First slide

Introduction to statistics

Question

There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test
Marks 0 1 2 3 4 5
Frequency x-2 x x² (x+1)² 2x x+1
where, x is a positive integer. Find the mean and standard deviation of the marks.

Easy

A

2.9, 1.14
B

2.8, 1.12
C

2.8, 1.18
D

None of these

Solution

First find x as given total number of students is 60
$\begin{array}{lr} i.e. x - 2 + x + x^{2} + (x + 1)^{2} + 2 x + x + 1 = 60 \\ \Rightarrow 5 x - 2 + x^{2} + x^{2} + 1 + 2 x + 1 = 60 \\ \Rightarrow 2 x^{2} + 7 x = 60 \\ \Rightarrow 2 x^{2} + 7 x - 60 = 0 \\ \Rightarrow 2 x^{2} + 15 x - 8 x - 60 = 0 \\ \Rightarrow x (2 x + 15) - 4 (2 x + 15) = 0 \\ \Rightarrow (2 x + 15) (x - 4) = 0 \\ \Rightarrow x = - \frac{15}{2}, x = 4 \\ \Rightarrow x = 4 (∵ x is positive) \end{array}$
$\begin{array}{l} Mean \bar{x} = \frac{{Σf}_{i} x_{i}}{N} = \frac{168}{60} = 2 .8 \\ \begin{aligned} ∴ Variance σ^{2} & = \sqrt{\frac{{Σf}_{i} x_{i}^{2}}{N} - (\bar{x})^{2}} = \sqrt{\frac{546}{60} - (2 .8)^{2}} \\ = \sqrt{9 .1 - 7 .84} = \sqrt{1 .26} = 1 .12 \end{aligned} \end{array}$

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