To find the mean using an assumed mean, we use the formula:

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To find the mean using an assumed mean, we use the formula:

\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i}

where $A$ is the assumed mean, $f_i$ is the frequency, and $d_i = x_i - A$ is the deviation of each data point $x_i$ from the assumed mean.

Identify the given data and the assumed mean.
The weights ( $x_i$ ) are 50 kg, 54 kg, 62 kg, and 65 kg.
The number of men ( $f_i$ ) for each weight are 15, 20, 25, and 30, respectively.
The assumed mean ( $A$ ) is 55 kg.

$x_i$ (Weight in kg)	$f_i$ (No. of men)	$d_i = x_i - 55$	$f_i d_i$
50	15	$50 - 55 = -5$	$15 \times (-5) = -75$
54	20	$54 - 55 = -1$	$20 \times (-1) = -20$
62	25	$62 - 55 = 7$	$25 \times 7 = 175$
65	30	$65 - 55 = 10$	$30 \times 10 = 300$

Calculate the sum of frequencies ( $\sum f_i$ ) and the sum of $f_i d_i$ ( $\sum f_i d_i$ ).
$\sum f_i = 15 + 20 + 25 + 30 = 90$ $\sum f_i d_i = -75 - 20 + 175 + 300 = -95 + 175 + 300 = 80 + 300 = 380$
Apply the formula for the mean.
$\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i}$ $\bar{x} = 55 + \frac{380}{90}$ $\bar{x} = 55 + \frac{38}{9}$ $\bar{x} = 55 + 4.222...$ $\bar{x} = 59.222...$
Rounding to two decimal places, the mean weight is 59.22 kg.

The mean weight is $59.22 \text{ kg}$ .

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