The following table gives the distribution of 100 families according to daily expenditure:
| Expenditure (in Rs) | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Number of families | 14 | $x$ | 27 | $y$ | 15 |
EXPLANATION
Measures of Central Tendency: Arithmetic Mean, Geometric Mean, Harmonic Mean, Median and Mode
- Step 1: Use total frequency $N = 100$:
$$14 + x + 27 + y + 15 = 100 \implies 56 + x + y = 100 \implies x + y = 44$$ - Step 2: Use Mode formula.
Given Mode = 24, which lies in the class interval 20-30. Hence, the modal class is 20-30:
Lower limit of modal class ($L$) = 20
Class width ($h$) = 10
Frequency of modal class ($f_1$) = 27
Frequency of pre-modal class ($f_0$) = $x$
Frequency of post-modal class ($f_2$) = $y$
Mode formula:
$$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$
$$24 = 20 + \left(\frac{27 - x}{2(27) - (x + y)}\right) \times 10$$
$$4 = \left(\frac{27 - x}{54 - 44}\right) \times 10 = \left(\frac{27 - x}{10}\right) \times 10 = 27 - x$$
$$4 = 27 - x \implies x = 27 - 4 = 23$$ - Step 3: Find $y$:
$$y = 44 - x = 44 - 23 = 21$$
Thus, $x = 23$ and $y = 21$.
- चरण 1: कुल आवृत्ति $N = 100$ से:
$$14 + x + 27 + y + 15 = 100 \implies x + y = 44$$ - चरण 2: बहुलक सूत्र का प्रयोग:
बहुलक = 24 वर्ग 20-30 में स्थित है, अतः बहुलक वर्ग 20-30 है:
$L = 20, h = 10, f_1 = 27, f_0 = x, f_2 = y$
बहुलक सूत्र:
$$\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - (x + y)} \times h$$
$$24 = 20 + \frac{27 - x}{54 - 44} \times 10$$
$$4 = \frac{27 - x}{10} \times 10 = 27 - x \implies x = 23$$ - चरण 3: $y$ का मान:
$$y = 44 - 23 = 21$$
अतः $x = 23$ और $y = 21$ (विकल्प 3)।
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