Question

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

Answer 1

The median height is 167.1 cm.

Height (in cms):          No. of students:(f)

160-162                                   15

163-165                                   118

166-168                                   142

169-171                                     127

172-174                                     18

Gap = 163 - 162 = 166 - 165 = 1.

Gap/2 = 1/2 = 0.5

To make a discontinuous function a continuous function subtract the Gap/2 from lower limit and add Gap/2 to the upper limit.

Height (in cms):          No. of students:(f)              Cumulative frequency (cf)

159.5 - 162.5                      15                                       15

162.5 - 165.5                     118                                      15+118=133

165.5 - 168.5                     42                                     133+142=275

168.5 - 171.5                      127                                     275+127=402

171.5 - 174.5                       18                                       402+18=420

N = 420

N/2 = 420/2 = 210

The cf just greater than N/2 = 275 belong to the class interval 165.5 - 168.5

The median class  = 165.5 - 168.5

h = height = 162.5 - 159.5 = 2

l = lower limit of the median class interval = 165.5

f = frequency of the median class interval = 142

F = cf of the class interval before the median class interval  = 133

We use a formula to calculate the Median

$Median = l + \dfrac{\frac{n}{2}-F}{f} * h$

Median = 165.5 + [ (210 - 133) / 142 ] × 2

= 165.5 + 77/142 × 2

= 165.5 + 77/71

= 167.1