To find the median, mode, and mean when class intervals and their corresponding frequencies are given, follow these steps:
Let's assume we have data in the form of class intervals and their respective frequencies.
Step 1: Organize the data Write down the class intervals and their corresponding frequencies in tabular form.
Step 2: Find the midpoint of each class interval
For each class interval, find its midpoint. The midpoint of a class interval is calculated by adding the lower and upper boundaries of the class and then dividing the sum by 2.
Step 3: Find the cumulative frequencies Calculate the cumulative frequency for each class interval. The cumulative frequency for a class interval is the sum of the frequencies of that class interval and all the previous class intervals.
Step 4: Find the median The median is the middle value of the data when arranged in ascending order.
To find the median:
Calculate the total frequency (sum of all frequencies).
Identify the median class, which is the class interval that contains the median value.
Find the cumulative frequency just before the median class (CF below).
Determine the median using the formula:
Median = Lower boundary of the median class + [(Total frequency / 2) - CF below] * Class width
Step 5: Find the mode
The mode is the value that occurs most frequently. In grouped data, the modal class is the class interval with the highest frequency.
Step 6: Find the mean
The mean is calculated using the formula: Mean = (Sum of (Midpoint * Frequency)) / Total frequency
Here's an example to illustrate the process:
Suppose we have the following class intervals and their frequencies:
CI F
10 - 20 5
20 - 30 12
30 - 40 18
40 - 50 8
50 - 60 7
Step 1: Organize the data
Already done in the table above.
Step 2: Find the midpoint of each class interval
The midpoint for each class is calculated as follows:
Midpoint of 10 - 20 class: (10 + 20) / 2 = 15
Midpoint of 20 - 30 class: (20 + 30) / 2 = 25
Midpoint of 30 - 40 class: (30 + 40) / 2 = 35
Midpoint of 40 - 50 class: (40 + 50) / 2 = 45
Midpoint of 50 - 60 class: (50 + 60) / 2 = 55
Step 3: Find the cumulative frequencies Calculate the cumulative frequency for each class:
Cumulative Frequency of 10 - 20 class: 5
Cumulative Frequency of 20 - 30 class: 5 + 12 = 17
Cumulative Frequency of 30 - 40 class: 17 + 18 = 35
Cumulative Frequency of 40 - 50 class: 35 + 8 = 43
Cumulative Frequency of 50 - 60 class: 43 + 7 = 50
Step 4: Find the median
Total frequency = 5 + 12 + 18 + 8 + 7 = 50 (sum of all frequencies)
Median class is the class interval with the cumulative frequency just greater than or equal to 50/2 = 25. In this case, it's the 30 - 40 class with a cumulative frequency of 35.
Median = Lower boundary of median class + [(Total frequency / 2) - CF below] * Class width
Median = 30 + [(25 - 17) / 18] * 10 = 30 + (8/18) * 10 = 30 + 4.44 ≈ 34.44
Step 5: Find the mode
The modal class is the 30 - 40 class with a frequency of 18, which is the highest frequency.
Mode = Midpoint of the modal class ≈ (30 + 40) / 2 = 35
Step 6: Find the mean
Mean = (Sum of (Midpoint * Frequency)) / Total frequency
Mean = (15 * 5 + 25 * 12 + 35 * 18 + 45 * 8 + 55 * 7) / 50
Mean = (75 + 300 + 630 + 360 + 385) / 50 Mean = 1750 / 50
Mean = 35
So, the median is approximately 34.44, the mode is 35, and the mean is 35.