The table shows information about the distances 570 students travelled to a university open day - Edexcel - GCSE Maths - Question 18 - 2018 - Paper 3

To estimate the median distance from the histogram, we need to find the cumulative frequency and determine the median position.

1. First, calculate the cumulative frequency:

   • For 0 < d ≤ 20: 120
   • For 20 < d ≤ 50: 120 + 90 = 210
   • For 50 < d ≤ 80: 210 + 120 = 330
   • For 80 < d ≤ 150: 330 + 140 = 470
   • For 150 < d ≤ 200: 470 + 100 = 570

2. The median position is given by:

   rac{N}{2} = rac{570}{2} = 285

   where N is the total frequency.

3. Locate the interval containing the 285th student:

   • The cumulative frequency reaches 210 at the interval 20 < d ≤ 50.
   • It exceeds 285 in the interval 50 < d ≤ 80 (cumulative frequency of 330).

4. Hence, the median lies within the interval 50 < d ≤ 80.

   • Using a proportional method, we can further refine this. Since 285 is 75 into the interval (285 - 210 = 75), we calculate the position within the interval:

   ext{Median} ext{ is approximately} = 50 + rac{75}{120}(80 - 50) = 50 + rac{75}{120} imes 30 = 50 + 18.75 = 68.75

Thus, the estimated median distance is about 68.75 miles.