Question

Find 'mean' and 'mode' of the following data : Frequency Distribution Table

Class	0 – 15	15 – 30	30 – 45	45 – 60	60 – 75	75 – 90
Frequency	11	8	15	7	10	9

Answer 1

To find the mean and mode, we first create a table with class mid-points (\(x_i\)) and \(f_i x_i\).

Class Interval	Frequency (fi)	Mid-point (xi)	fixi
0 – 15	11	7.5	82.5
15 – 30	8	22.5	180.0
30 – 45	15	37.5	562.5
45 – 60	7	52.5	367.5
60 – 75	10	67.5	675.0
75 – 90	9	82.5	742.5
Total	Σfi = 60		Σfixi = 2610.0

Mean Calculation: The mean \(\bar{x}\) is given by the formula \(\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i}\). \[ \bar{x} = \frac{2610}{60} = \frac{261}{6} = 43.5 \] Mode Calculation: The class with the highest frequency is the modal class. Highest frequency = 15, which corresponds to the class interval 30 -- 45. So, the modal class is 30 -- 45. Lower limit of the modal class (\(l\)) = 30. Frequency of the modal class (\(f_1\)) = 15. Frequency of the class preceding the modal class (\(f_0\)) = 8. Frequency of the class succeeding the modal class (\(f_2\)) = 7. Class size (\(h\)) = 15. The mode is given by the formula: Mode = \(l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h\) Mode = \(30 + \left(\frac{15 - 8}{2(15) - 8 - 7}\right) \times 15\) Mode = \(30 + \left(\frac{7}{30 - 15}\right) \times 15\) Mode = \(30 + \left(\frac{7}{15}\right) \times 15\) Mode = \(30 + 7 = 37\) \[ \boxed{\text{Mean} = 43.5, \text{Mode} = 37} \]