Question

The following is the distribution of height of students of a certain class in a certain city.
Height (in cms):
160−162
163−165
166−168
169−171
172−174
No. of students:
15
118
142
127
18
Find the median height.

Answer 1

SOLUTION :  

★★Here, the data is not in continuous form( exclusive) so we convert it into continuous form (inclusive).  For this subtract 0.5 from lower limit of each class interval and add 0.5 to upper limit of each class interval.

CUMULATIVE FREQUENCY TABLE is in the attachment.  

Hence, the Median height is 167.13 cm .

MEDIAN: Median is defined as the middle most or the Central observation , when the observations are arranged either in ascending or descending order of their magnitudes.

★★Median is that value of the given observation which divides it into exactly two parts.i.e 50% of the observations lie below the median and the remaining are above the median.

MEDIAN for the GROUPED data : For this we find the Cumulative frequency(cf) of all the classes and n/2 , where n =  number of observations. Now find the class whose Cumulative frequency is greater than and nearest to n/2 and this class is called median class,then use  the following formula calculating the median.

MEDIAN = l + [(n/2 - cf )/f ] ×h

Where,

l = lower limit of the median class

n = number of observations  

cf = cumulative frequency  of class interval preceding the  median class

f = frequency  of median class

h = class  size

★★ CUMULATIVE FREQUENCY:

Cumulative frequency is defined as a consecutive sum of frequencies.

**The Cumulative frequency of first observation is the same as its frequency since there is no frequency before it.

HOPE THIS ANSWER WILL HELP YOU.  

Answer 2

Answer :

The median height is 167.13 .

Step-by-step explanation :

Median -

It is defined as the middle observation or the observation which comes at centre when the observations are arranged in either ascending order or descending order.

Median is the value which differentiate observation into 50% - 50% each.

Median for grouped data -

To calculate median for grouped data, we find cumulative frequency and divide number of observations by 2.

Then, we find the class whose cumulative frequency is greater than and nearest to half of  n and this class is known as median class. After this we use the following formula,

$Median=l+\frac{\frac{n}{2}-Cf}{f}\times h$

Where, l is lower limit of median class, n is number of observations, Cf is the cumulative frequency, f is frequency of median class and h is the class size.

Note - The data given above is not in exclusive form, so we will convert it into inclusive form. To convert, subtract 0.5 from lower limit and add 0.5 to upper limit.

Cumulative Frequency Table -

$\begin{tabular}{|c|c|c|}\cline{1-3}Height & Number\:of\:students & Cumulative\:frequency\\ \cline{1-3}159.5-162.5 & 15 & 15\\ \cline{1-3}162.5-165.5 & 118 & 15+118=133\\ \cline{1-3}165.5-168.5 & 142 & 133+142=275\\ \cline{1-3}168.5-171.5 & 127 & 275+127=402\\ \cline{1-3}171.5+174.5 & 18 & 402+18=420\\ \cline{1-3} & n=420 & \\ \cline{1-3}\end{tabular}$

Here,

n/2 = 210, f = 142, h = 3, Cf = 133 & l = 165.5

$Median=l+\frac{\frac{n}{2}-Cf}{f}\times h$

$Median=165.5+\frac{210-133}{142}\times 3$

$Median=165.5+\frac{77\times 3}{142}$

$Median=165.5+\frac{231}{142}$

$Median=165.5+1.63$

$Median=167.13$