To find the variance and standard deviation of Adimas' grades, let's follow these steps:
1. Calculate the squared differences between each grade and the mean:
The grades are:
[tex]\( 76, 87, 65, 88, 67, 84, 77, 82, 91, 85, 90 \)[/tex]
The mean ([tex]\(\bar{x}\)[/tex]) is 81.
Calculate the squared difference for each grade:
[tex]\[
(76 - 81)^2 = (-5)^2 = 25
\][/tex]
[tex]\[
(87 - 81)^2 = 6^2 = 36
\][/tex]
[tex]\[
(65 - 81)^2 = (-16)^2 = 256
\][/tex]
[tex]\[
(88 - 81)^2 = 7^2 = 49
\][/tex]
[tex]\[
(67 - 81)^2 = (-14)^2 = 196
\][/tex]
[tex]\[
(84 - 81)^2 = 3^2 = 9
\][/tex]
[tex]\[
(77 - 81)^2 = (-4)^2 = 16
\][/tex]
[tex]\[
(82 - 81)^2 = 1^2 = 1
\][/tex]
[tex]\[
(91 - 81)^2 = 10^2 = 100
\][/tex]
[tex]\[
(85 - 81)^2 = 4^2 = 16
\][/tex]
[tex]\[
(90 - 81)^2 = 9^2 = 81
\][/tex]
2. Sum the squared differences:
[tex]\[
25 + 36 + 256 + 49 + 196 + 9 + 16 + 1 + 100 + 16 + 81 = 785
\][/tex]
3. Calculate the variance ([tex]\(\sigma^2\)[/tex]):
The variance is calculated by averaging these squared differences. Since there are 11 grades, we divide the sum of the squared differences by 11:
[tex]\[
\sigma^2 = \frac{785}{11} \approx 71.36
\][/tex]
So the variance, rounded to the nearest hundredth, is:
[tex]\[
\sigma^2 = 71.36
\][/tex]
4. Calculate the standard deviation ([tex]\(\sigma\)[/tex]):
The standard deviation is the square root of the variance:
[tex]\[
\sigma = \sqrt{71.36} \approx 8.45
\][/tex]
So the standard deviation, rounded to the nearest hundredth, is:
[tex]\[
\sigma = 8.45
\][/tex]
In summary:
- The variance of Adimas' grades ([tex]\(\sigma^2\)[/tex]), rounded to the nearest hundredth, is:
[tex]\[
\sigma^2 = 71.36
\][/tex]
- The standard deviation of Adimas' grades ([tex]\(\sigma\)[/tex]), rounded to the nearest hundredth, is:
[tex]\[
\sigma = 8.45
\][/tex]
