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Mean Deviation

100 surnames ...

Question

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

$\begin{matrix} N u m b e r o f l e t t e r s & N u m b e r o f s u r n a m e s 1 - 4 & 6 4 - 7 & 30 7 - 10 & 40 10 - 13 & 16 13 - 16 & 4 16 - 19 & 4 \end{matrix}$

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, find the modal size of the surnames. [4 MARKS]

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Solution

Concept: 1 Mark
Application: 3 Marks

$\begin{matrix} N u m b e r o f & N u m b e r o f & C u m u l a t i v e l e t t e r s & s u r n a m e s & f r e q u e n c y 1 - 4 & 6 & 6 4 - 7 & 30 & 36 7 - 10 & 40 & 76 10 - 13 & 16 & 92 13 - 16 & 4 & 96 16 - 19 & 4 & 100 \end{matrix}$

Now, n = 100

So, $\frac{n}{2} = \frac{100}{2} = 50$

This observation lies in the class 7 - 10. So, 7 - 10 is the median class.

Therefore, l = 7, h = 3, f = 40, cf = 36

$∴ M e d i a n = l + (\frac{\frac{n}{2} - c f}{f}) \times h = 7 + (\frac{50 - 36}{40}) \times 3$

$= 7 + \frac{21}{20} = 7 + 1.05 = 8.05$

Hence, the median number of letters in the surnames is 8.05

Mean. Take a = 8.5, h = 3

$\begin{matrix} N u m b e r o f & N u m b e r o f & C l a s s & d_{i} = x_{i} - 8.5 & u_{i} = \frac{x_{i} - 8.5}{3} & f_{i} u_{i} l e t t e r s & s u r n a m e s & m a r k (f_{i}) 1 - 4 & 6 & 2.5 & - 6 & - 2 & - 12 4 - 7 & 30 & 5.5 & - 3 & - 1 & - 30 7 - 10 & 40 & 8.5 & 0 & 0 & 0 10 - 13 & 16 & 11.5 & 3 & 1 & 16 13 - 16 & 4 & 14.5 & 6 & 2 & 8 16 - 19 & 4 & 17.5 & 9 & 3 & 12 T o t a l & \sum f_{i} = 100 & \sum f_{i} u_{i} = - 6 \end{matrix}$

Using the step-deviation method.

$¯ x = a + (\frac{\sum f_{i} u_{i}}{\sum f_{i}}) \times h = 8.5 + (\frac{- 6}{100}) \times 3$

$= 8.5 - 0.18 = 8.32$

Hence, the mean number of letters in the surnames is 8.32

Mode: Since the maximum number of surnames have number of letters in the interval 7 - 10, the modal class is 7 - 10

Therefore, $l = 7, h = 3, f_{1} = 40, f_{0} = 30, f_{2} = 16$

$∴ M o d e = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h = 7 + (\frac{40 - 30}{2 \times 40 - 30 - 16}) \times 3$

$= 7 + \frac{30}{34} = 7 + 0.88 = 7.88$

Hence, the modal size of the surnames is 7.88

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100

surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Number of letters	$1 - 4$	$4 - 7$	$7 - 10$	$10 - 13$	$13 - 16$	$16 - 19$
Number of surnames	$6$	$30$	$40$	$16$	$4$	$4$

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

100 surnames were randomly picked from a local directory and the distribution of a number of letters of the English alphabet in the surname was obtained as follows:

Determine the median and mean a number of letters in the surname. Also, find the modal size of surnames

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