Question

The following table shows the ages of the patients admitted in a hospital during a year.

Age in years:	0-15	15-25	25-35	35-45	45-55	55-65
Number of patients:	6	11	21	23	14	5

Find the mode and the mean of the data given above. Compare and intercept the two measures of central tendency.

Answer 1

Hint: We use the given information of the number of students in each age group to calculate the mean and the mode by substituting the values from the data in the formulas.
* Mean of any data is given by dividing the sum of the observations by the number of observations. If we have a grouped data then mean is given by
\[\overline x = \dfrac{{\sum\nolimits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\nolimits_{i = 1}^n {{f_i}} }}\],
Where \[{x_i} = \](upper class limit-lower class limit)/2 and \[{f_i}\] is the frequency of the class limit
* Mode of any data is the value that is most repeated. For calculation of mode we have to find the class interval having the highest frequency which is called modal class. If class with maximum frequency is l-m where ‘l’ is the lower limit of the modal class and ‘m’ is the higher limit, the class size is ‘h’,
\[{f_0}\]: Frequency of proceeding modal class
\[{f_1}\]: Frequency of proceeding class
\[{f_2}\]: Frequency of succeeding modal class
Then mode is given by \[l \times \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h\]

Complete Step by Step Solution:
We are given the ages of the patients admitted in a hospital during a year.
Since we are given a grouped data, we will find the frequency of each class and class mark for each respective class.
We form table of the given data which has class mark (\[{x_i}\]) and then we find the frequency \[{f_i}\] and then multiply each class mark with respective frequency (\[{f_i}{x_i}\]).

Age (In years)	Number of patients(\[{f_i}\])	Class mark (\[{x_i}\])	\[{f_i}{x_i} = {f_i} \times {x_i}\]
5-15	6	\[\dfrac{{5 + 15}}{2} = 10\]	\[6 \times 10 = 60\]
15-25	11	\[\dfrac{{15 + 25}}{2} = 20\]	\[11 \times 20 = 220\]
25-35	21	\[\dfrac{{25 + 35}}{2} = 30\]	\[21 \times 30 = 630\]
35-45	23	\[\dfrac{{35 + 45}}{2} = 40\]	\[23 \times 40 = 920\]
45-55	14	\[\dfrac{{45 + 55}}{2} = 50\]	\[14 \times 50 = 700\]
55-65	5	\[\dfrac{{55 + 65}}{2} = 60\]	\[5 \times 60 = 300\]
Total:	\[\sum {{f_i} = 80} \]		\[\sum {{f_i}{x_i} = 2830} \]

We substitute the values in formula for mean
Mean: \[\overline x = \dfrac{{2830}}{{80}}\]
Cancel same factors from numerator and denominator
\[ \Rightarrow \overline x = \dfrac{{283}}{8}\]
\[ \Rightarrow \overline x = 35.375\]...............… (1)
Now we calculate mode of the given observations
From the observations, highest frequency is 23
Modal class is 35-45
Lower limit of the modal class: \[l = 35\]
Class interval: \[h = 10\]
Frequency of proceeding modal class: \[{f_0} = 21\]
Frequency of proceeding class: \[{f_1} = 23\]
Frequency of succeeding modal class: \[{f_2} = 14\]
Then mode is given by \[l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h\]
Substitute the values in the formula for mode
\[ \Rightarrow \]Mode\[ = l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h\]
\[ \Rightarrow \]Mode\[ = 35 + \dfrac{{23 - 21}}{{2 \times 23 - 21 - 14}} \times 10\]
\[ \Rightarrow \]Mode\[ = 35 + \dfrac{2}{{46 - 35}} \times 10\]
\[ \Rightarrow \]Mode\[ = 35 + \dfrac{2}{{11}} \times 10\]
\[ \Rightarrow \]Mode\[ = 35 + \dfrac{{20}}{{11}}\]
Take LCM in RHS
\[ \Rightarrow \]Mode\[ = \dfrac{{385 + 20}}{{11}}\]
\[ \Rightarrow \]Mode\[ = \dfrac{{405}}{{11}}\]
\[ \Rightarrow \]Mode\[ = 36.8181\]................… (2)
\[\therefore \]Mean of the given data is 35.375 and mode of the given data is 36.8181

\[\therefore \]Average age of the patient admitted in the hospital is 35.375 years and maximum and repeated numbers of patients in the hospital are of age group 36.8181 years.

Note: Many students make mistake while calculating mode of the given data as they put the value of \[{f_0}\] as frequency of the modal class as usually \[{f_0}\] denotes mostly initial data, but keep in mind \[{f_0}\] is the frequency of preceding modal class which means class before the modal class. Also, write both the values of mean and mode up to two decimal places.
Students are likely to make mistakes in the calculation part of the table as they directly make columns for \[{f_i}{x_i}\] and don’t calculate \[{x_i}\]. Keep in mind we need to calculate the class mark as well. Also, many students directly apply the formula of mean as sum of observations divided by number of observations, this is wrong as the data given is not normal data it is grouped data.