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

Steps to Solve

1. Calculate the deviations from the assumed mean for Problem 5

The assumed mean is 25. Calculate $d_i = x_i - A$, where $x_i$ is the size, and $A$ is the assumed mean (25). $d_1 = 30 - 25 = 5$ $d_2 = 29 - 25 = 4$ $d_3 = 28 - 25 = 3$ $d_4 = 27 - 25 = 2$ $d_5 = 26 - 25 = 1$ $d_6 = 25 - 25 = 0$ $d_7 = 24 - 25 = -1$ $d_8 = 23 - 25 = -2$ $d_9 = 22 - 25 = -3$ $d_{10} = 21 - 25 = -4$

2. Calculate $f_i d_i$ for Problem 5

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

3. Calculate $\sum f_i d_i$ for Problem 5

Sum all the $f_i d_i$ values. $\sum f_i d_i = 10 + 16 + 15 + 6 + 2 + 0 - 1 - 8 - 15 - 28 = -3$

4. Calculate $\sum f_i$ for Problem 5

Sum all the frequencies. $\sum f_i = 2 + 4 + 5 + 3 + 2 + 7 + 1 + 4 + 5 + 7 = 40$

5. Apply the short-cut formula for Problem 5

The formula for the average using the short-cut method is: $\text{Average} = A + \frac{\sum f_i d_i}{\sum f_i}$, where $A$ is the assumed mean. $\text{Average} = 25 + \frac{-3}{40} = 25 - 0.075 = 24.925$

6. Calculate $f_i x_i$ for Problem 6

Multiply each mark $x_i$ by its corresponding number of students $f_i$. $f_1 x_1 = 8 \times 15 = 120$ $f_2 x_2 = 4 \times 20 = 80$ $f_3 x_3 = 7 \times 22 = 154$ $f_4 x_4 = 3 \times 23 = 69$ $f_5 x_5 = 8 \times 27 = 216$ $f_6 x_6 = 7 \times 35 = 245$ $f_7 x_7 = 5 \times 18 = 90$

7. Calculate $\sum f_i x_i$ for Problem 6

Sum all the $f_i x_i$ values. $\sum f_i x_i = 120 + 80 + 154 + 69 + 216 + 245 + 90 = 974$

8. Calculate the average for Problem 6

The average is $\frac{\sum f_i x_i}{\sum f_i}$. We are given that the total number of students ($\sum f_i$) is 42. $\text{Average} = \frac{974}{42} \approx 23.19$