Question

Find the median of the following distribution:

x	5	3	8	7	9	11
fi	38	31	27	36	25	35

Answer 1

Step 1: Arrange data and Calculate cumulative frequency.

First, we arrange the values of x in ascending order and calculate the cumulative frequency (cf) for each.

x (Value)	fi (Frequency)	cf (Cumulative Frequency)
3	31	31
5	38	69 (31 + 38)
7	36	105 (69 + 36)
8	27	132 (105 + 27)
9	25	157 (132 + 25)
11	35	192 (157 + 35)

Step 2: Find the total number of observations.

The total number of observations (N) is the sum of all frequencies:

N = Σf_i = 192

Step 3: Determine the median position.

Since N (192) is an even number, the median is the average of the `(N/2)^(th)` and `(N/2 + 1)^(th)` observations.

`N/2 = 192/2` = 96^th observation

`N/2 + 1` = 97^th observation

Step 4: Identify the median value.

Now, look at the cumulative frequency (cf) column to find where the 96^th and 97^th observations fall:

The cf just greater than 96 and 97 is 105.

The value of corresponding to the cumulative frequency of 105 is 7.

Since both the 96^th and 97^th terms are 7, the average is simply 7.

The median of the distribution is 7.