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- - 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-

We can find cumulative frequencies with their respective class intervals as below: [ Number of letters amp;Frequency (fi) amp;Cumulative frequency; ...

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Standard XII

Mathematics

Median

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 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.

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Solution

We can find cumulative frequencies with their respective class intervals as below:
$\begin{matrix} N u m b e r o f l e t t e r s & F r e q u e n c y (f i) & C u m u l a t i v e f r e q u e n c y 1 - 4 & 6 & 6 4 - 7 & 30 & 30 + 6 = 36 7 - 10 & 40 & 36 + 40 = 76 10 - 13 & 16 & 76 + 16 = 92 13 - 16 & 4 & 92 + 4 = 96 16 - 19 & 4 & 96 + 4 = 100 T o t a l (n) & 100 \end{matrix}$
Now, we observe that cumulative frequency just greater than $c l a s s \frac{n}{2} (i . e, \frac{100}{2} = 50)$ is of interval 7 - 10
Median class = 7 - 10
Lower limit $l$ of median class = 7
Cumulative frequency $c f$ of class preceding median class = 36
Frequency $f$ of median class = 40
Class size $h$ = 3
$M e d i a n = l + (\frac{\frac{n}{2} - c f}{f}) \times h$
$= 7 + (\frac{50 - 36}{40}) \times 3$
$= 7 + \frac{14 \times 3}{40}$
$= 8.05$
Now, we can find the class marks of given class intervals by using relation:
$C l a s s m a r k = \frac{u p p e r c l a s s l i m i t + l o w e r c l a s s l i m i t}{2}$
Taking 11.5 as assumed mean $a$ , we can find d_i, u_i and f_iu_i according to step deviation method as below.
$\begin{matrix} N u m b e r o f l e t t e r s & N u m b e r o f s u r n a m e s & x_{i} & x_{i} - a & u_{i} = \frac{x_{i} - a}{3} & f_{i} u_{i} 1 - 4 & 6 & 2.5 & - 9 & - 3 & - 18 4 - 7 & 30 & 5.5 & - 6 & - 2 & - 60 7 - 10 & 40 & 8.5 & - 3 & - 1 & - 40 10 - 13 & 16 & 11.5 & 0 & 0 & 0 13 - 16 & 4 & 14.5 & 3 & 1 & 4 16 - 19 & 4 & 17.5 & 6 & 2 & 8 T o t a l & 100 & - 106 \end{matrix}$
$\sum f_{i} u_{i} = - 106$
$\sum f_{i} = 100$
$M e a n ¯ x = a + (\frac{\sum f_{i} u_{i}}{\sum f_{i}}) \times h$
$= 11.5 + (\frac{- 106}{100}) \times 3$
$= 11.5 - 3.18 = 8.32$
We know that:
3 median = mode + 2 mean
3(8.05) = mode + 2(8.32)
24.15 - 16.64 = mode
7.51 = mode
So, median number and mean number of letters in surnames is 8.05 and 8.32 respectively while modal size of surnames is 7.51.

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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 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	6	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 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.

Q. 100 surnames were randomly picked up from a local telephone directory and the distribution of number of letters of the English alphabet 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 and mean number of letters in the surname. Also, find the modal size of surnames.

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