Definition to Population Variance

Population variance is defined as the square of the mean deviation value by the total number of data. Population variance is calculated 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.

Population Variance Values - Example Problems:

Population Variance Values - Problem 1:

calculate the population variance for the given data set. 2, 4, 3, 3, 5, 6, 5

Solution:

Mean:  Calculate the mean for the given data

 mu = (sum_(K=1) ^n (x_k))/N

using the above formula find the average for the given values.

 mu = (2+4+3+3+5+6+5)/7

mu = 28/ 7

 mu = 4

Population Variance: Calculate the population variance value from the mean.

sigma^2 =( sum_(k=1)^n(x_k - mu)^2) / N

substitute the mean values to find the deviation values from the given values.

 sigma^2 = ((2-4)^2+(4-4)^2+(3-4)^2+(3-4)^2+(5-4)^2+(6-4)^2+(5-4)^2)/7

sigma^2 = 12/7

sigma^2 = 1.7142857142857

Hence the population variance value is calculated using the mean value.

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Population Variance Values - Problem 2:

Find the value for the population variance of the given data set. 367, 378, 365, 366.

Solution:

Mean: Calculate the mean for the given data set using the formula,

 mu = (sum_(K=1) ^n (x_k))/N

Find the mean value that is average of the giave data set

mu = (367+378+365+366)/4

mu = 1476 / 4

mu = 369

Population variance: Calculate the population variance value from the mean.

sigma^2 =( sum_(k=1)^n(x_k - mu)^2) / N

substitute the mean values to find the deviation values from the given values.

sigma^2 =((367-369)^2 + (378-369)^2 + (365 - 369)^2+(366-369)^2) / 4

sigma^2 = 110/4

sigma^2 = 27.5

Hence the population variance value is calculated using the mean value.

Population Variance Values - Problem 3:

Find the value for the population variance of the given data set. 36, 37, 36, 39.

Solution:

Mean: Calculate the mean for the given data set using the formula,

 mu = (sum_(K=1) ^n (x_k))/N

Find the mean value that is average of the giave data set

mu = (36+37+36+39) / 4

mu = 148 / 4

mu = 37

My previous blog post was on Natural Numbers please express your views on the post by commenting.

Population variance: Calculate the population variance value from the mean.

sigma^2 =( sum_(k=1)^n(x_k - mu)^2) / N

substitute the mean values to find the deviation values from the given values.

"sigma^2 = ((36-37)^2 +(37-37)^2+(36-37)^2+(39-37)^2) / 4

sigma^2 = 6/4

sigma^2 = 1.5

Hence the population variance value is calculated using the mean value.

Population Variance Values - Practice Problems:

Find the value for the population variance of the given data set. 45, 47, 49, 43.

Answer: Population Variance = 5

Find the value for the population variance of the given data set. 87, 89, 78, 82, 89.

Answer: Population Variance = 18.8