Calculate the average of the following discrete series using the short-cut method, assuming a mean of 25: Size: 30, 29, 28, 27, 26, 25, 24, 23, 22, 21 Frequency: 2, 4, 5, 3, 2, 7,... Calculate the average of the following discrete series using the short-cut method, assuming a mean of 25: Size: 30, 29, 28, 27, 26, 25, 24, 23, 22, 21 Frequency: 2, 4, 5, 3, 2, 7, 1, 4, 5, 7 Read more

Steps to Solve

1. Calculate the deviations $d_i$ from the assumed mean

We are given that the assumed mean $A = 25$. The deviation $d_i$ is calculated as $d_i = x_i - A$, where $x_i$ represents each size value.

$d_1 = 30 - 25 = 5$ $d_2 = 29 - 25 = 4$ $d_3 = 28 - 25 = 3$ $d_4 = 27 - 25 = 2$ $d_5 = 26 - 25 = 1$ $d_6 = 25 - 25 = 0$ $d_7 = 24 - 25 = -1$ $d_8 = 23 - 25 = -2$ $d_9 = 22 - 25 = -3$ $d_{10} = 21 - 25 = -4$

2. Calculate the product of frequencies and deviations $f_i d_i$

Multiply each frequency $f_i$ with the corresponding deviation $d_i$.

$f_1 d_1 = 2 \times 5 = 10$ $f_2 d_2 = 4 \times 4 = 16$ $f_3 d_3 = 5 \times 3 = 15$ $f_4 d_4 = 3 \times 2 = 6$ $f_5 d_5 = 2 \times 1 = 2$ $f_6 d_6 = 7 \times 0 = 0$ $f_7 d_7 = 1 \times -1 = -1$ $f_8 d_8 = 4 \times -2 = -8$ $f_9 d_9 = 5 \times -3 = -15$ $f_{10} d_{10} = 7 \times -4 = -28$

3. Calculate the sum of the products of frequencies and deviations $\sum f_i d_i$

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

4. Calculate the sum of the frequencies $\sum f_i$

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

5. Apply the short-cut method formula to calculate 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:

$$ \text{Mean} = 25 + \frac{-3}{40} $$ $$ \text{Mean} = 25 - 0.075 $$ $$ \text{Mean} = 24.925 $$