Find the mean of the following frequency distribution using the direct method.
Class-interval $0 - 10$ $10 - 20$ $20 - 30$ $30 - 40$ $40 - 50$
Frequency $12$ $16$ $6$ $7$ $9$

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Solution

Finding the mean:
Completing the table,
Class-interval Mid value $x_{i}$ Frequency $f_{i}$ $f_{i} x_{i}$
$0 - 10$ $5$ $12$ $60$
$10 - 20$ $15$ $16$ $240$
$20 - 30$ $25$ $6$ $150$
$30 - 40$ $35$ $7$ $245$
$40 - 50$ $45$ $9$ $405$
$Σ f_{i} = 50$ $Σ f_{i} x_{i} = 1100$
The formula to find the mean using the direct method is given by $\bar{x} = \frac{Σ f_{i} x_{i}}{Σ f_{i}}$ .
where $f_{i}$ is the frequency of observations and $x_{i}$ is the mid-value of the class-interval.
So, substituting the values in the formula,
$\Rightarrow \bar{x} = \frac{1100}{50} ∴ \bar{x} = 22$
Therefore, the mean of the frequency distribution is $22$ .

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Find the arithmetic mean of the following frequency distribution using the step deviation method.

Class	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
Frequency	$9$	$12$	$15$	$10$	$14$

Class Interval	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
Frequency	$7$	$8$	$12$	$13$	$10$

Class Interval	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$
Frequency	$10$	$6$	$8$	$12$	$5$	$9$

Class Interval	$0 - 9$	$10 - 19$	$20 - 29$	$30 - 39$	$40 - 49$	$50 - 50$	$60 - 60$
Frequency	$8$	$10$	$22$	$26$	$18$	$9$	$7$

The arithmetic mean of the following frequency distribution is $25$ . Determine the value of $p$ .

Class-interval	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$
Frequency	$5$	$18$	$15$	$p$	$6$

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