Question

Calculate the median for the following data:

Age (in years)	19-25	26-32	33-39	40-46	47-53	54-60
Frequency	35	96	68	102	34	4

Answer 1

Hint: Here in this question, we have to find the median. The question is related to the statistics topic. The median is the middle value of the sorted elements. Here we use formula $ M = l + \left\{ {h \times \left( {\dfrac{{\dfrac{N}{2} - cf}}{f}} \right)} \right\} $ where l is the class interval, h is difference of class intervals, N is the total number of frequency, cf is the cumulative frequency and f is the frequency of particular class interval.

Complete step-by-step answer:
Now we convert the given data into exclusive form so we have,

Class-interval	frequency(f)	Cumulative frequency(cf)
18.5-25.5	35	35
25.5-32.5	96	131
32.5-39.5	68	199
39.5-46.5	102	301
46.5-53.5	34	336
53.5-60.5	4	340
	$ N = \sum = 340 $

The total of frequency is 340 and now we will find the mid value so we divide the N by 2 so we have
Therefore, we have
 $ \Rightarrow \dfrac{N}{2} = \dfrac{{340}}{2} = 170 $
We got the value 170. so, we have to choose the class interval frequency by considering the cumulative frequency. We have to look into the cumulative frequency column and notice that value greater than 170. So 199 is the greater number than the 170.
Now consider the class interval 32.5-39.5 and the frequency 68
Apply the formula to find the median
 $ M = l + \left\{ {h \times \left( {\dfrac{{\dfrac{N}{2} - cf}}{f}} \right)} \right\} $
Here the lower value of the class interval 32.5-39.5 is 32.5. Therefore $ l = 32.5 $
The difference between the class interval is 7, therefore $ h = 7 $
Here we consider the cumulative frequency value of previous class
Therefore, by applying the values to the formula we have
 $
  M = l + \left\{ {h \times \left( {\dfrac{{\dfrac{N}{2} - cf}}{f}} \right)} \right\} \\
   \Rightarrow M = 32.5 + \left\{ {7 \times \left( {\dfrac{{\dfrac{{340}}{2} - 131}}{{68}}} \right)} \right\} \;
  $
On simplification we have
 $
   \Rightarrow M = 32.5 + \left\{ {7 \times \left( {\dfrac{{170 - 131}}{{68}}} \right)} \right\} \\
   \Rightarrow M = 32.5 + \left\{ {7 \times \dfrac{{39}}{{68}}} \right\} \\
   \Rightarrow M = 32.5 + 4.01 \\
   \Rightarrow M = 36.51 \;
  $
Therefore, the median for the given following data is $ M = 36.51 $
So, the correct answer is “ $ M = 36.51 $ ”.

Note: The median is the mid value of the given data. To calculate the median, we have a formula $ M = l + \left\{ {h \times \left( {\dfrac{{\dfrac{N}{2} - cf}}{f}} \right)} \right\} $ , by substituting the values we can obtain the Required solution or median for the given following data. Here cumulative frequency is calculated by adding upon the above frequencies.