Question

For the following frequency distribution, find:

1. median 
2. lower quartile
3. upper quartile 
4. interquartile range

Variate	10	18	20	22	25	27	28
Frequency	4	6	8	9	7	8	6

Answer 1

To find the median and quartiles, we first need to calculate the cumulative frequency (cf) for the given data set. 

Frequency distribution table

Variable (x)	Frequency (f)	Cumulative Freuqency (cf)
10	4	4
18	6	10
20	8	18
22	9	27
25	7	34
27	8	42
28	6	48

The total number of observations is N = 48.

i. Median

Since N is even 48, the median is the average of the `(N/2)^(th)` and `(N/2 + 1)^(th)` terms.

Position: `48/2` = 24^th and 25^th terms.

Looking at the cumulative frequency table, the values from the 19^th to the 27^th position are all 22.

Median = 22

ii. Lower Quartile (Q₁)

For a discrete frequency distribution, the lower quartile is the value at the `(N/4)^(th)` position.

Position: `48/4` = 12^th term.

The cumulative frequency just greater than 12 is 18, which corresponds to the variate 20.

Lower Quartile (Q₁) = 20

iii. Upper Quartile (Q₃)

The upper quartile is the value at the `((3N)/4)^(th)` position.

Position: `(3 xx 48)/4` = 36^th term

The cumulative frequency just greater than 36 is 42, which corresponds to the variate 27.

Upper Quartile (Q₃) = 27

iv. Interquartile Range (IQR)

The interquartile range is the difference between the upper and lower quartiles.

Formula: IQR = Q₃ – Q₁

Calculation: 27 – 20 = 7

Interquartile Range = 7