Question

The following are the paddy yields (in kg/plot) for 14 plots: 30, 32, 35, 38, 40, 42, 48, 49, 52, 55, 58, 60, 62, 65. The 25th percentile (Q1) yield in kg/plot will be:

• A. 37.25
• B. 38.00
• C. 40.00
• D. 48.50

Hint:

Use the rank formula (n+1) times p/100 to find the position, then interpolate between the two bracketing values.

Answer 1

The Correct Option is A

Step 1: Arrange the data in ascending order and assign ranks. The 14 paddy yield values are already in ascending order: 30, 32, 35, 38, 40, 42, 48, 49, 52, 55, 58, 60, 62, 65, with ranks 1 through 14.
Step 2: For a data set of \(n\) values, the position of the \(p\)th percentile is found using the rank formula \(R = \frac{p}{100}(n+1)\). Here \(n = 14\) and \(p = 25\), so \(R = \frac{25}{100}(14+1) = 0.25 \times 15 = 3.75\).
Step 3: A rank of 3.75 means the 25th percentile lies between the 3rd and 4th observations, three quarters of the way from the 3rd value towards the 4th value. The 3rd value is 35 and the 4th value is 38.
Step 4: Interpolate: \(Q_1 = X_3 + 0.75(X_4 - X_3) = 35 + 0.75(38-35) = 35 + 2.25 = 37.25\).
Step 5: Option 2 (38.00) would only be right if the rank were rounded up to 4 instead of interpolated. Options 3 and 4 correspond to much higher ranks and are far from position 3.75. The correctly interpolated value 37.25 kg/plot matches option 1.