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Concept: Median of Grouped Data

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  • Computation of Measures of Central Tendency - Median of Grouped Data
  • cumulative frequency column

notes

Example : A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained: 

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

Find the median height.

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. 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 frequency of 145 - 150 is 29 – 11 = 18, for 150 - 155, it is 40 – 29 = 11, and so on. So, our frequency distribution table with the given cumulative frequencies becomes:

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

Now n = 51. So, n/2=51/2 =25.5. This observation lies in the class 145 - 150. Then,

l (the lower limit) = 145,
cf (the cumulative frequency of the class preceding 145 - 150) = 11,
f (the frequency of the median class 145 - 150) = 18,
h (the class size) = 5.

Using the formula, Median `= l +((n/2-cf)/f)xxh` we have

Median `= 145 +((25.5-11)/18)xx5`

`=145+72.5/18=149.03`

So, the median height of the girls is 149.03 cm.
This means that the height of about 50% of the girls is less than this height, and 50% are taller than this height.

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