(Q1) A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and data was obtained as shown in table. Find their median.

Height(in cm) Number of girls
Less than 140 4
Less than 145 11
Less than 150 29
Less than 155 40
Less than 160 46
Less than 165 51

Solution : To calculate the median height, we need to find the class intervals and their corresponding frequencies. The given distribution being of the less than type, 140, 145, 150, . . ., 165 give the upper limits of the corresponding class intervals. So, the classes should be below 140, 140 - 145, 145 - 150, . . ., 160 - 165.

Class Intervals Frequency Cumulative Frequency
Below 140 4 4
140-145 7 11
145-150 18 29
150-155 11 40
155-160 6 46
160-165 5 51

Observe that from the given distribution, we find that there are 4 girls with height less than 140, i.e., the frequency of class interval below 140 is 4 . Now, there are 11 girls with heights less than 145 and 4 girls with height less than 140. Therefore, the number of girls with height in the interval 140 - 145 is 11 - 4 = 7. Similarly, the frequencies can be calculated as shown in table.

Number of observations, n = 51

n
2
 =
51
2
 = 25.5th observation, which lies in the class 145-150.

∴ 145-150 is the Median Class

Then, l (the lower boundary) = 145,

cf (the cumulative frequency of the class preceding 145-150) = 11

f(the frequency of the median class 145-150) = 18 and

h(the class size) = 5.

Using the formula, Median = l +
(
n
2
 - cf )
f
 × h
Median = +
( - )
18
 ×
   = +
18
 = .03

So, the median height of the girls is .03 cm. This means that the height of about 50% of the girls is less than this height, and that of other 50% is greater than this height.