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Q6:

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:

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.

Solution :

Given: A frequency distribution of the number of letters in 100 surnames.

Number of lettersNumber of surnames ($f_i$)
1 - 46
4 - 730
7 - 1040
10 - 1316
13 - 164
16 - 194

To Find: The Median, Mean, and Mode of the given data.

Step 1: Calculation of Mean ($\bar{x}$)

We use the Assumed Mean Method. Let the assumed mean $a = 11.5$. The class size $h = 3$.

Class IntervalFrequency ($f_i$)Class Mark ($x_i$)$d_i = x_i - a$$u_i = \frac{x_i - a}{h}$$f_i u_i$
1-462.5-9-3-18
4-7305.5-6-2-60
7-10408.5-3-1-40
10-131611.5000
13-16414.5314
16-19417.5628
Total$\sum f_i = 100$---$\sum f_i u_i = -106$

Formula for Mean: $\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$

$\bar{x} = 11.5 + \left( \frac{-106}{100} \right) \times 3 = 11.5 - 3.18 = 8.32$

Step 2: Calculation of Median

Total frequency $N = 100$. So, $N/2 = 50$.

Class IntervalFrequency ($f_i$)Cumulative Frequency ($cf$)
1-466
4-73036
7-104076

The cumulative frequency just greater than 50 is 76, so the median class is $7-10$.

Lower limit ($l$) = 7, $cf$ of preceding class = 36, $f$ of median class = 40, $h = 3$.

Median = $l + \left( \frac{N/2 - cf}{f} \right) \times h = 7 + \left( \frac{50 - 36}{40} \right) \times 3 = 7 + \left( \frac{14}{40} \right) \times 3 = 7 + 1.05 = 8.05$

Step 3: Calculation of Mode

The modal class is the class with the highest frequency, which is $7-10$ ($f_1 = 40$).

$l = 7, f_1 = 40, f_0 = 30, f_2 = 16, h = 3$.

Mode = $l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h = 7 + \left( \frac{40 - 30}{2(40) - 30 - 16} \right) \times 3$

Mode = $7 + \left( \frac{10}{80 - 46} \right) \times 3 = 7 + \left( \frac{10}{34} \right) \times 3 = 7 + 0.88 = 7.88$

Final Answer: The Mean number of letters is 8.32, the Median is 8.05, and the Mode is 7.88.


More Questions from Class 10 Mathematics Statistics EXERCISE 13.3


CBSE Solutions for Class 10 Mathematics Statistics


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