Question

In a class test, the marks scored by 18 students are 31, 53, 42, 25, 30, 26, 45, 38, 41, 35, 29, 47, 35, 28, 29, 23, 31, 21. Find:

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

Answer 1

Given:

Marks 18 students: 31, 53, 42, 25, 30, 26, 45, 38, 41, 35, 29, 47, 35, 28, 29, 23, 31, 21.

Number of observations n = 18.

Step-wise calculation:

1. Arrange in ascending order with positions:

21, 23, 25, 26, 28, 29, 29, 30, 31, 31, 35, 35, 38, 41, 42, 45, 47, 53

2. For even n, quartile positions are at `n/4` and `(3n)/4`; if these are fractional, take the average of the surrounding observations.

Q₁ position = `n/4`

= `18/4`

= 4.5 

Average of 4th and 5th observations

= `(26 + 28)/2` 

= 27

Q₃ position = `(3n)/4`

= `3 xx 18/4`

= 13.5 

Average of 13th and 14th observations

= `(38 + 41)/2`

= 39.5

3. Interquartile range = Q₃ – Q₁

= 39.5 – 27

= 12.5

Lower quartile (Q₁) = 27.

Upper quartile (Q₃) = 39.5.

Interquartile range (IQR) = 12.5.