Calculate the arithmetic mean of the following series. Take 15 as the assumed average and use the short-cut method. Size: 20, 19, 18, 17, 16, 15, 14, 13, 12, 11 Frequency: 1, 2, 4... Calculate the arithmetic mean of the following series. Take 15 as the assumed average and use the short-cut method. Size: 20, 19, 18, 17, 16, 15, 14, 13, 12, 11 Frequency: 1, 2, 4, 8, 11, 10, 7, 4, 2, 1
Understand the Problem
The question asks to calculate the arithmetic mean of a given series of numbers, using 15 as the assumed average and applying the short-cut method.
Answer
$\bar{X} = 15.54$
Answer for screen readers
$\bar{X} = 15.54$
Steps to Solve
- Calculate the deviations from the assumed mean
The assumed mean is 15. We calculate the deviation $d_i$ for each size $x_i$ using the formula $d_i = x_i - A$, where $A$ is the assumed mean.
$d_1 = 20 - 15 = 5$ $d_2 = 19 - 15 = 4$ $d_3 = 18 - 15 = 3$ $d_4 = 17 - 15 = 2$ $d_5 = 16 - 15 = 1$ $d_6 = 15 - 15 = 0$ $d_7 = 14 - 15 = -1$ $d_8 = 13 - 15 = -2$ $d_9 = 12 - 15 = -3$ $d_{10} = 11 - 15 = -4$
- Multiply the deviations by their corresponding frequencies
Multiply each deviation $d_i$ by its corresponding frequency $f_i$: $f_1d_1 = 1 \times 5 = 5$ $f_2d_2 = 2 \times 4 = 8$ $f_3d_3 = 4 \times 3 = 12$ $f_4d_4 = 8 \times 2 = 16$ $f_5d_5 = 11 \times 1 = 11$ $f_6d_6 = 10 \times 0 = 0$ $f_7d_7 = 7 \times -1 = -7$ $f_8d_8 = 4 \times -2 = -8$ $f_9d_9 = 2 \times -3 = -6$ $f_{10}d_{10} = 1 \times -4 = -4$
- Calculate the sum of the products of deviations and frequencies
Calculate $\sum f_i d_i = 5 + 8 + 12 + 16 + 11 + 0 - 7 - 8 - 6 - 4 = 27$
- Calculate the sum of the frequencies
$\sum f_i = 1 + 2 + 4 + 8 + 11 + 10 + 7 + 4 + 2 + 1 = 50$
- Apply the short-cut method formula
The formula for the arithmetic mean using the short-cut method is:
$\bar{X} = A + \frac{\sum f_i d_i}{\sum f_i}$
where: $\bar{X}$ is the arithmetic mean, $A$ is the assumed mean, $\sum f_i d_i$ is the sum of the products of the deviations and their corresponding frequencies, $\sum f_i$ is the sum of the frequencies.
$\bar{X} = 15 + \frac{27}{50} = 15 + 0.54 = 15.54$
$\bar{X} = 15.54$
More Information
The short-cut method is a way to simplify the calculation of the arithmetic mean, especially when dealing with large numbers or a wide range of values. It reduces the computational complexity by working with deviations from an assumed mean rather than the original data.
Tips
A common mistake is to incorrectly calculate the deviations or to make errors in summing the frequencies or the products of frequencies and deviations. Double-checking these calculations is essential for accuracy.
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