Question

find the median of the following distribution.​

Answer 1

Answer:

In this question, we are given a continuous series and we have to find the median of the data. For this, we will find the sum of frequencies (N) and then use it to locate the class interval in which our median lies. Then we will find the cumulative frequency of the data. Using cumulative frequency and the sum of frequency, we will find the median class which will be the class containing (N2)th item. At last, we will apply the formula given below to find the median.

Median=l1+N2−c.ff×i

Where, l1 is the lower limit of median class, c.f. is cumulative frequency of class preceding the median class, f is the simple frequency of median class and i is the class interval of the median class.

Complete step-by-step solution:

Here, we are given continuous series hence, the median cannot be located straight forward. In this case, the median lies in between the lower and upper limits of a class interval. To get the exact value of the median, we have to interpolate the median with the help of the formula given as

Me=l1+N2−c.ff×i

Where, “Me” is the median, l1 is the lower limit of median class, c.f. is cumulative frequency of class preceding the median class, f is the simple frequency of median class, N is the sum of frequencies and i is the class interval of the median class. Let us move step by step for finding all the required terms for the formula to calculate the median.

Step 1: Let us calculate cumulative frequencies and make a frequency distribution table. The cumulative frequency for any row is the sum of preceding all the frequencies. Hence, we get the table below:

Class Frequency (f) Cumulative Frequency (c.f.)

0-10 5 5

10-20 10 5+10=15

20-30 20 5+10+20=35

30-40 7 5+10+20+7=42

40-50 8 5+10+20+7+8=50

50-60 5 5+10+20+7+8+5=55

Now, the sum of frequencies will be given by the last cumulative frequency. Hence, N = 55.

Step 2: Now, we will find the median item which will be (N2)th item.

As N = 55, therefore, (N2)th item = (552)th item 27.5th item

Hence, N2=27.5⋯⋯⋯⋯⋯(1)

Step 3: Now, we will find the median class by inspecting cumulative frequencies. We will