To calculate the standard deviation of a population, we first need to calculate the variance using the provided formula:
[tex]\[
\sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}
\][/tex]
Given:
- Depths of dives: [tex]\(60, 58, 53, 49, 60\)[/tex]
- Mean ([tex]\(\mu\)[/tex]): [tex]\(56\)[/tex]
Let's break down the solution step-by-step:
Step 1: Calculate the numerator of the variance formula
First, find the squared differences from the mean for each data point:
1. [tex]\((60 - 56)^2 = 4^2 = 16\)[/tex]
2. [tex]\((58 - 56)^2 = 2^2 = 4\)[/tex]
3. [tex]\((53 - 56)^2 = (-3)^2 = 9\)[/tex]
4. [tex]\((49 - 56)^2 = (-7)^2 = 49\)[/tex]
5. [tex]\((60 - 56)^2 = 4^2 = 16\)[/tex]
Now, sum these squared differences:
[tex]\[
16 + 4 + 9 + 49 + 16 = 94
\][/tex]
So, the numerator evaluates to [tex]\( \boxed{94} \)[/tex].
Step 2: Calculate the denominator of the variance formula
The denominator is the number of data points (N), which is the size of the data set. We have 5 depths:
[tex]\[
N = 5
\][/tex]
So, the denominator evaluates to [tex]\( \boxed{5} \)[/tex].
Step 3: Calculate the variance
Using the values obtained for the numerator and denominator:
[tex]\[
\sigma^2 = \frac{94}{5} = 18.8
\][/tex]
So, the variance equals [tex]\( \boxed{18.8} \)[/tex].
These calculations allow us to understand the spread of the given data set around the mean depth of 56 feet.
