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Averages - Fundamentals

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Average can be defined as a single value that is meant to typify a data set. It gives a measure of the middle or expected value of the data set. Though there are many measures of central tendency, average typically refers to the arithmetic mean and is defined as the ratio of sum of items to the number of items in a dataset.

Average = Total value of all the items / Number of items

For example, let’s take the heights of the students in a class. If the number of students is say 5, and their heights are 168, 170, 169, 174 and 166, then average can be calculated using the above formula as:
Average = 847/5 = 169.4

We can also solve the above problem by using the deviations method, which uses the concept that the sum of individual deviations from the average is always equal to zero.

In the above example, for using the deviations method, we need to assume a value as the average. Depending on the closeness of the assumed average, the calculations will become simpler. Hence, it is always better to assume some number which is in between the smallest and the largest values in the data set as the real average will also lie between these values.

Now, let’s assume that the average in the above case is 170. Now, actual average will be the sum of the assumed average and the average of deviations from the assumed average.

Hence, Actual average = Assumed average + Average of deviations of the data points from the assumed average
 = 170 + [(-2) + 0 + (-1) + 4 + (-4)] / 5
= 170 - 3/5 = 169.4

Change in averages
If the values of all the elements in a group are increased or decreased by the same value, the average of the group will also increase or decrease by the same value.
If the values of all the elements in a group are multiplied or divided by the same value, the average of the group will also get multiplied or divided by the same value.

Important Concepts in Averages

1. Instead of using the normal approach towards solving problems, deviation method is much faster and accurate, because it removes large numbers from the equation and looks at only the smaller deviations from the assumed mean. These deviations can be extended to solve any problem involving averages.
2. Given no other information, assume that all the numbers are at the same level as their average.
3. When a new number higher than the average is added to the group, then the average is bound to increase and vice versa. The increase in average will amount to
= Difference between the new number and average / (Original number of items + 1)
4. When a new number lesser than the average is added to the group, then the average is bound to decrease and vice versa. The decrease in average will amount to
= Difference between average and the new number / (Original number of items + 1)
5. When a number is replaced by a greater number average will increase and if it is replaced by a smaller number average will decrease. The increase or decrease in average can be calculated using the below formula.
Net increase in the sum (Diff. between new value & original value) / Original number of items
6. Typically, people tend to confuse things when they encounter a few cases where the result of their calculations for average returns counterintuitive numbers.



When we give it a superficial look, it is intuitive that the overall admission success rate for women should be much better than the admission success rate for men. However, when you calculate the weighted average admission success rate with the weights being the number of applicants, the results can be summed up as follows:

As we can see, there seems to be a significant difference between the overall admission success rate for women and men, with men being more successful. This is very counterintuitive to the original presumption and hence, a candidate should always guard against such problems which produce counter-intuitive results.



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