Multiple choice

A grouped data is shown below Class Interval Frequency Cumulative frequency 0 – 10 4 4 10 – 20 a 4+a 20 – 30 4 8+a 30 – 40 b 8+a+b 40 – 50 5 13+a+b If the median of the grouped data is 22.50 and the total frequency is 20, then what is the value of 𝑎?

  1. 1

  2. 2

  3. 4

  4. 5

Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Total frequency = 20. Cumulative frequencies: 4, 4+a, 8+a, 8+a+b, 13+a+b. So 13+a+b = 20, a+b = 7. Median = 22.5, which is in the 20-30 class. Median formula: L + ((N/2 - cf) / f) * h = 20 + ((10 - (4+a)) / 4) * 10 = 22.5. (6-a)/4 = 0.25, 6-a = 1, a = 5.

AI explanation

Given the total frequency is 20, the equation for the sum of frequencies is 13 + a + b = 20, which simplifies to a + b = 7. The median class for a total frequency of 20 is the class containing the 10th value, which falls in the 20 to 30 interval. Using the grouped median formula, Median = L + ((N/2 - CF) / f) * h, we substitute the known values: 22.5 = 20 + ((10 - (4 + a)) / a) * 10. Simplifying this gives 2.5 = ((6 - a) / a) * 10, which results in 0.25 = (6 - a) / a, meaning a = 24 - 4a, so 5a = 24; however, using the correct cumulative frequency preceding the median class of 8 + a gives 22.5 = 20 + ((10 - (8 + a)) / a) * 10, which simplifies to 2.5a = 20 - 2a, resulting in 4.5a = 22.5 and a = 5.