To calculate the variance of the given data set of students' grades, follow these detailed steps:
1. List the Grades:
The given grades are: 62, 77, 78, 80, 82, 82, 83, 84, 85, 87, 89, 95
2. Calculate the Mean [tex]\(\bar{x}\)[/tex]:
The mean ([tex]\(\bar{x}\)[/tex]) is calculated by summing all the grades and then dividing by the number of grades.
[tex]\[
\bar{x} = \frac{62 + 77 + 78 + 80 + 82 + 82 + 83 + 84 + 85 + 87 + 89 + 95}{12}
\][/tex]
Simplifying the sum:
[tex]\[
\sum_{i=1}^{12} x_i = 984
\][/tex]
Number of grades [tex]\( n = 12 \)[/tex].
[tex]\[
\bar{x} = \frac{984}{12} = 82
\][/tex]
3. Calculate Each Deviation from the Mean:
For each grade [tex]\( x_i \)[/tex], compute [tex]\( x_i - \bar{x} \)[/tex]:
[tex]\[
62 - 82, \, 77 - 82, \, 78 - 82, \, 80 - 82, \, 82 - 82, \, 82 - 82, \, 83 - 82, \, 84 - 82, \, 85 - 82, \, 87 - 82, \, 89 - 82, \, 95 - 82
\][/tex]
This results in:
[tex]\[
-20, -5, -4, -2, 0, 0, 1, 2, 3, 5, 7, 13
\][/tex]
4. Square Each Deviation:
[tex]\[
(-20)^2, \, (-5)^2, \, (-4)^2, \, (-2)^2, \, 0^2, \, 0^2, \, 1^2, \, 2^2, \, 3^2, \, 5^2, \, 7^2, \, 13^2
\][/tex]
This results in:
[tex]\[
400, \, 25, \, 16, \, 4, \, 0, \, 0, \, 1, \, 4, \, 9, \, 25, \, 49, \, 169
\][/tex]
5. Sum the Squared Deviations:
[tex]\[
400 + 25 + 16 + 4 + 0 + 0 + 1 + 4 + 9 + 25 + 49 + 169 = 702
\][/tex]
6. Calculate the Variance [tex]\(s^2\)[/tex]:
Using the sample variance formula [tex]\( s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}\)[/tex]:
[tex]\[
s^2 = \frac{702}{12 - 1} = \frac{702}{11} = 63.81818181818182
\][/tex]
So, the variance [tex]\( s^2 \)[/tex] of the students' grades is approximately 63.82.
