The median of cumulative frequency is 67.86 and the interquartile range is 32.86.
The mean is "the middle number is sorter, ascending or descending, list of numbers and can be more descriptive of that data set than the average".
The interquartile range is "measure of where the 'middle fifty' is in data set".
According to the question,
Score Mid value Frequency Cumulative frequency
10 ≤ x < 20 15 1 1
20≤x < 30 25 3 4
30≤x < 40 35 6 10
40≤x < 50 45 15 25
50≤x < 60 55 14 39
60≤x < 70 65 28 67
70≤x < 80 75 18 85
80≤x < 90 85 11 96
90≤x < 100 95 4 100
Uniform interval is 10.
a) The median of examination mark
Frequency = 1+3+6+15+14+28+18+11+4 = 100
Median = [tex]l+\frac{(\frac{N}{2} - m )}{f}[/tex] × c where l = lower limit of the median class, N = total frequency, m = cumulative frequency of the pre median class,
f = frequency of the median class, c = uniform class interval.
= 50 + [tex]\frac{\frac{100}{2} - 25 }{14}[/tex] × 10
= 50 + [tex]\frac{50 - 25}{14}[/tex] × 10
= 50 +[tex]\frac{25}{14}[/tex] × 10
= 50 + 1.786 × 10
= 50 + 17.86
= 67.86.
Thus, the median of examination mark is 67.86.
b) The number of students score less than 65 marks.
In order to find the number of students score less than 65 marks sum of all frequency below 65.
= 1 + 3 + 6 + 15 + 14
= 39
Thus, the number of students score less than 65 marks is 39.
c) The number of students scored between 50 and 80 marks
In order to find the number of students scored between 50 and 80 marks only if add frequency between 50 and 80 marks.
= 14 + 28 + 18 + 11
= 71
Thus, the number of students scored between 50 and 80 marks is 71.
d) The number of students failed, given that the pass mark is 50.
In order to find the number of students failed, given that the pass mark is 50 only if add frequency below 50.
= 1 + 3 + 6 + 15
= 25.
Thus, the number of students failed, given that the pass mark is 50 is 25.
f) The interquartile range
Interquartile range = Q₃ - Q₁ [In words, difference between upper quartile range and lower quartile range]
Q₁ = l₁ + [tex]\frac{\frac{N}{4} - m }{f}[/tex] × c where l₁ is the lower limit of the Q₁ class, m₁ is the cumulative frequency of the pre median class, f₁ is the Frequency of the Q₁ class, c is the class interval, N is the total frequency.
[tex]\frac{N}{4}[/tex] = [tex]\frac{100}{4} = 25[/tex]; [tex]\frac{3N}{4}[/tex] = 75
= 40 + [tex]\frac{25 - 25}{6}[/tex] * 10 = 40
Q₁ = 40
Q₃ = l₃ + [tex]\frac{3\frac{N}{4} - m3 }{f3}[/tex] × c
= 60 + [tex]\frac{75 - 39}{28}[/tex] × 10
= 60 + [tex]\frac{36}{28}[/tex] × 10
= 60 + 1.2857 × 10
= 60 + 12.857
= 72.857
Interquartile range = upper quartile range - lower quartile range
= 72.857 - 40
= 32.8571 ≈ 32.86.
Thus, interquartile range is 32.86.
Hence, a) The median of examination mark is 67.86.
b) The number of students score less than 65 marks is 39.
c) The number of students scored between 50 and 80 marks is 71.
d) The number of students failed, given that the pass mark is 50 below 50 is 25.
f) The interquartile range is 32.86.
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