population Variance = 62.98

Example 6: Find the variance of the first 69 natural numbers.

Solution:

Standard deviation of first n natural numbers is [Tex]\sqrt{\frac{n^2-1}{12}}   [/Tex].

Thus, Variance = {n²-1}/12

Here, n = 69

Thus, 

Var = {69²-1}/12

Var = 396.67

FAQs on Variance and Standard Deviation

Q1: What do you mean by variance?

Answer:

Variance is the squared deviation of items/values in a statistical series from its arithmetic mean. Higher value of variance means higher deviation of the observations from their arithmetic mean and vice- versa.

Q2: How does the variance differ from the standard deviation?

Answer:

Variance is the squared deviation of items/values in a statistical series from its arithmetic mean, whereas standard deviation is the square root of the variance.

Q3: Is variance a better statistical tool than the standard deviation?

Answer:

Variance is a much better tool mathematically to express deviation in data because of its many properties.

Q4: What is the formula to find variance for ungrouped data?

Answer:

The formula to find variance for ungrouped data is

Var(X) = [Tex]\Sigma\dfrac{(X_i-\bar{X})^2}{N-1}[/Tex]

Q5: What is the symbol of variance?

Answer:

Variance is denoted by symbols: σ², s², or Var(X).

Q6: Explain the relationship between variance and standard deviation.

Answer:

The relationship between variance and standard deviation is given by,

Variance = (Standard Deviation)²