GMAT Tip of the Week: What Does the Median Mean?

GMAT Tip of the WeekStatistics-based GMAT questions can be tricky, particularly for those who haven’t been formally trained in stats or for those whose knowledge of statistics is more incomplete than they realize. One concept for which many students have blind spots is that of the median, so let’s take a moment to identify and explain a few of these common knowledge gaps.

For starters, everyone taking the GMAT likely knows that the median is the “middle number” in a set of data, and that to find that middle number you have to first sort the values in order. So for a set:

{2, 5, 1, 3, 4}

The median isn’t 1, the middle number as it’s displayed above, but rather 3, the middle number once the data set has been sorted in order as: {1, 2, 3, 4, 5}.

Further, everyone likely knows that if there are an even number of terms, the median is the average of the two middle terms. So in the set {2, 4, 6, 8, 10, 12}, the median is the average of 6 and 8, which is 7.

But here are a few things that people who flip through that section of their prep book and think “median, yeah that’s easy” often tend to overlook and miss:

1) The median doesn’t have to be BETWEEN two other numbers

Take the set {7, 10, 12, x, 17, 19, 22}, in which x is defined as the median. You might look at that and think “oh, x has to be between 12 and 17”. But that’s not necessarily true. If x were 12, the set would list as: {7, 10, 12, (x =) 12, 17, 19, 22} and 12 would still be the 4th out of 7 numbers. x could match either 12 or 17.

This can be extremely important on Data Sufficiency questions. If the question were “Is x > 12?” and you were told that x is the median of that set, you’d be very much tempted to say that it has to be greater than 12. But remember – the median of a 7 term set doesn’t have to be between the 3rd and 5th terms; it could match one of those.

2) The median also means “an equal number of terms above and below”

This definition works with “the middle number” but it adds another level of conceptual understanding that can help you on challenging problems. Consider the problem:

Sets A, B, and C are combined together to create set J. What is the median of set J?

1. The medians of sets A, B, and C are all 25
2. Sets A, B, and C each have the same number of terms

You might very well be tempted to think that you need statement 2, but as it turns out statement 1 is sufficient alone based on that extra definition above. If sets A, B, and C each equally divide their terms above and below 25, then 25 will remain the median of the new combined set. You can see it with numbers:

Set A: {21, 23, 25, 27, 29} — two terms below 25, two terms above 25, one term is exactly 25
Set B: {24, 26} — one term below 25, one term above 25

Just seeing this, you should note that the new set will now have three terms below 25, three terms above 25, and the middle term is 25. Or if all the sets had an even number of values:

Set B: {24, 26}
Set C: {10, 20, 30, 40}

Note that when you combine them, the “innermost” pair of numbers that will form the middle two both come from Set B, keeping the median at 25. Any way you do this, the average will stay at 25.

So remember – in addition to “the middle term”, median also means “an equal number of values above and below”.

3) In an evenly spaced set, the median equals the mean

When a set is evenly spaced (such as “consecutive even numbers” or “consecutive multiples of 7”, the median and the mean will be the same. So in a set like:

{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}

if you want to find the average, you don’t actually have to add up all 15 values and divide by 15, you can instead just find the 8th value and that will be the average. The average will be 16.

Where this can be extremely helpful is when you’re asked to determine the sum of a set of evenly spaced values, such as in the question:

What is the sum of all the even numbers between 0 and 100, inclusive?

You can here use the average calculation that Average = Sum of values / Number of values, solving for the sum. You know that the average will be the middle number, 50, so then you just have to find the number of values (which is 51, calculated as the range (100) divided by 2 (since you only want every other number), plus one for “inclusive”). The answer is then 51*50, or 2550.

Knowing that you can use the median to your advantage this way can save you valuable time.

In summary, don’t let your knowledge of Median stop at just “the middle number”. It’s more than that, and savvy test takers can raise their score well above the GMAT’s median value by taking advantage of a more thorough understanding.

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By Brian Galvin