Calculating mode by grouping method

To calculate mode by the grouping method, we need to follow the below steps:

Step 1: Determine the class intervals

In this question, the class intervals are:

10 (15-24), 10 (25-34), 10 (35-44), 10 (45-54), 10 (55-64), 10 (65-74), 10 (75-84), 10 (85-94)

Step 2: Find the frequency for each class interval

The frequency for each class interval is given in the question:

5, 9, 13, 21, 20, 15, 8, 3

Step 3: Find the midpoint for each class interval

To find the midpoint for each class interval, we add the lower limit and upper limit of each interval and divide the sum by 2. The midpoints for each class interval are:

20, 30, 40, 50, 60, 70, 80, 90

Step 4: Find the class interval with the highest frequency

The class interval with the highest frequency is 45-54, with a frequency of 21.

Step 5: Calculate the mode

To calculate the mode, we use the formula:

Mode = L + (f1-f0) / (2*f1-f0-f2) * w

Where:
L = Lower limit of the class interval with the highest frequency (45 in this case)
f1 = Frequency of the class interval with the highest frequency (21 in this case)
f0 = Frequency of the class interval preceding the class interval with the highest frequency (13 in this case)
f2 = Frequency of the class interval succeeding the class interval with the highest frequency (20 in this case)
w = Width of the class interval (10 in this case)

Substituting the values in the formula, we get:

Mode = 45 + (21-13) / (2*21-13-20) * 10
Mode = 45 + 8 / 18 * 10
Mode = 45 + 4.44
Mode = 49.44

Therefore, the mode by grouping method is 49.44.