To solve for the variance of the given data set, we follow the steps below:
1. Collect the Data:
The depths of the dives are:
[tex]\[
60, 58, 53, 49, 60
\][/tex]
2. Calculate the Mean:
The mean of the data set, [tex]\(\mu\)[/tex], is given as:
[tex]\[
\mu = 56
\][/tex]
3. Calculate the Numerator:
The numerator of the variance formula is calculated by summing the squared deviations of each data point from the mean:
[tex]\[
(60 - 56)^2 + (58 - 56)^2 + (53 - 56)^2 + (49 - 56)^2 + (60 - 56)^2
\][/tex]
Simplifying each term:
[tex]\[
(60 - 56)^2 = 4^2 = 16
\][/tex]
[tex]\[
(58 - 56)^2 = 2^2 = 4
\][/tex]
[tex]\[
(53 - 56)^2 = (-3)^2 = 9
\][/tex]
[tex]\[
(49 - 56)^2 = (-7)^2 = 49
\][/tex]
[tex]\[
(60 - 56)^2 = 4^2 = 16
\][/tex]
Summing these values:
[tex]\[
16 + 4 + 9 + 49 + 16 = 94
\][/tex]
Therefore, the numerator evaluates to:
[tex]\[
94
\][/tex]
4. Calculate the Denominator:
The denominator of the variance formula is the number of data points, [tex]\(N\)[/tex]. Given the data set:
[tex]\[
60, 58, 53, 49, 60
\][/tex]
There are 5 data points, so:
[tex]\[
N = 5
\][/tex]
Therefore, the denominator evaluates to:
[tex]\[
5
\][/tex]
5. Calculate the Variance:
The variance [tex]\(\sigma^2\)[/tex] is the numerator divided by the denominator:
[tex]\[
\sigma^2 = \frac{94}{5} = 18.8
\][/tex]
Therefore, the variance equals:
[tex]\[
18.8
\][/tex]
To summarize:
- Numerator: [tex]\(94\)[/tex]
- Denominator: [tex]\(5\)[/tex]
- Variance: [tex]\(18.8\)[/tex]
