Variance Analysis Statistics
Introduction to variance analysis statistics:
Variance is calculated from the set of numbers is used in the statistics. It is a statistical data which defines the variance value. Variance is the variability of the data. Variance is calculated from the mean value of the given data set. Here we are going to see about the variance analysis of statistics in sample variance and population variance with the example problem and the practice problems related to the functions in the data set.
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Definition of Sample Variance Analysis:
Variance is the measure of summation of the total squared mean difference of deviation value which is divided by the total number of values subtracted by one. For calculating the variance we have to manipulate the mean value at first. Variance is calculated by using the mean value of the given data set.
Mean is the average of the given total numbers
Formula for measuring the sample mean is,
'barx = (sum_(K=1) ^n (x_k)) / N'
Where as
'barx ' is the variable for sample mean of the given data set.
'x_k' is the values that is given in the data set.
N is the total values in the given data set.
Formula for measuring the variance of a set of numbers is,
'S^2 = (sum_(K=1) ^n (x_k - barx)^2) /( N-1)'
Where as,
'S^2' is the variable for variance of a set of numbers.
Definition of Population Variance Analysis:
Population variance of the data set is defined as the square of the mean difference of the deviation value by the total number of data. Population variance is measured to find the variance in the probability of data.
Formula for finding the
'sigma^2 =( sum_(k=1)^n(x_k - mu)^2) / N'
Where as
'sigma^2' - symbol for population variance
'mu' - It is the mean of the given data.
N - Total number of values given in data set.
For finding the population variance we have to find the sample mean. For the take the average for the given values.
' mu = (sum_(K=1) ^n (x_k))/N'
This mean value is used in the population variance to find its value.
Variance Analysis Statistics - Example Problems:
Variance analysis statistics - Problem 1:
Find the sample variance of the given data set. 3, 4, 6, 5, 4, 5, 8.
Solution:
Mean:
Formula for measuring the mean value of the given data set is given by,
'barx = (sum_(k=1)^n (x_k)) / N'
=' (3+4+6+5+4+5+8) / 7'
= '35 / 7'
'barx' = 5
Sample Variance:
Formula for measuring the variance is given by,
'"" S^2 = (sum_(K=1)^n (x_k - barx)) /(N-1) '
= '((3-5)^2+(4-5)^2+(6-5)^2+(5-5)^2+(4-5)^2+(5-5)^2+(8-5)^2)/(7-1)'
= '16/6'
'S^2' = 2.66666667
Variance analysis statistics - Problem 2:
Measure the population variance for the given data set. 23, 24, 23, 22, 25, 26, 25
Solution:
Population Mean: Calculate the population mean for the given data set.
' mu = (sum_(K=1) ^n (x_k))/N'
using the above formula find the population mean for the given values.
' mu = (23+24+23+22+25+26+25)/7'
'mu = 168/ 7'
' mu' = 24
Population Variance: Measure the population variance value from the Population mean value.
'sigma^2 =( sum_(k=1)^n(x_k - mu)^2) / N'
substitute the population mean values to find the deviation values from the given values.
'sigma^2' '= ''((23-24)^2+(24-24)^2+(23-24)^2+(22-24)^2+(25-24)^2+(26-24)^2+(25-24)^2) / 7'
'sigma^2 = 12/7'
'sigma^2 = 1.7142857142857 '
Variance Analysis Statistics - Practice Problems:
Problem 1:
Calculate the sample variance for the given data set. 23,24,25,26,26,27
Answer: Sample Variance = 2.1666666666667
Problem 2:
Find the value for the population variance of the given data set. 87, 89, 78, 82, 89.
Answer: Population Variance = 18.8
by: Omkar Nayak
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