Question

The time taken by 50 students to complete a 100 meter race is given below. Find the standard deviation.

Time taken (sec)	8.5 – 9.5	9.5 – 10.5	10.5 – 11.5	11.5 – 12.5	12.5 – 13.5
Number of students	6	8	17	10	9

Answer 1

Hint: We will calculate the standard deviation of the given data by drawing the table. We will calculate ${x_i}$as the mid value of the time taken by the students, and we will suppose A = 11, and h (${x_n} - {x_{n - 1}}$ ) will also be calculated by the table and then, we can calculate the value of the deviation d as: $d = \dfrac{{{x_i} - A}}{h}$. The standard deviation is given by the formula: $\sigma = h \times \sqrt {\dfrac{{\sum {{f_i}{d_i}^2} }}{N} - {{\left( {\dfrac{{\sum {{f_i}{d_i}} }}{N}} \right)}^2}} $ where N is the total number of students,${f_i}{d_i}$ and ${f_i}{d_i}^2$will be calculated from the table.

Complete step-by-step answer:
We are given that 50 students participated in a 100 meter race and the time taken by them was recorded.
We need to find the standard deviation by the data given in tabular form as:

Time taken (sec)	8.5 – 9.5	9.5 – 10.5	10.5 – 11.5	11.5 – 12.5	12.5 – 13.5
Number of students	6	8	17	10	9

From this table, we can calculate${x_i}$, h =${x_n} - {x_{n - 1}}$and then $d = \dfrac{{{x_i} - A}}{h}$as:
Here, A = 11 and h = 10 – 9 = 11 – 10 = 1

Time taken(in sec)	Mid value${x_i}$	${f_i}$	$d = \dfrac{{{x_i} - A}}{h}$	${f_i}{d_i}$	${f_i}{d_i}^2$
8.5 – 9.5	9	6	-2	-12	24
9.5 – 10.5	10	8	-1	-8	8
10.5 – 11.5	11	17	0	0	0
11.5 – 12.5	12	10	1	10	10

12.5 – 13.5	13	9	2	18	36
		N = 50		$\sum {{f_i}{d_i}} $ = 8	$\sum {{f_i}{d_i}^2} $ = 78

Now, we know that the standard deviation is given by the formula: $\sigma = h \times \sqrt {\dfrac{{\sum {{f_i}{d_i}^2} }}{N} - {{\left( {\dfrac{{\sum {{f_i}{d_i}} }}{N}} \right)}^2}} $
Putting the values of h, $\sum {{f_i}{d_i}} $ and $\sum {{f_i}{d_i}^2} $from the table in this equation, we get
\[
   \Rightarrow \sigma = h \times \sqrt {\dfrac{{\sum {{f_i}{d_i}^2} }}{N} - {{\left( {\dfrac{{\sum {{f_i}{d_i}} }}{N}} \right)}^2}} = 1 \times \sqrt {\dfrac{{78}}{{50}} - {{\left( {\dfrac{8}{{50}}} \right)}^2}} \\
   \Rightarrow \sigma = \sqrt {\dfrac{{78}}{{50}} - \dfrac{{64}}{{2500}}} = \sqrt {\dfrac{{3900 - 64}}{{2500}}} = \sqrt {\dfrac{{3836}}{{2500}}} = \sqrt {1.5344} = 1.238 \\
 \]
Therefore, the standard deviation of the given data is 1.238$ \approx $1.24.
Note: In this question, you may go wrong while calculating the values of the table i.e., you need to be careful while putting the values in the formulae to calculate d, ${f_i}{d_i}$and${f_i}{d_i}^2$. You should also take care of the values while you put them to calculate the standard deviation in its formula.