Question

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

Heights (in cm)	Number of girls
less than 140	04
less than 145	11
less than 150	29
less than 155	40
less than 160	46
less than 165	51

Find the median height of girls. If the mode of the above distribution is 148.05, find the mean using the empirical formula.

Answer 1

Height (in cm)	Number of girls (fi)	Cumulative frequency
0 − 140	4	4
140 − 145	11 − 4 = 7	4 + 7 = 11
145 − 150	29 − 11 = 18	11 + 18 = 29
150 − 155	40 − 29 = 11	29 + 11 = 40
155 − 160	46 − 40 = 6	40 + 6 = 46
160 − 165	51 − 46 = 5	46 + 5 = 51

Median = `l + ((N/2 - cf)/f) xx h`

Here, `N/2 = 51/2` = 25.5

∴ 145 − 150 is the median class.

And, l = Lower limit of median class = 145

h = Class interval = 145 − 140 = 5

cf = Cumulative frequency of the class before median class = 11

f = Frequency of the median class = 18

Putting values in the formula:

Median = `l + ((N/2 - cf)/f) xx h`

= `145 + (25.5 - 11)/18 xx 5`

= `145 + 14.5/18 xx 5`

= 145 + 4.03

= 149.03

Therefore, median height is 149.03 cm.

Finding the mean using empirical formula:

Given that Mode = 148.05 and Median = 149.03

3 × Median = Mode + 2 × Mean

3 × 149.03 = 148.05 + 2 × Mean

447.09 − 148.05 = 2 × Mean

299.04 = 2 × Mean

Mean = `299.04/2`

Mean = 149.52