Question

The following table shows the marks obtained by 200 students at an examination in mathematics. Find the mean of the data.

Marks	0- 10	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Number of students	05	10	11	20	27	38	40	29	14	06

Answer 1

Hint:For solving this problem we create a table of some columns that contain number of students as \[{{f}_{i}}\], midpoint of marks given as \[{{x}_{i}}\], product of number of students and midpoint of marks as \[{{f}_{i}}{{x}_{i}}\], then we calculate the value of mean using the formula \[\bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\].

Complete step-by-step solution
Let us assume that the given number of students in each interval is represented as \[{{f}_{i}}\].
Now let us create the table that contain number of students as \[{{f}_{i}}\], midpoint of marks given as \[{{x}_{i}}\], product of number of students and midpoint of marks as \[{{f}_{i}}{{x}_{i}}\].

Marks	Number of students\[{{f}_{i}}\]	Midpoint of marks\[{{x}_{i}}\]	\[{{f}_{i}}{{x}_{i}}\]
0-10	05	5	25
10-20	10	15	150
20-30	11	25	275
30-40	20	35	700
40-50	27	45	1215
50-60	38	55	2090
60-70	40	65	2600
70-80	29	75	2175
80-90	14	85	1190
90-100	06	95	570

Now, we know that the mean of the continuous data can be calculated by using the formula
\[\bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}...........equation(i)\]
Let us find the unknown values.
\[\sum{{{f}_{i}}}\] is calculated by adding all the students, which in turn gives the total number of students.
In the question, we are given that the total number of students is 200.
Therefore, we can write \[\sum{{{f}_{i}}}=200\].
Now let us find the value of \[\sum{{{f}_{i}}{{x}_{i}}}\].
 By adding all the values in the column \[{{f}_{i}}{{x}_{i}}\] we get
\[\begin{align}
& \Rightarrow \sum{{{f}_{i}}{{x}_{i}}}=25+150+275+700+1215+2090+2600+2175+1190+570 \\
 & \Rightarrow \sum{{{f}_{i}}{{x}_{i}}}=10990 \\
\end{align}\]
Now by substituting the values of \[\sum{{{f}_{i}}}\] and \[\sum{{{f}_{i}}{{x}_{i}}}\] in equation (i) we get
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}} \\
 & \Rightarrow \bar{x}=\dfrac{10990}{200} \\
 & \Rightarrow \bar{x}=54.95 \\
\end{align}\]
Therefore the mean of given data is 54.95.

Note: Students may make mistakes in calculating the value of \[{{f}_{i}}{{x}_{i}}\]. As the data is written in one line type students may do multiplication mistake instead of multiplying the corresponding \[{{f}_{i}}\] and \[{{x}_{i}}\] values, in a hurry they multiply the value of \[{{x}_{i}}\] and different value of \[{{f}_{i}}\] which results in the wrong answer. The calculation part only needs to be taken care of.