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 array - c - c - Whine Number of letters - Number of surnames lilhline 1-4 - 6 Wllline4-7 - 30 Whline 7-10 - 40 Whline 10-13 - 16 Whline13-16 - 4 Whline16-19 - 4 Whlinelend array 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-

Concept: 1 Mark Application: 3 Marks [ Number of amp; Number of amp; Cumulative; letters amp; surnames amp; frequency; 1 4 ...

You visited us 4 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard X

Mathematics

Measure of Central Tendency

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]

Open in App

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

Suggest Corrections

Similar questions

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabet in the surnames was obtained as follows:
$\begin{matrix} N u m b e r o f l e t t e r s & 1 - 4 & 4 - 7 & 7 - 10 & 10 - 13 & 13 - 16 & 16 - 19 N u m b e r o f s u r n a m e s & 6 & 30 & 40 & 16 & 4 & 4 \end{matrix}$

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.

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

$\begin{matrix} Number of letters & 1 - 4 & 4 - 7 & 7 - 10 & 10 - 13 & 13 - 16 & 16 - 19 Number of surnames & 6 & 30 & 40 & 16 & 4 & 4 \end{matrix}$
Determine the median number of letters in the surnames.

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

Join BYJU'S Learning Program