from the following freequency distribution table determine median and modeClass: 0-50,50-100,100-150,150-200,200-250,250-300,300-350,350-400,400-450,450-500Frequency: 78,123,187,82,51,47,13,9,6,4
Question
from the following freequency distribution table determine median and modeClass: 0-50,50-100,100-150,150-200,200-250,250-300,300-350,350-400,400-450,450-500Frequency: 78,123,187,82,51,47,13,9,6,4
Solution
To find the median and mode from a frequency distribution table, follow these steps:
Step 1: Calculate the cumulative frequency. This is done by adding each frequency from the top down.
Here's the cumulative frequency for your data:
Class: 0-50, 50-100, 100-150, 150-200, 200-250, 250-300, 300-350, 350-400, 400-450, 450-500 Frequency: 78, 123, 187, 82, 51, 47, 13, 9, 6, 4 Cumulative Frequency: 78, 201, 388, 470, 521, 568, 581, 590, 596, 600
Step 2: Find the median. The median is the value that separates the higher half from the lower half of the data set. It is found using the formula: Median Class = L + [(N/2 - CF)/f] * c
Where: L = Lower boundary of the median group N = Total number of data CF = Cumulative frequency of the group before the median group f = Frequency of the median group c = Width of the interval
From the cumulative frequency, N = 600. So, N/2 = 300. The median group is the group where the cumulative frequency surpasses N/2 for the first time, which is the 100-150 group.
So, L = 100, CF = 201 (from the previous group), f = 187, and c = 50 (150-100).
Substituting these values into the formula gives:
Median Class = 100 + [(300 - 201)/187] * 50 = 126.26
Step 3: Find the mode. The mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode, or no mode at all. The mode is found using the formula: Mode = L + [(d1)/(d1 + d2)] * c
Where: L = Lower boundary of the modal group d1 = Difference between the frequency of the modal group and the previous group d2 = Difference between the frequency of the modal group and the next group c = Width of the interval
The modal group is the group with the highest frequency, which is the 100-150 group.
So, L = 100, d1 = 187 - 123 = 64, d2 = 187 - 82 = 105, and c = 50.
Substituting these values into the formula gives:
Mode = 100 + [(64)/(64 + 105)] * 50 = 123.08
So, the median of the data is approximately 126.26 and the mode is approximately 123.08.
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