To find the arithmetic mean of the given frequency distribution using the step deviation method, follow these steps:
1. List the given data:
- Weights (in kgs): 50, 55, 60, 65, 70
- Frequency (number of men): 15, 20, 25, 30, 10
2. Identify the midpoints of each class:
- In this case, the weights themselves are the midpoints:
[tex]\[50, 55, 60, 65, 70\][/tex]
3. Choose an assumed mean:
- Let's choose our assumed mean ([tex]\(A\)[/tex]) as 60 kg.
4. Determine the step (interval size):
- The interval between the weights is 5 kg.
5. Calculate the deviations ([tex]\(d_i\)[/tex]) from the assumed mean:
- The deviation for each midpoint from the assumed mean, divided by the step size, is:
[tex]\[
d_i = \frac{(x_i - A)}{\text{step}}
\][/tex]
Where:
[tex]\[
x_i = \text{midpoint of the }i\text{-th class}
\text{, step} = 5
\][/tex]
6. Compute [tex]\(d_i\)[/tex] for each class:
[tex]\[
d_1 = \frac{(50 - 60)}{5} = -2.0
\][/tex]
[tex]\[
d_2 = \frac{(55 - 60)}{5} = -1.0
\][/tex]
[tex]\[
d_3 = \frac{(60 - 60)}{5} = 0.0
\][/tex]
[tex]\[
d_4 = \frac{(65 - 60)}{5} = 1.0
\][/tex]
[tex]\[
d_5 = \frac{(70 - 60)}{5} = 2.0
\][/tex]
7. Multiply each deviation by the corresponding frequency ([tex]\(f_i\)[/tex]) to get [tex]\(f_i \cdot d_i\)[/tex]:
[tex]\[
f_1 \cdot d_1 = 15 \cdot -2 = -30.0
\][/tex]
[tex]\[
f_2 \cdot d_2 = 20 \cdot -1 = -20.0
\][/tex]
[tex]\[
f_3 \cdot d_3 = 25 \cdot 0 = 0.0
\][/tex]
[tex]\[
f_4 \cdot d_4 = 30 \cdot 1 = 30.0
\][/tex]
[tex]\[
f_5 \cdot d_5 = 10 \cdot 2 = 20.0
\][/tex]
8. Sum the frequencies [tex]\((N)\)[/tex]:
[tex]\[
N = 15 + 20 + 25 + 30 + 10 = 100
\][/tex]
9. Sum [tex]\(f_i \cdot d_i\)[/tex]:
[tex]\[
\sum (f_i \cdot d_i) = -30 + -20 + 0 + 30 + 20 = 0
\][/tex]
10. Calculate the arithmetic mean ([tex]\(\overline{x}\)[/tex]) using the step deviation formula:
[tex]\[
\overline{x} = A + \text{step} \cdot \left(\frac{\sum (f_i \cdot d_i)}{N}\right)
\][/tex]
[tex]\[
\overline{x} = 60 + 5 \cdot \left(\frac{0}{100}\right)
\][/tex]
[tex]\[
\overline{x} = 60 + 5 \cdot 0
\][/tex]
[tex]\[
\overline{x} = 60
\][/tex]
Therefore, the arithmetic mean of the given frequency distribution is [tex]\(60\)[/tex] kgs.
