Mean, median, mode, range | Statistics and data interpretation | Quantitative reasoning

These basic statistical terms are extremely important in the GRE.

Definitions

Mean: The “simple average” of a set
Median: The middle number in a set
Mode: The number that occurs the most frequently in a set
Range: The difference between the largest and smallest numbers in a set
Interquartile range (IQR): We’ll cover this soon when we discuss standard deviation; for now, just don’t confuse it with the normal range!

Strictly speaking, a list and a set have different meanings. The elements in a list are ordered, whereas the elements in a set can be in any order. These terms are almost always interchangeable unless we’re working with probabilities.

Let’s go through some examples.

How to calculate the mode of a list or set

The mode is the most frequently occurring number in the set.

What is the mode of Set A $[2, 4, 4, 6, 7, 13]$ ?

(spoiler)

Answer: 4

There are two 4s and only one of each other number. This means 4 is the most frequently occurring number, and therefore is the mode.

How to calculate the range of a list or set

The range is the difference between the minimum and maximum numbers.

What is the range of Set A $[2, 4, 4, 6, 7, 13]$ ?

(spoiler)

Answer: 11

The minimum here is 2, and the maximum is 13, so the range is $13 - 2 = 11$ .

How to calculate the mean of a list or set

The mean is calculated by adding all the numbers and dividing by the total number of elements.

What is the mean of Set A $[1, 2, 3, 4, 6]$ ?

(spoiler)

Answer: 3.2

In our case, the sum of all numbers is $1 + 2 + 3 + 4 + 6 = 16$ , and there are $5$ numbers, so the mean is $16/5 = 3.2$

How to calculate the median of a list or set

The median is the middlemost value when the numbers are put in order.

What is the median of Set A $[1, 2, 3, 4, 6]$ ?

(spoiler)

Answer: 3

In our case, 3 occurs in the center of the list. Even if the list were in a different order, the median would not change. You would still put it in order first before finding the middlemost value.

What if we wanted to find the median of the list $[1, 2, 3, 4]$ ? There is no singular number in the center of the list, right? In this case, you need to find the mean of the two center numbers to find the median. The two middlemost numbers are $2$ and $3$ , and their mean is $(2 + 3) /2$ , so the median is $2.5$ .

Change in mean questions

A common question type asks for a new mean after a new number is added to a set. Or, you might be asked what number needs to be added to a set in order to change the mean to a specific value. Let’s start with the first variation:

If you scored a mean of 85 points on your first 4 quizzes, what would be the average of all five quizzes if you scored 90 on the final quiz?

To solve this question, we first need to determine how many points we collected over the first 4 quizzes.

$85 \times 4 = 340$

Then add the extra 90 points to this sum to find the sum points of all 5 quizzes.

$340 + 90 = 430$

To find the average of those 5 quizzes, we can divide this sum by 5.

$430/5 = 86$

So, the average score rises to 86 points when we add an additional quiz with a score of 90 points.

Let’s try a slightly different version using the same concepts.

You currently have an average score of 97 on your first four quizzes. If you wanted to get an average grade of at least 95 on the five quizzes, what score will you need on the fifth quiz?

Take a moment to see if you can solve it using the technique above.

(spoiler)

Answer: 87 or higher

First, find how many points are needed to score a 95 average. Multiply the average by the total number of quizzes.

$95 \times 5 = 475$

Next, we need to determine how many points we have scored so far by multiplying our current average by the number of quizzes we have already taken.

$97 \times 4 = 388$

Because we already have a total of 388 points, and we have only one more quiz to reach 475, the points needed on the final quiz to get an average of 95 is the difference between 475 and 388.

$475 - 388 = 87$

The final quiz must be at least an 87 for your average to be 95 or higher. You can also backtest this result, double-checking that it’s correct, by solving for the average given that the final score is 87.

$(97 + 97 + 97 + 97 + 87) /5 = 475/5 = 95$

It checks out!

How to calculate a weighted average

Weighted averages are used when the inputs are not of equal importance. But a weighted average, rather, sums up each part of the group but assigns a multiplier between 0 and 1 to each part of the group and then sums them up. For example, if an input is weighted at 25%, then one should multiply that value by .25. The rest of the values should also be multiplied by some number between 0 and 1. Finally, all of these values need to add up to one.

Have you ever taken a course that weighs your various scores differently? Perhaps the final was 50% of the grade, attendance was 10% of the grade, and homework was 40% of the grade. This is an example of weighted averages used in the real world. When solving for a weighted average, you should follow the steps below.

Make sure the weights add up to one.
Write an equation that solves for the sum of the products of each value and its respective weight
Multiply each value by its own weight
Add up all of the products

Try using the steps above to solve this practice problem.

A student just scored an 85 on the final of their economics course. Their average homework score was 82 and they attended 90% of the lectures. What is the student’s final grade if the professor weighs the final at 20% of the grade, attendance at 20%, and homework at 60%?

Are you ready for the answer?

(spoiler)

The student’s final grade is $84.2$

Let’s pull out the key information from the question:

$20%$ of the grade was the final, with a score of $85$
$60%$ of the grade was homework, with a score of $82$
$20%$ of the grade was attendance, with a score of $90$

And now we can put this into an equation and solve it.

$grade = 85 (.2) + 82 (.6) + 90 (.2) = 17 + 49.2 + 18 = 84.2$