Calculate the mean mark, standard deviation, and coefficient of variation for the given frequency…

Question

Calculate the mean mark, standard deviation, and coefficient of variation for the given frequency distribution of student marks, assuming a mean of 62.

ScanSolve AI Tutor · Accepted Answer

To solve this problem, we first need to set up a frequency distribution table with midpoints, deviations from the assumed mean, and their products.

The given data is:

Marks	No. of students (f)
50-54	5
55-59	8
60-64	11
65-69	10
70-74	8
75-79	4

Note: The question states "The marks scored by 50 students", but the sum of the given frequencies is $5+8+11+10+8+4 = 46$ . We will proceed with $\sum f = 46$ as per the provided table.

We will use an assumed mean $A = 62$ as instructed.

Create a table to calculate $\sum f$ , $\sum fd$ , and $\sum fd^2$ .

The midpoint ( $x$ ) for each class is calculated as (lower limit + upper limit) / 2.
The deviation ( $d$ ) is $x - A = x - 62$ .

Marks	f	x	d = x - 62	fd	fd^2
50-54	5	52	-10	-50	500
55-59	8	57	-5	-40	200
60-64	11	62	0	0	0
65-69	10	67	5	50	250
70-74	8	72	10	80	800
75-79	4	77	15	60	900
Total	$\sum f = 46$			$\sum fd = 100$	$\sum fd^2 = 2650$

a) (i) Mean mark

Use the formula for the mean using an assumed mean.
$\bar{x} = A + \frac{\sum fd}{\sum f}$
Substitute the values from the table.
$\bar{x} = 62 + \frac{100}{46}$ $\bar{x} = 62 + 2.173913...$ $\bar{x} = 64.173913...$
Round the mean to two decimal places.
$\bar{x} \approx 64.17$
The mean mark is $\boxed{\text{64.17}}$ .

a) (ii) Standard deviation
Use the formula for standard deviation using an assumed mean.
$s = \sqrt{\frac{\sum fd^2}{\sum f} - \left(\frac{\sum fd}{\sum f}\right)^2}$
Substitute the values from the table.
$s = \sqrt{\frac{2650}{46} - \left(\frac{100}{46}\right)^2}$ $s = \sqrt{57.608695... - (2.173913...)^2}$ $s = \sqrt{57.608695... - 4.726086...}$ $s = \sqrt{52.882608...}$ $s = 7.27204...$
Round the standard deviation to two decimal places.
$s \approx 7.27$
The standard deviation is $\boxed{\text{7.27}}$ .

b) Calculate the number of students who scored more than 68 marks.

To find the number of students who scored more than 68 marks, we need to consider the classes above 68.
The class intervals are given as discrete values (e.g., 65-69). When dealing with continuous data for interpolation, we use class boundaries.
The class boundaries for 65-69 are 64.5 and 69.5.

Calculate the mean mark, standard deviation, and coefficient of variation for the given frequency distribution of student marks, assuming a mean of 62.

Step-by-step solution

Create a table to calculate $\sum f$ , $\sum fd$ , and $\sum fd^2$ .

Use the formula for the mean using an assumed mean.

Substitute the values from the table.

Round the mean to two decimal places.

Use the formula for standard deviation using an assumed mean.

Substitute the values from the table.

Round the standard deviation to two decimal places.

Still stuck on this one?

More Mathematics Questions

Calculate the mean mark, standard deviation, and coefficient of variation for the given frequency distribution of student marks, assuming a mean of 62.

Step-by-step solution

Create a table to calculate ∑f\sum f∑f, ∑fd\sum fd∑fd, and ∑fd2\sum fd^2∑fd2.

Use the formula for the mean using an assumed mean.

Substitute the values from the table.

Round the mean to two decimal places.

Use the formula for standard deviation using an assumed mean.

Substitute the values from the table.

Round the standard deviation to two decimal places.

Still stuck on this one?

More Mathematics Questions

Create a table to calculate $\sum f$ , $\sum fd$ , and $\sum fd^2$ .