A polling agency interviewed 200 persons. The age distribution of those persons was recorded as under: Calculate mean age of the persons by Direct Method.

Understand the Problem

The user wants to calculate the mean age of persons based on the data provided. We are given age ranges and their frequencies in a survey. To find the mean, you'll need to find the midpoint of each age range, multiply by the frequency, sum these values, and then divide by the total number of persons (200).

Answer

$35.8$ years

Steps to Solve

Find the midpoint of each age range.

To find the midpoint, add the lower and upper limits of each range and divide by 2.

80-89: $(80+89)/2 = 84.5$
70-79: $(70+79)/2 = 74.5$
60-69: $(60+69)/2 = 64.5$
50-59: $(50+59)/2 = 54.5$
40-49: $(40+49)/2 = 44.5$
30-39: $(30+39)/2 = 34.5$
20-29: $(20+29)/2 = 24.5$
10-19: $(10+19)/2 = 14.5$

Multiply each midpoint by its corresponding frequency.

Multiply each midpoint calculated in step 1 by its frequency

84.5 * 2 = 169
74.5 * 2 = 149
64.5 * 6 = 387
54.5 * 20 = 1090
44.5 * 56 = 2492
34.5 * 40 = 1380
24.5 * 42 = 1029
14.5 * 32 = 464

Calculate the sum of these products.

Sum all the values we computed in step 2. $169 + 149 + 387 + 1090 + 2492 + 1380 + 1029 + 464 = 7160$

Divide the sum by the total number of persons.

Divide the sum we computed in step 3 by the total number of people interviewed, which is given as 200. $7160 / 200 = 35.8$

The mean age of the persons is $35.8$ years.

More Information

The mean is a measure of central tendency which is obtained by dividing the sum of the values by the number of values. In this case, we were dealing with grouped data, which is why we had to approximate each value in the group by the midpoint of the group.

Tips

A common mistake is using the interval boundaries instead of the midpoints. Also, arithmetic errors in calculating the midpoints, the products, or the final division can occur.

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