GMAT Statistics: Weighted Average and Evenly Spaced Sets
Arithmetic mean is a mathematical name for the average, a concept we should all be familiar with from early school maths lessons. The GMAT takes our basic understanding of the mean and tests it in multiple ways. Once we understand the basics of the arithmetic mean, we will be in a good position to answer any related questions the GMAT sets us. In this post we will discuss the concepts of weighted mean and mean of evenly spaced sets.
This article continues on from our introduction to the Arithmetic Mean, where we described the formula for the arithmetic mean as:
We will stick to the worded version of the formula as many GMAT related questions rely on our ability to understand how the equation actually works.
The weighted average questions are rather common on the GMAT. They are very similar to regular arithmetic mean questions, except in this type of question not all values in the set contribute equally to the mean.
To clarify things, let's look at an example:
There are two art classes, one with 4 students and one with 6. The average grade in Class A is 60 and the average grade in Class B is 70. The grades in each class on a test were:
Class A: 55, 65, 73, 47
Class B: 72, 79, 67, 65, 80, 57
What was the average/mean grade of all students combined?
The traditional way to solve this equation is to combine all the numbers:
A quicker way to achieve the same answer is to make use of each class's individual mean and 'weight' them. Looking back at out original equation and rearranging, the sum of the values in a set is defined by:
Sum of values in a set = Mean × Number of values in the set
The sum of values of class A can therefore be calculated as:
Sum of values in Class A = Mean × Number of values in the set
Sum of values in Class A = 60 × 4
Likewise the sum of values in Class B can be expressed as:
Sum of values in Class B = 70 × 6
Replacing this into our original equations gives us an easier route to the answer:
All we did is simplified the route to calculating the sum of values of the whole set. This is known as the weighted mean or weighted average. Quite often we will be given two distinct sets of data with different averages and asked to work out the combined average. The process is straightforward. We simply need to rearrange the original equation and understand that the sum of a set is defined as:
Sum of a set = Mean × Number of values in the set
Let us look at another example:
A shopkeeper sold 5 shirts. Three of the shirts cost $50 apiece; the other two shirts cost $100 apiece. What was the average cost of a shirt sold?
Again we can simply use the weighted mean method to calculate the answer quickly:
However, numbers and sizes in the examples above have been simplified. The GMAT will often give you much larger sets and increasingly difficult values for the weighted averages.
In such cases, we can use a rather cool shortcut to find the answer without complex calculations. When you are presented with two different averages, one is always greater than the other. If we think of the above example, there are five shirts, all of which have a value of at least $50. So we already know the average of all five shirts will be above $50. This understanding alone can help eliminate certain answer choices, but we can find an exact answer as well.
As we have determined, each shirt must cost at least $50, we are only interested in how much more than $50 the total average will be. We firstly look for the lowest shirt price. In this case it is $50, which we will now call our baseline cost.
We are only interested in the difference of each cost from this baseline, so subtract our baseline from each shirt cost. ($50 – baseline ($50) = 0 and $100 - baseline = $50)
Now using only the cost difference from our baseline, the equation becomes:
This shortcut does not give us the new average, but very quickly tells us how much larger than the baseline the new average will be. The equation above tells us that the new average is $20 above the baseline.
To recap the steps in this shortcut for weighted average:
1. Note your lowest average, as you will use this as your baseline.
2. Calculate the difference between each set mean and the baseline mean
3. Use the standard arithmetic mean equation but use difference from baseline calculated in step 2
4. Add the resulting answer in step 3 to the baseline you calculated in step 1 to get the total mean.
Evenly Spaced Sets
Another interesting shortcut we can use in arithmetic mean calculation involves evenly spaced sets. Evenly spaced sets are any type of set, where the distance between consecutive numbers is always constant.
This could be:
1, 2, 3, 4, 5 (difference 1)
100, 200, 300, 400, 500 (difference 100)
1, 3, 5, 7, 9 (difference 2)
To calculate the mean (or average) of any such set, you would normally add up all the values and divide by the total number of values in the set. Rather easy, until we get to GMAT type questions, which will typically involve large sets as in this example:
What is the average (arithmetic mean) of all integers from 200 to 400 inclusive?
Quite impractical to add up all those values! Thankfully, there is a really easy to remember formula to calculate the mean of any evenly spaced set. The formula is simple:
In this question the mean is therefore:
We can prove this works with an example, using the set:
1, 2, 3, 4, 5
The average using the traditional method is calculated as:
Using our shortcut we would simply calculate:
Try it with any evenly spaced set; it's a great shortcut and a very handy tool to add to your GMAT arsenal.
By setting rather strict time limits, GMAT tests your ability to apply the most efficient approach, so it is vital that you learn all relevant shortcuts and situations where they can be applied.