In probability distribution, the variance is frequently used to measure the spread of distance of data values. It is usually used to watch the relation of the data values with one another in the given set of distribution.
It is the best technique for finding the relation of the given data set rather than arranging the data values and finding the quartiles. In this article, we will learn the definition, formulas, and solved examples of the variance.
What is the variance?
In statistics, a measure of dispersion or measure of how far a set of numbers is spread out from the observations and the mean value. The square of the standard deviation is said to be the variance.
The observation of the variance is used to types of the data set. One is the sample data and the other is the population data. The measurement of the whole set of the data values is said to be the population set of data and the sample from the whole observation is the approximated value.
1. Population variance
In the probability distribution, the population variance is helpful in finding the spread of the total observation of the data set from the whole. In this kind of the variance, the deviation is to be calculated by taking the mean and after that find the take the quotient of the statistical sum of squares and the total number of the data values.
In simple words, the square of the population standard deviation gives the population variance. The process of the population variance is similar to the process of population standard deviation.
The formula for the population variance is:
Population variance = σ2 = ∑ (xi – μ)2/n
- In the above expression of population variance, the “σ2” notation is used to denote the population variance of the given data.
- Where “n” is the total number of observations in a given data set.
- xi is the population data set that is separated by commas.
- The “μ” is the population means of the variance in a given set of observations.
- (xi – μ)2 is the sum of squares of the deviation.
2. Sample variance
In the probability distribution, the sample variance is helpful in finding the spread of the sample observation of the data set taken from the whole. In this kind of the variance, the deviation is to be calculated by taking the mean and after that find the take the quotient of the statistical sum of squares and one decrease in the total number of the data values.
In simple words, the square of the sample standard deviation gives the sample variance. The process of the sample variance is similar to the process of sample standard deviation.
The formula for the sample variance is:
Sample variance = s2 = ∑ (xi – x̅)2/n – 1
- In the above expression of sample variance, the “s2” notation is used to denote the sample variance of the given data.
- Where “n” is the total number of observations in a given data set.
- xi is the sample data set that is separated by commas.
- The “x̅” is the sample mean of the variance in a given set of observations.
- (xi – x̅)2 is the sum of squares of the deviation.
How to find the variance?
The problems of the sample and population variance can be solved easily either by using their corresponding formulas or a variance calculator. Follow the steps below to calculate the problems of sample and population variance manually.
- In the initial step, you have to determine the sample or population mean of the data observations (x̅ or μ) by dividing the sum of data values and the total number of data values.
- Once you determine the mean of the sample or population data, then determine the deviation of the mean of the sample and population from the data values one by one. The process of the deviation is also said to be the subtraction of observations of the data set from the mean. After that find the squares of each difference term to make all the values positive.
- Determine the sum of squares of the deviation terms.
- After that, take the quotient of the statistical sum of squares by total numbers “n” for population variance or by n – 1 for sample variance.
Example I: For sample variance
Evaluate the sample variance of 4, 7, 8, 11, 12, 17, 19, 23, 34, 35.
Solution
Step-1: First of all, find the sample mean of variance of the comma separated observations of sample data.
Sum of sample values = 4 + 7 + 8 + 11 + 12 + 17 + 19 + 23 + 34 + 35
= 170
Total number of observation = n = 10
Sample mean of data set = x̅ = 170/10 = 85/5
= 17
Step-2: Now subtract the sample mean from all the given observations individually.
x1 – x̅ = 4 – 17 = -13
x2 – x̅ = 7 – 17 = -10
x3 – x̅ = 8 – 17 = -9
x4 – x̅ = 11 – 17 = -6
x5 – x̅ = 12 – 17 = -5
x6 – x̅ = 17 – 17 = 0
x7 – x̅ = 19 – 17 = 2
x8 – x̅ = 23 – 17 = 6
x9 – x̅ = 34 – 17 = 17
x10 – x̅ = 35 – 17 = 18
Step-3: Determine the squares of the deviations to make all the entries positive.
(x1 – x̅)2 = (-13)2 = 169
(x2 – x̅)2 = (-10)2 = 100
(x3 – x̅)2 = (-9)2 = 81
(x4 – x̅)2 = (-6)2 = 36
(x5 – x̅)2 = (-5)2 = 25
(x6 – x̅)2 = (0)2 = 0
(x7 – x̅)2 = (2)2 = 4
(x8 – x̅)2 = (6)2 = 36
(x9 – x̅)2 = (17)2 = 289
(x9 – x̅)2 = (18)2 = 324
Step-4: Now calculate the summation of the statistical sum of squares.
∑ (xi – x̅)2 = 169 + 100 + 81 + 36 + 25 + 0 + 4 + 36 + 289 + 324
= 1064
Step-5: Now divide the above summation value by n – 1.
∑ (xi – x̅)2 / n – 1 = 1064 / 10 – 1
= 1064 / 9
= 118.22
To avoid such lenthy steps to find the sample variance, you can use a variance calculator with steps to get the result in a fraction of seconds
Example II: For population variance
Calculate the population variance of 1, 5, 11, 14, 18, 27, 35, 38, 40, 41.
Solution
Step-1: First of all, find the population mean of variance of the comma separated observations of sample data.
Sum of population values = 1 + 5 + 11 + 14 + 18 + 27 + 35 + 38 + 40 + 41
= 230
Total number of observation = n = 10
Mean of population data set = μ = 230/10 = 46/2
= 23
Step-2: Now subtract the sample mean from all the given observations individually.
x1 – μ = 1 – 23 = -22
x2 – μ = 5 – 23 = -18
x3 – μ = 11 – 23 = -12
x4 – μ = 14 – 23 = -9
x5 – μ = 18 – 23 = -5
x6 – μ = 27 – 23 = 4
x7 – μ = 35 – 23 = 12
x8 – μ = 38 – 23 = 15
x9 – μ = 40 – 23 = 17
x10 – μ = 41 – 23 = 18
Step-3: Determine the squares of the deviations to make all the entries positive.
(x1 – μ)2 = (-22)2 = 484
(x2 – μ)2 = (-18)2 = 324
(x3 – μ)2 = (-12)2 = 144
(x4 – μ)2 = (-9)2 = 81
(x5 – μ)2 = (-5)2 = 25
(x6 – μ)2 = (4)2 = 16
(x7 – μ)2 = (12)2 = 144
(x8 – μ)2 = (15)2 = 225
(x9 – μ)2 = (17)2 = 289
(x10 – μ)2 = (18)2 = 324
Step-4: Now calculate the summation of the statistical sum of squares.
∑ (xi – μ)2 = 484 + 324 + 144 + 81 + 25 + 16 + 144 + 225 + 289 + 324
= 2056
Step-5: Now divide the above summation value by n.
∑ (xi – μ)2 / n = 992 / 10
= 496 / 5
= 99.2
Summary
In this article, we have learned all the basics of variance like definition, working, types, formulas, and solved examples. Now after reading the above post, you can solve any problem of sample and population variance easily.