Let's solve the problem of finding the population standard deviation for the given dataset using a step-by-step approach.
### Step 1: Calculate the total number of data points
The total number of data points, [tex]\(N\)[/tex], is the sum of all the frequencies:
[tex]\[ N = 9 + 4 + 6 + 8 + 3 + 4 + 2 = 36 \][/tex]
### Step 2: Calculate the weighted mean
The weighted mean [tex]\(\mu\)[/tex] is found by taking the sum of each data point multiplied by its frequency, and then dividing by the total number of data points:
[tex]\[
\mu = \frac{\sum (x_i \cdot f_i)}{N}
\][/tex]
where [tex]\(x_i\)[/tex] is the data point and [tex]\(f_i\)[/tex] is the frequency.
The sum of the data points multiplied by their frequencies is:
[tex]\[
\sum (x_i \cdot f_i) = (12 \cdot 9) + (13 \cdot 4) + (14 \cdot 6) + (25 \cdot 8) + (33 \cdot 3) + (34 \cdot 4) + (36 \cdot 2) = 751
\][/tex]
So, the weighted mean [tex]\(\mu\)[/tex] is:
[tex]\[
\mu = \frac{751}{36} \approx 20.861
\][/tex]
### Step 3: Calculate the variance
The variance [tex]\(\sigma^2\)[/tex] is found by taking the sum of the squared differences between each data point and the mean, multiplied by their frequencies, and then dividing by the total number of data points:
[tex]\[
\sigma^2 = \frac{\sum [f_i \cdot (x_i - \mu)^2]}{N}
\][/tex]
The sum of the squared differences multiplied by frequencies is:
[tex]\[
\sum [f_i \cdot (x_i - \mu)^2] = (9 \cdot (12 - 20.861)^2) + (4 \cdot (13 - 20.861)^2) + (6 \cdot (14 - 20.861)^2) + (8 \cdot (25 - 20.861)^2) + (3 \cdot (33 - 20.861)^2) + (4 \cdot (34 - 20.861)^2) + (2 \cdot (36 - 20.861)^2) = 2964.306
\][/tex]
So, the variance [tex]\(\sigma^2\)[/tex] is:
[tex]\[
\sigma^2 = \frac{2964.306}{36} \approx 82.342
\][/tex]
### Step 4: Calculate the population standard deviation
The population standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance:
[tex]\[
\sigma = \sqrt{82.342} \approx 9.074
\][/tex]
Therefore, the population standard deviation for the given set of data, rounded to the nearest thousandth, is:
[tex]\[
\sigma \approx 9.074
\][/tex]
