Question

On the sports day of a school, 300 students participated. Their ages are given in the following distribution.

Age (in years)	5-7	7-9	9-11	11-13	13-15	15-17	17-19
Number of students	67	33	41	95	36	13	15

Find the mean and mode of the date.

Answer 1

Hint: Here in this question we will use the concept and formula of the mean and mode to find the value of mean and mode. To calculate mean, we will find the mid-value of each class interval. Then we will find the product of the mid-value and frequency of the corresponding class. We will then substitute these values in the formula of the mean to get its value. Then for mode we will select the modal class from the given data and modal class is the class with the maximum frequency. Further, we will apply the formula of the mode to find its value.

Formula used:
We will use the following formulas:
1) Mean \[ = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\] where, \[{f_i}\] is the frequency of the class and \[{x_i}\] is the mid-value of the class interval.
2) Formula of the mode \[ = L + h\left( {\dfrac{{{f_m} - {f_1}}}{{2{f_m} - {f_1} - {f_2}}}} \right)\] where, \[L\] is the lower limit of the modal class, \[h\] is the size of the class interval, \[{f_m}\] is the frequency of the model class, \[{f_1}\] is the frequency of the class preceding the modal class, \[{f_2}\] is the frequency of class succeeding the modal class.

Complete step by step solution:
First, we will calculate the value of the mean. We will find the value of the mid-value of the class interval and then we will find the value of the product of the mid-value and the corresponding value of the frequency. Therefore, we get

Class interval	Frequency \[{f_i}\]	Mid-value \[{x_i}\]	\[{f_i}{x_i}\]
5-7	67	6	402
7-9	33	8	264
9-11	41	10	410
11-13	95	12	1140
13-15	36	14	504
15-17	13	16	208
17-19	15	18	270
	\[\sum {{f_i} = 300} \]		\[\sum {{f_i}{x_i} = 3198} \]

Now we will put the value from the table into the formula of the mean. Therefore, we get
Mean \[ = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\]
\[ \Rightarrow \] Mean \[ = \dfrac{{3198}}{{300}}\]
Dividing the terms, we get
\[ \Rightarrow \] Mean \[ = 10.66\]
Now we will find the model class from the data. Modal is the class with the maximum frequency in the given data. So from the data we can see that class 11-13 is the class with the maximum frequency i.e. 95. So, modal class is 11-13.
Lower limit of the modal class is 11. Therefore,
\[L = 11\]
Size of the class interval in the given data is 2. Therefore,
\[h = 2\]
Frequency of the modal class is 95. Therefore,
\[{f_m} = 95\]
Frequency of the class preceding the modal class i.e. class 9-11 is 41. Therefore,
 \[{f_1} = 41\]
Frequency of the class succeeding the modal class i.e. class 13-15 is 36. Therefore,
\[{f_2} = 36\]
Now, we will put these values in the formula mode \[ = L + h\left( {\dfrac{{{f_m} - {f_1}}}{{2{f_m} - {f_1} - {f_2}}}} \right)\]. Therefore, we get
Mode \[ = 11 + 2\left( {\dfrac{{95 - 41}}{{2\left( {95} \right) - 41 - 36}}} \right)\]
Simplifying the expression, we get
\[ \Rightarrow \] Mode \[ = 11 + 2\left( {\dfrac{{54}}{{190 - 77}}} \right)\]
Subtracting the terms in the denominator, we get
\[ \Rightarrow \] Mode \[ = 11 + 2\left( {\dfrac{{54}}{{113}}} \right)\]
Simplifying the expression further, we get
\[ \Rightarrow \] Mode \[ = 11 + 0.95\]
Adding the terms, we get
\[ \Rightarrow \] Mode \[ = 11.95\]

Hence, the mean of the data is \[10.66\] and mode of the data is \[11.95\].

Note:
Statistics is the science of collecting some data in the form of the number and studying it to forecast or predict its future possibility. Some definitions we should know
Mean is equal to the ratio of sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers.
Mode is the most common or most repeating number.
Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.