# Statistics - GMAT Math Study Guide

1. Definitions
2. Mean (Overview) · Mean (In-Depth)
3. Median (Overview) · Median (In-Depth)
4. Mode (Overview) · Mode (In-Depth)
5. Range (Overview)
6. Standard Deviation (Overview) · Standard Deviation (In-Depth)
7. Types of GMAT Problems

## Definitions

• Mean (aka Average or Arithmetic Mean) - The sum of all of the numbers in a list divided by the number of items in that list.
For example, the mean of the numbers 2, 6, 13 is 7 since the sum of the numbers (2+6+13=21) divided by the number of terms (3) is 7.
• Median - The number that is located in the middle of a set of numbers when that set is ordered sequentially from the smallest to the largest.
For example, the median of the numbers 2, 6, 4, 8, 10 is 6 since the middle of the set of numbers when ordered sequentially (i.e., 2, 4, 6, 8, 10) is 6.
• Mode - The value that appears most within a set of numbers.
For example, the mode of the set composed of 2, 10, 15, 21, 10, 2, 10 is 10 since the number 10 appears most often (i.e., 3 times).
• Range - The length of the interval containing all the data points in a set (i.e., the distance from the smallest to largest data point).
For example, the range of the set 10, 15, 38, 150 is 150-10 = 140.
• Standard Deviation - A measure of how spread out or dispersed the data in a set are relative to the set's mean.
For example, a data set with a standard deviation of 10 is more spread out than a data set with a standard deviation of 5.

## Mean

The formula for the mean of a list of numbers is the sum of all the numbers in the list divided by the number of numbers in the list. A = average (or arithmetic mean)
N = the number of terms (e.g., the number of items or numbers being averaged)
S = the sum of the numbers in the set of interest (e.g., the sum of the numbers being averaged)

Consider the following example:

If a student reads 2 books during each of the first two weeks of January, three books during each of the final two weeks of January, and a book a week for each of the four weeks of February, what is the average number of books the student read per week during January and February?

S = total number of books read = (2*2) + (3*2) + (1*4) = 14
N = the number of weeks = 2 + 2 + 4 = 8
A = 14/8 = 1.75

## Median

The median is the middle number in a set that is ordered sequentially (i.e., from smallest to largest). The most common mistake people make in calculating the median is forgetting to order the numbers in a set sequentially from smallest to largest before finding the middle number.

Set S: 2, 4, 6, 3, 9

It would be a mistake to conclude that the median is 6 since we must first order the numbers from smallest to largest.

Ordered Sequentially: 2, 3, 4, 6, 9
Median: 4

If there is an even number of elements, then the median is the average of the two elements in the middle. For example:

The median of the set {1, 2, 3, 4, 5, 6, 8, 10, 11} is 5 as there are 4 elements above and 4 elements below it.
The median of the set {5, 10, 15, 20, 25, 30} is 17.5, which is the average of 15 and 20.

## Mode

The mode of a set of data is the number (or numbers) that appear most frequently in the set of data. For example:

Set K: 2, 4, 6, 8, 5, 3, 4
The mode is 4 since this number appears twice (more than any other number).

It is possible to have multiple modes but impossible for a set of data to have no mode. For example:

The set {1, 2, 2, 1, 2} has one mode, 2
The set {1, 2, 2, 1, 2, 1} has two modes, 1 and 2

## Range

The range is the distance from the smallest to largest number in a set. For example:

Set R: 14, 10, 19, 143, 180
Range = largest - smallest = 180 - 10 = 170

Another Example:

The range for the set {1, 29, 3, 17, 54, 11} is 54-1 = 53

## Standard Deviation

The standard deviation measures the dispersion (i.e., the extent to which the data are spread out) relative to the mean.

While it is extremely rare that you need the formula for the standard deviation, it is offered here in order to give a better understanding of what the standard deviation of a data set measures.  (For students with a background in statistics, different notation can be used depending on whether one is calculating the population or sample standard deviation. For details on how to calculate the standard deviation and an example of finding the standard deviation of a set, please see the detailed standard deviation study guide.)

The most important part of understanding standard deviations is knowing that as the standard deviation increases, the dispersion of the data increases.

The standard deviation of the red graph is much larger than that of the blue graph.  The following is an example of how the standard deviation might be tested:

In a normally distributed set of data, about 99% of data is contained within the area encompassed by 2.5 standard deviations under the mean to 2.5 standard deviations above the mean. If Set S is normally distributed with a mean of 10 and a standard deviation of 1, what interval encompasses about 99% of data?
Lower Boundary: Mean - 2.5(Standard Deviation) = 10 - 2.5(1) = 7.5
Upper Boundary: Mean + 2.5(Standard Deviation) = 10 + 2.5(1) = 12.5

## Types of GMAT Problems

1. Calculating the Mean With Large Values

Make an approximation as to what the average will be. For each number in the set of data, determine the difference from the guessed average. Sum the deviations and divide by the total number of elements in the data set to find the average difference. Add the average difference to the guessed average to determine the true average.

Find the average of 111, 71, 98, 105, 92, 87
 A) 90 B) 98 C) 94 D) 95 E) 96
1. Make an approximation as to what the average is.
Average ≅ 95
Note: There is nothing magical about choosing 95. If you choose 96 or 93 or another number, you would still arrive at the correct answer.
2. Calculate the difference between 95 and each value in the set of data.
111-95 = 16
71-95 = -24
98-95 = 3
105-95 = 10
92-95 = -3
87-95 = -8
3. Add all of the differences together and divide by the number of numbers in the set, 6
Sum of Differences: 16-24+3+10-3-8 = -6
Sum of Differences/Numbers: -6/6 = -1
4. Add -1, the sum of differences/numbers, to 95, the initial estimate
95 + (-1) = 95-1 = 94
5. In case you are struggling with the choice of 95, consider how this problem could be solved by choosing other numbers as an estimate of the mean.
Estimate: 96
Sum of Differences: -12
Sum of Differences/6: -2
Mean: 96 + (-2) = 94

Estimate: 100
Sum of Differences: -36
Sum of Differences/6: -6
Mean: 100 + (-6) = 94