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In this 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in 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 number of median letters in the surnames. Find the number of mean letters in the surnames and also, find the size of modal in the surnames.

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Detailed Solution

To calculate median:

Class Interval	Frequency	Cumulative Frequency
1-4	6	6
4-7	30	36
7-10	40	76
10-13	16	92
13-16	4	96
16-19	4	100

Given:

n = 100 & $\frac{n}{2}$ = 50

Median class = 7-10

Therefore, l = 7, C_f = 36, f = 40 & h = 3

Ncert solutions class 10 chapter 14-6

Median = 7+( $\frac{50 - 36}{40}$ ) × 3

Median = 7+ $\frac{42}{40}$

Median=8.05

Calculate the Mode:

Modal class = 7-10,

Where, l = 7, f₁ = 40, f₀ = 30, f₂ = 16 & h = 3

Ncert solutions class 10 chapter 14-7

Mode = 7+( $\frac{40 - 30}{80 - 30 - 16}$ ) × 3

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

= 7.88

Therefore mode = 7.88

Calculate the Mean:

Class Interval	f_i	x_i	f_ix_i
1-4	6	2.5	15
4-7	30	5.5	165
7-10	40	8.5	340
10-13	16	11.5	184
13-16	4	14.5	51
16-19	4	17.5	70
	Sum f_i = 100		Sum f_ix_i = 825

Mean = x̄ = $\frac{\sum_{} f_{i} x_{i}}{\sum_{} f_{i}}$

Mean = $\frac{825}{100}$ = 8.25

Therefore, mean = 8.25

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