Sure, let's find the mean, variance, and standard deviation for the given data set step-by-step using [tex]\( x_i \)[/tex] and [tex]\( f_i \)[/tex]:
1. Mean (Weighted Average):
The mean is calculated by taking the sum of each [tex]\( x_i \)[/tex] value multiplied by its corresponding frequency [tex]\( f_i \)[/tex], then dividing by the total sum of the frequencies.
[tex]\[
\text{Total Weight} = \sum f_i = 5 + 18 + 42 + 27 + 8 = 100
\][/tex]
[tex]\[
\text{Weighted Sum} = \sum (x_i \cdot f_i) = (61 \cdot 5) + (64 \cdot 18) + (67 \cdot 42) + (70 \cdot 27) + (73 \cdot 8) = 305 + 1152 + 2814 + 1890 + 584 = 6745
\][/tex]
[tex]\[
\text{Mean} = \frac{\text{Weighted Sum}}{\text{Total Weight}} = \frac{6745}{100} = 67.45
\][/tex]
2. Variance:
The variance is calculated by taking the sum of each frequency [tex]\( f_i \)[/tex] multiplied by the square of the difference between each [tex]\( x_i \)[/tex] and the mean, then dividing by the total sum of the frequencies.
[tex]\[
\text{Variance Sum} = \sum f_i \cdot (x_i - \text{Mean})^2
\][/tex]
[tex]\[
\begin{aligned}
& = 5 \cdot (61 - 67.45)^2 + 18 \cdot (64 - 67.45)^2 + 42 \cdot (67 - 67.45)^2 + 27 \cdot (70 - 67.45)^2 + 8 \cdot (73 - 67.45)^2 \\
& = 5 \cdot (-6.45)^2 + 18 \cdot (-3.45)^2 + 42 \cdot (-0.45)^2 + 27 \cdot (2.55)^2 + 8 \cdot (5.55)^2 \\
& = 5 \cdot 41.6025 + 18 \cdot 11.9025 + 42 \cdot 0.2025 + 27 \cdot 6.5025 + 8 \cdot 30.8025 \\
& = 208.0125 + 214.245 + 8.505 + 175.5675 + 246.42 \\
& = 852.75
\end{aligned}
\][/tex]
[tex]\[
\text{Variance} = \frac{\text{Variance Sum}}{\text{Total Weight}} = \frac{852.75}{100} = 8.5275
\][/tex]
3. Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{8.5275} \approx 2.9202
\][/tex]
To summarize:
- Total weight (sum of frequencies, [tex]\( f_i \)[/tex]): 100
- Weighted sum: 6745
- Mean (weighted average, [tex]\( \bar{x} \)[/tex]): 67.45
- Sum for variance calculation: 852.75
- Variance ([tex]\( \sigma^2 \)[/tex]): 8.5275
- Standard deviation ([tex]\( \sigma \)[/tex]): approximately 2.9202
