Calculate the average of the discrete series using the short-cut method by taking 25 as the assumed average.

Steps to Solve

1. Calculate the deviations from the assumed mean

Let $A$ be the assumed mean, which is $25$. The deviation $d_i$ for each size $x_i$ is calculated as $d_i = x_i - A$.

Here are all the deviations: $30 - 25 = 5$ $29 - 25 = 4$ $28 - 25 = 3$ $27 - 25 = 2$ $26 - 25 = 1$ $25 - 25 = 0$ $24 - 25 = -1$ $23 - 25 = -2$ $22 - 25 = -3$ $21 - 25 = -4$

2. Multiply each deviation by its corresponding frequency

Multiply each $d_i$ by its corresponding frequency $f_i$ to get $f_i d_i$.

Here are the $f_i d_i$ values: $5 * 2 = 10$ $4 * 4 = 16$ $3 * 5 = 15$ $2 * 3 = 6$ $1 * 2 = 2$ $0 * 7 = 0$ $-1 * 1 = -1$ $-2 * 4 = -8$ $-3 * 5 = -15$ $-4 * 7 = -28$

3. Calculate the sum of the products of frequency and deviations

Calculate $\sum f_i d_i$ by adding up all the values calculated in Step 2.

$\sum f_i d_i = 10 + 16 + 15 + 6 + 2 + 0 - 1 - 8 - 15 - 28 = -3$

4. Calculate the sum of the frequencies

Calculate $\sum f_i$. $\sum f_i = 2 + 4 + 5 + 3 + 2 + 7 + 1 + 4 + 5 + 7 = 40$

5. Apply the short-cut formula for calculating the mean

The formula for the mean using the short-cut method is: $\text{Mean} = A + \frac{\sum f_i d_i}{\sum f_i}$

Substitute the values we calculated: $\text{Mean} = 25 + \frac{-3}{40} = 25 - 0.075 = 24.925$