Calculate the average mean of the runs by the Step-deviation Method, given the following data: Runs: 5-15, 15-25, 25-35, 35-45, 45-55 Frequency: 10, 12, 17, 19, 22 Calculate arit... Calculate the average mean of the runs by the Step-deviation Method, given the following data: Runs: 5-15, 15-25, 25-35, 35-45, 45-55 Frequency: 10, 12, 17, 19, 22 Calculate arithmetic mean of the following frequency distribution: Values: less than 10, 10-20, 20-30, 30-40, 40-50, 50-60, more than 60 Frequency: 5, 12, 18, 22, 6, 4, 3 Read more

Steps to Solve

Here are the steps to solve the two problems.

Problem 1: Calculating the average mean of runs.

1. Find the midpoints $x_i$ of each class interval.

The midpoint is calculated as (lower limit + upper limit) / 2.

• 5-15: $(5 + 15) / 2 = 10$
• 15-25: $(15 + 25) / 2 = 20$
• 25-35: $(25 + 35) / 2 = 30$
• 35-45: $(35 + 45) / 2 = 40$
• 45-55: $(45 + 55) / 2 = 50$

2. Choose an assumed mean $a$.

Let's take $a = 30$.

3. Calculate the step deviations $u_i = (x_i - a) / h$, where $h$ is the class size.

Here, $h = 10$.

• $u_1 = (10 - 30) / 10 = -2$
• $u_2 = (20 - 30) / 10 = -1$
• $u_3 = (30 - 30) / 10 = 0$
• $u_4 = (40 - 30) / 10 = 1$
• $u_5 = (50 - 30) / 10 = 2$

4. Multiply the frequencies $f_i$ by the step deviations $u_i$.

   • $f_1u_1 = 10 \times -2 = -20$
   • $f_2u_2 = 12 \times -1 = -12$
   • $f_3u_3 = 17 \times 0 = 0$
   • $f_4u_4 = 19 \times 1 = 19$
   • $f_5u_5 = 22 \times 2 = 44$

5. Calculate $\sum f_i u_i$ .

$\sum f_i u_i = -20 + (-12) + 0 + 19 + 44 = 31$

6. Calculate $\sum f_i$ .

$\sum f_i = 10 + 12 + 17 + 19 + 22 = 80$

7. Apply the step-deviation formula to find the mean.

$\text{Mean} = a + h \times (\sum f_i u_i / \sum f_i)$ $\text{Mean} = 30 + 10 \times (31 / 80) = 30 + 10 \times 0.3875 = 30 + 3.875 = 33.875$

Problem 2: Calculating the arithmetic mean of the frequency distribution.

1. Convert the 'less than' and 'more than' frequencies to class intervals.

The intervals are:

• Less than 10: 0-10, Frequency = 5
• 10-20, Frequency = 12
• 20-30, Frequency = 18
• 30-40, Frequency = 22
• 40-50, Frequency = 6
• 50-60, Frequency = 4
• More than 60: 60-70 (assuming a class size of 10), Frequency = 3

2. Find the midpoints $x_i$ of each class interval.

• 0-10: $(0 + 10) / 2 = 5$
• 10-20: $(10 + 20) / 2 = 15$
• 20-30: $(20 + 30) / 2 = 25$
• 30-40: $(30 + 40) / 2 = 35$
• 40-50: $(40 + 50) / 2 = 45$
• 50-60: $(50 + 60) / 2 = 55$
• 60-70: $(60 + 70) / 2 = 65$

3. Multiply the frequencies $f_i$ by the midpoints $x_i$.

   • $5 \times 5 = 25$
   • $12 \times 15 = 180$
   • $18 \times 25 = 450$
   • $22 \times 35 = 770$
   • $6 \times 45 = 270$
   • $4 \times 55 = 220$
   • $3 \times 65 = 195$

4. Calculate $\sum f_i x_i$.

$\sum f_i x_i = 25 + 180 + 450 + 770 + 270 + 220 + 195 = 2110$

5. Calculate $\sum f_i$ .

$\sum f_i = 5 + 12 + 18 + 22 + 6 + 4 + 3 = 70$

6. Compute the arithmetic mean.

$\text{Mean} = \sum f_i x_i / \sum f_i$ $\text{Mean} = 2110 / 70 = 30.1428 \approx 30.14$