Find the average of the workers from the following distribution table. Find the values of x and y, if the total frequency is 100.

Steps to Solve

1. Identify the frequencies and values in the table

From the second part of the problem, the frequencies are given as:

• More than 150: 0
• More than 140: 10
• More than 130: 29
• More than 120: 60
• More than 110: 104
• More than 100: 134
• More than 90: 151
• More than 80: 160

Let ( x ) be the frequency for the range 500 - 600 and ( y ) be the frequency for the range 400 - 500.

2. Set up the frequency equation

We know the total frequency is 100. Thus, we have an equation:

$$ 0 + 10 + 29 + 60 + 104 + 134 + 151 + x + y = 100 $$

3. Simplify the equation

First, compute the total of given frequencies:

$$ 0 + 10 + 29 + 60 + 104 + 134 + 151 = 488 $$

So now we can simplify the total frequency equation:

$$ 488 + x + y = 100 $$

4. Solve for ( x + y )

Rearranging the equation from the previous step gives:

$$ x + y = 100 - 488 $$

This becomes:

$$ x + y = -388 $$

This indicates that the values need to be adjusted since frequencies cannot be negative.

5. Finding x and y directly

Assuming the frequencies must sum to 100, observe that total frequencies cannot exceed 100, thus this must be verified before concluding.

6. Average Calculation

The average can be computed using the formula:

$$ \text{Average} = \frac{\sum (f \cdot x)}{N} $$

Where ( f ) is the frequency and ( x ) is the mid-value of each group.

Add the contribution only with a valid frequency (N = 100) for calculation.