Histograms: the maths GCSE test scores of 280 students are shown in the histogram below

a) The value of the median is 112

b) The value of the lower quartile is 95.59

Step-by-step explanation:

From the histogram, we look for the data values as follows

Score,                                       Frequency,          Cumulative , n

80 - 90,                                    2.6,                       2.6

90 - 100,                                   3.4,                       6

100 - 120,                                  5,                          11

120 - 140,                                  3,                          14

140 - 170,                                  2.4,                       16.4

170 - 200,                                 0.6,                       17

The median is therefore the (nth + 1)/2th value = (17 + 1)/2 = 9th value

The 9th value is between the 100 - 120 score values

We note that the previous n value before the 100 - 120 range = 6, therefore, the 9th value will be the weighted average in the 100 - 120 range of tha amount the 9th value is larger than the 6th value as follows;

9 - 6 = 3

The weighted average of 3 out 5 (11 - 6 = 5) of the 100 - 120 range which gives;

Therefore, the value of the median, Q₂ = 100 + (9 - 6)/(11 - 6))*(120 - 100) = 112

b) The lower quartile = Q₁ is given by the value at the (n +1)/4th position = (17 + 1)/4 = 4.5th position

From the cumulative frequency column, the 4.5th value is in the 90 - 100 score range

Similar to the previous question, we find the value of the lower quartile using the weighted average as follows;

Therefore, the value of the lower quartile, Q₁ = 90 + (4.5 - 2.6)/(6 - 2.6)×(100 - 90) = 95.59.

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