The following table shows the ages of the patients admitted in a hospital during a year:Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Solution: Here, Maximum frequency is $23$.
So, Modal class will be corresponding class to $23$ that is , $35-45$.
Now, Mode can be given by formula,
$Mode=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\ast h$
Here, $l=$ Lower limit of modal class $=35$
$f_1=$ Frequency of Modal class $=23$
$f_2=$ Frequency of Pre Modal class $=21$
$f_3=$ Frequency of Succeeding Modal class $=14$
$h=$ Class interval $=10$
Putting these values in Mode formula,
$Mode=35+\frac{23-21}{46-21-14}\ast 10=35+\frac{2}{11}\ast 10=35+1.8=36.8$ $Mode=36.8$ means maximum number of patients are of age $36.8$ years.

Now, we will calculate mean for the given data.
Here, we can use step deviation method to find the mean of given data.
First we have to construct a table for the given data. Please refer to video for creating complete table.
We know, Mean,$\left(\overline{X}\right)=a+\frac{\sum f_i{u}_i}{\sum f_i}\ast h$
Here, $a=$ assumed mean $=30$
$\sum f_i{u}_i=430$
$\sum f_i=80$
$h=$ step size $=1$
Putting these values in formula,
$\overline{X}=30+\left(430\right)\ast \frac{1}{80}=30+\frac{43}{8}=30+5.38=35.38$
$Mean=35.38$ means average age of the patients is $35.38$ years.