Question

Find the variance of the following data:
12. 22. 28. 24. 21. 16. 31, 14
41.54
45.63
25.33
39.25​

Answer 1

Answer:

39.25Step-by-step explanation:

Answer 2

The variance of the following data is 39.25

Step-by-step explanation:

Given:

$x_{i}$= 12, 22, 28, 24, 21, 16, 31, 14

To find:

variance($\sigma^2$)

Formula:

$(\sigma^2)=\sum \frac{(x_{i}-\bar x)^2}{N}$

Solution:

Let's calculate the mean of the observations first,N=8

Mean($\bar x$)= sum of all observations/number of observations

∴ $\bar x$= 12+22+28+24+21+16+31+14/8

     =168/8

∴ mean($\bar x$)=21

Next we will calculate $x_{i}-\bar x$ for all $x_{i}$ values

$x_{i}-\bar x= 12-21 =-9\\\\x_{i}-\bar x=22-21=1\\\\x_{i}-\bar x=28-21=7\\\\x_{i}-\bar x=24-21=3\\\\x_{i}-\bar x=21-21=0\\\\x_{i}-\bar x=16-21=-5\\\\x_{i}-\bar x=31-21=10\\\\x_{i}-\bar x=14-21=-7$

The sum of $(x_{i}-\bar x)$ will be equal to

-9+1+7+3+0+(-5)+10+(-7)

∴ ∑$(x_{i}-\bar x)$=0

Let's square the $(x_{i}-\bar x)$ values

∴ $(x_{i}-\bar x)^2$=(-9)²=81

                =(1)²=1

                =(7)²=49

                =(3)²=9

                =(0)²=0

                =(-5)²=25

                =(10)²=100

                =(-7)²=49

Next we sum all the $(x_{i}-\bar x)^2$ values

∴ ∑$(x_{i}-\bar x)^2$= 81+1+49+9+0+25+100+49

∴ ∑ $(x_{i}-\bar x)^2$= 314

Putting all the values in the formula

$(\sigma^2)=\sum \frac{(x_{i}-\bar x)^2}{N}$

$(\sigma^2)=\frac{314}{8}$

$(\sigma^2)=39.25$

∴ The variance is 39.25