Certainly! Let's calculate the Geometric Mean (GM) and Individual Interval Means (IIM) for the given frequency data step-by-step.
### Given Data:
Frequency Data: [tex]\(10, 20, 145.77\)[/tex]
Class Intervals: [tex]\( (10, 20), (20, 30), (30, 40), (40, 50), (50, 60) \)[/tex]
### 1. Geometric Mean (GM):
The Geometric Mean (GM) is calculated using the formula:
[tex]\[ GM = (x_1 \cdot x_2 \cdot x_3 \cdots x_n)^{\frac{1}{n}} \][/tex]
#### Steps:
1. Multiply all the values given in the frequency data:
[tex]\[ 10 \times 20 \times 145.77 = 29154 \][/tex]
2. Take the n-th root of the product, where [tex]\( n \)[/tex] is the number of values (3 in this case):
[tex]\[ GM = 29154^{\frac{1}{3}} \][/tex]
This gives:
[tex]\[ GM \approx 30.777455813955374 \][/tex]
### 2. Individual Interval Mean (IIM):
The Individual Interval Mean (IIM) is calculated for each class interval using the midpoint of the class interval, weighted by the frequency of that interval, and then normalized by the total frequency.
#### Steps:
1. Calculate the total frequency:
[tex]\[ 10 + 20 + 145.77 = 175.77 \][/tex]
2. Determine the midpoint of each class interval that has a corresponding frequency:
- For [tex]\(10-20\)[/tex]: Midpoint [tex]\( = \frac{10 + 20}{2} = 15 \)[/tex]
- For [tex]\(20-30\)[/tex]: Midpoint [tex]\( = \frac{20 + 30}{2} = 25 \)[/tex]
- For [tex]\(30-40\)[/tex]: Midpoint [tex]\( = \frac{30 + 40}{2} = 35 \)[/tex]
3. Calculate the weighted midpoints:
[tex]\[
\begin{align*}
\text{Weighted Midpoint for }(10-20) &= 15 \times \frac{10}{175.77} \approx 0.8533879501621436 \\
\text{Weighted Midpoint for }(20-30) &= 25 \times \frac{20}{175.77} \approx 2.8446265005404787 \\
\text{Weighted Midpoint for }(30-40) &= 35 \times \frac{145.77}{175.77} \approx 29.026284348864998 \\
\end{align*}
\][/tex]
This gives:
[tex]\[ IIM \approx [0.8533879501621436, 2.8446265005404787, 29.026284348864998] \][/tex]
### Summary:
- Geometric Mean (GM): [tex]\( \approx 30.777455813955374 \)[/tex]
- Individual Interval Means (IIM): [tex]\( \approx [0.8533879501621436, 2.8446265005404787, 29.026284348864998] \)[/tex]
These are the Geometric Mean (GM) and Individual Interval Means (IIM) for the given frequency data and class intervals.
