Let's analyze the formulas and apply the given data to find the necessary statistical measures.
Given data: [tex]\( 62, 77, 78, 80, 82, 82, 83, 84, 85, 87, 89, 95 \)[/tex]
1. Number of observations (n):
The total number of grades is [tex]\( n = 12 \)[/tex].
2. Mean (average) of the grades:
The mean [tex]\(\bar{x}\)[/tex] is calculated as:
[tex]\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
\][/tex]
Adding up the grades:
[tex]\[
62 + 77 + 78 + 80 + 82 + 82 + 83 + 84 + 85 + 87 + 89 + 95 = 984
\][/tex]
So,
[tex]\[
\bar{x} = \frac{984}{12} = 82.0
\][/tex]
3. Sample variance (s²):
The sample variance formula is:
[tex]\[
s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}
\][/tex]
Plugging in the values results:
[tex]\[
s^2 = 63.81818181818182
\][/tex]
4. Sample standard deviation (s):
The sample standard deviation is the square root of the sample variance:
[tex]\[
s = \sqrt{s^2} = \sqrt{63.81818181818182} = 7.988628281387351
\][/tex]
5. Population variance ([tex]\(\sigma²\)[/tex]):
The population variance formula is:
[tex]\[
\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{N}
\][/tex]
Here, [tex]\(N\)[/tex] is the population size, which in this context is the same as [tex]\(n\)[/tex].
[tex]\[
\sigma^2 = 58.5
\][/tex]
6. Population standard deviation ([tex]\(\sigma\)[/tex]):
The population standard deviation is the square root of the population variance:
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{58.5} = 7.648529270389178
\][/tex]
Summarized results:
- Mean grade ([tex]\(\bar{x}\)[/tex]): [tex]\(82.0\)[/tex]
- Sample variance ([tex]\(s^2\)[/tex]): [tex]\(63.81818181818182\)[/tex]
- Sample standard deviation ([tex]\(s\)[/tex]): [tex]\(7.988628281387351\)[/tex]
- Population variance ([tex]\(\sigma^2\)[/tex]): [tex]\(58.5\)[/tex]
- Population standard deviation ([tex]\(\sigma\)[/tex]): [tex]\(7.648529270389178\)[/tex]
Hence, these are the statistical measures for the given set of data.
