Suppose that we want to calculate the mean of a given frequency distribution. The shortcut method is a method that allows us to significantly reduce our calculations when finding the mean of the distribution. In this article, we explain the shortcut method formula and procedure and illustrate it via examples.
Suppose that we are given a frequency distribution with data values x_i and frequencies f_i respectively. Then the mean can be calculated by the usual method by the formula, \text{Mean }=\frac{\sum f_ix_i}{\sum f_i}. The quantity in the numerator can become very large making our calculations long and tedious. In such cases, we can shorten our calculations by applying the shortcut method which we explain below.
Shortcut Method for Mean Formula:
The formula to calculate the mean via the shortcut method is given as, \text{Mean }=A + \frac{\sum f_id_i}{\sum f_i}.
- Here, x_i represents the data values with frequencies f_i respectively.
- The number A is the assumed mean. It is a random data value chosen by the person doing the calculation. It is usually chosen to be the smallest or middlemost data value.
- The values d_i represent the difference between the actual data values and the assumed mean. We have that d_i = x_i - A.
Procedure to Apply Shortcut Method to Find Mean:
- If the distribution is grouped, then find the class midpoint for intervals. The class midpoints will be our x_i values in this case.
- Choose a random number as the assumed mean. It is usually taken to be the smallest or the middlemost data value.
- Subtract the value of A from the x_i‘s.
- Calculate the assumed mean by applying the formula. Let us now look at some examples.
Example 1:
We can find the mean of the following frequency distribution by shortcut method as follows,
| Data values (x_i) | Frequency (f_i) |
| 100 | 2 |
| 120 | 3 |
| 140 | 4 |
| 160 | 6 |
| 180 | 9 |
Let us take the assumed mean to be the smallest value, A = 100. Note that we are free to take any other value as the assumed mean.
| Data values (x_i) | Frequency (f_i) | (d_i = x_i - A) | (f_id_i) |
| 100 | 2 | 0 | 0 |
| 120 | 3 | 20 | 60 |
| 140 | 4 | 40 | 160 |
| 160 | 6 | 60 | 360 |
| 180 | 9 | 80 | 720 |
| \sum f_i = 24 | \sum f_id_i = 1300 |
The mean can now be calculated by the formula, \text{Mean }=A + \frac{\sum f_id_i}{\sum f_i} \text{Mean }=100 + \frac{1300}{24} = 100 + 54.1667 = 154.1667
Example 2:
Consider the following grouped frequency distribution,
| Class Intervals | Frequency (f_i) |
| 10-20 | 4 |
| 20-30 | 3 |
| 30-40 | 2 |
| 40-50 | 5 |
| 50-60 | 1 |
We first find the class marks (x_i) by taking the midpoints of the class intervals.
| Class Intervals | Class Mark (x_i) | Frequency (f_i) |
| 10-20 | 15 | 4 |
| 20-30 | 25 | 3 |
| 30-40 | 35 | 2 |
| 40-50 | 45 | 5 |
| 50-60 | 55 | 1 |
Let us take the assumed mean to be the middlemost value, A = 35.
| Data values (x_i) | Frequency (f_i) | (d_i = x_i - 35) | (f_id_i) |
| 15 | 4 | -20 | -80 |
| 25 | 3 | -10 | -30 |
| 35 | 2 | 0 | 0 |
| 45 | 5 | 10 | 50 |
| 55 | 1 | 20 | 20 |
| \sum f_i = 15 | \sum f_id_i = -40 |
The mean can now be calculated by the formula, \text{Mean }=A + \frac{\sum f_id_i}{\sum f_i} \text{Mean }=35 + \frac{(-40)}{15} = 35 - 2.1667 = 32.3333
