Calculate the average of the following discrete series. Use the shortcut method by taking 25 as the assumed average. Also, calculate the average of the marks secured by 42 students... Calculate the average of the following discrete series. Use the shortcut method by taking 25 as the assumed average. Also, calculate the average of the marks secured by 42 students in Economics with the following distribution:

Understand the Problem

The question requires calculating the average of a discrete series using the shortcut method, assuming 25 as the assumed average. Then it asks for the average of marks secured by students with given mark distribution.

Answer

Average of the discrete series $= 24.925$ Average of marks secured by students $= \frac{331}{42} \approx 7.88$

Steps to Solve

Calculate the deviations $d_i$ from the assumed mean

Here, we will calculate $d_i = x_i - A$, where $x_i$ represents the 'Size' values and $A$ is the assumed mean, which is 25.

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

Calculate the product of frequencies and deviations $f_i d_i$

Multiply each deviation $d_i$ by its corresponding frequency $f_i$.

$f_1d_1 = 2 \times 5 = 10$ $f_2d_2 = 4 \times 4 = 16$ $f_3d_3 = 5 \times 3 = 15$ $f_4d_4 = 3 \times 2 = 6$ $f_5d_5 = 2 \times 1 = 2$ $f_6d_6 = 7 \times 0 = 0$ $f_7d_7 = 1 \times -1 = -1$ $f_8d_8 = 4 \times -2 = -8$ $f_9d_9 = 5 \times -3 = -15$ $f_{10}d_{10} = 7 \times -4 = -28$

Calculate the sum of the products $\sum f_i d_i$

Sum all the $f_i d_i$ values we calculated.

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

Calculate the sum of the frequencies $\sum f_i$

Sum all the frequencies.

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

Apply the shortcut method formula to find the average

The formula for the shortcut method is:

$\text{Average} = A + \frac{\sum f_i d_i}{\sum f_i}$

Plug in the calculated values:

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

Calculate the average of marks secured by 42 students

Sum the marks and divide by the number of students. $\text{Average} = \frac{15 + 20 + 22 + 23 + 25 + 40 + 28 + 29 + 30 + 99}{42} = \frac{331}{42} \approx 7.88$

Average of the discrete series using the shortcut method: $24.925$ Average of marks secured by students: $\frac{331}{42} \approx 7.88$

More Information

The shortcut method simplifies average calculations for grouped data by using an assumed mean and deviations. The average marks calculation provides a simple average of given marks.

Tips

Incorrectly calculating deviations: Ensure you subtract the assumed mean correctly from each data point.
Errors in multiplying and summing: Double-check your multiplication and summation to avoid arithmetic errors.
Incorrectly applying the formula: Make sure you add the correction factor $\frac{\sum f_i d_i}{\sum f_i}$ to the assumed mean, not subtract or multiply it.
Misunderstanding assumed mean: Forgetting that shortcut method is best applicable for large datasets. Using normal average method could be faster for smaller dataset.

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