Question

The marks ( out of 100 ) obtained by a group of students in a science test are 85, 78, 90, 85, 39, 48, 58, 95, 81 and 75. Find the:

1. Highest and lowest marks obtained by student
2. Mean marks obtained
3. No. of students who obtained marks more than mean marks.

Answer 1

We will first arrange all the data in the ascending order. After that, pick up the first number, it will be the lowest marks and the last number will be the highest marks. Now, subtract the lowest from highest to get the range. Now, find the mean using the formula and then see how many observations cross it to get the number of students who obtained more than mean marks.

Complete step-by-step answer:
Part 1:
First, let us arrange the data in the ascending order. We have 85, 78, 90, 85, 39, 48, 58, 95, 81 and 75.
We have this in ascending order as 39, 48, 56, 75, 76, 81, 85, 85, 90, 95.
Now, we can clearly see that the first and last entries are 39 and 95 respectively.
$\therefore$ Lowest Marks = 39
$\therefore$ Highest Marks = 95
Part 2:
We have found that:
$\Rightarrow$ Lowest Marks = 39
And, Highest Marks = 95
We know that Range = Highest Observation – Lowest Observation
$\Rightarrow$ Range of the data is = 95 – 39
Simplifying the calculation to get:-
$\Rightarrow$ Range of the data is = 56
$\therefore$ the required range of the given data is 56.
Part 3:
We have marks of some students given by 85, 78, 90, 85, 39, 48, 58, 95, 81 and 75.
We see that we have here the marks of 10 students with us. So, the number of observations is 10.
Now, we know that the formula of mean is given by the sum of all the observations divided by the number of observations.
$\therefore Mean = \dfrac{{85 + 78 + 90 + 85 + 39 + 48 + 58 + 95 + 81 + 75}}{{10}}$
Simplifying the numerator to get:-
$\Rightarrow Mean = \dfrac{{730}}{{10}}$
Simplifying the RHS further to get:-
$\Rightarrow Mean = 73$
$\therefore$ the mean of the given data is 73.
Now, the data in ascending order was 39, 48, 56, 75, 76, 81, 85, 85, 90, 95.
We see that we have 75, 76, 81, 85, 85, 90 and 95 which is 7 entries above 73.
$\therefore$ 7 students scored more than the mean marks.

Note: The students must commit to the memory:
Mean is given by the sum of all the observations divided by the number of observations.
Range = Highest Observation – Lowest Observation
The students must also note that if one entry would have been exactly equal to the mean which is 73, then that entry would not be counted as the student getting more than mean marks, because that is just equal to the mean marks, not greater than it.
The students should also note that the average and mean represent the same thing. Here, if we see that the class had an average of 73 marks. So, the class overall performed well.