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Question

A survey regarding the height (in cm) of 51 girls of class X of a school was conducted
and the following data was obtained:
Height in cmNumber of GirlsLess than 1404Less than 14511Less than 15029Less than 15540Less than 16046Less than 16551
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, 155, 160, 165 give the upper limits of the corresponding class intervals. So, the classes should be below 140, 140-145, 145-150, 150-155, 155-160, 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 given cumulative frequencies becomes:

Class interval Frequency Cumulative frequency
Below 140

140 – 145

145 – 150

150 – 155

155 – 160

160 – 165

4

7

18

11

6

5

4

11

29

40

46

51

Now n = 51. an odd number that is  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 space the space formula comma space Median space equals space straight l plus left parenthesis fraction numerator begin display style straight n over 2 minus cf end style over denominator straight f end fraction right parenthesis cross times straight h Median space equals space 145 plus left parenthesis fraction numerator begin display style 25.5 minus 11 end style over denominator 18 end fraction right parenthesis cross times 5 space space space space space space space space space space space space space space space equals space 145 space plus space fraction numerator 72.5 over denominator 18 end fraction space space space space space space space space space space space space space space space equals space 149.03

So, the median height of the girls is 149.03 cm

This means that the height of about 50% of the girls in less than this height, and 50% are taller than this height.


Mathematics
RD Sharma
Standard X

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