GRE Exam: A Simple Guide To Solving Statistics Problems

Statistics refers to the properties of a set of data. For the purposes of the GRE, you can think of data as numerical pieces of information. For example, the number of students in a class is a data point, as is the average grade for a class, or the range of scores in a class.

Though statistics is a broad field within mathematics, on the GRE, you will be expected to understand the following statistical concepts:
- Mean
- Median
- Mode
- Range
- Standard deviation

Let’s look at a set of data to understand these concepts:
 

Example: The test scores for seven students in a class are 72, 90, 72, 83, 81, 63, and 94.

The median refers to the data point that has an equal number of data points greater than it and less than it. In other words, the median is the middle value in a set when the data points are listed in increasing order.

To determine the median, list the data points from least to greatest.
Listed in increasing order, the data points in the preceding list will read: 63, 72, 72, 81, 83, 90, 94.

Since there are seven data points, the median is the fourth data point.

In this case, the middle value is 81.

When the number of terms in a set is even, the median will be the average of the two middle terms.

Example: What is the median of 2, 8, 10, 11, 12, and 14?

SOLUTION: Since there are six data points, the median will be the average of the middle two values. In this case, the middle two values are 10 and 11.
To determine the median, find the average of 10 and 11. .
- The range refers to the positive difference between the largest and smallest value in a set. In the earlier example, the range is 94 – 63 = 31.
- The mode is the data point that appears most frequently in a set. In the earlier example, the mode is 72.
- The average of a set is the sum of the data points/the number of data points.

Put more simply:

where A = average of the set, S = sum of the set, and N = number of data points in the set.

Averages
Of all statistical topics, averages are tested most frequently. Unfortunately, most average questions will not be as simple as the preceding one. Usually, the GRE will give you the average for a set and expect you to solve for the values of one or more data points in a set or the number of items in a set.

Regardless of how the question is framed, you will always use the average formula: .

If the word “average” appears on the GRE, it will always be followed by “(arithmetic mean).”

For example, a question might ask: “What is the average (arithmetic mean) of 6 and 8?”

Don’t be swayed by the term arithmetic mean—it’s just a fancy term for average.

Example: If a company’s average yearly revenue over a 10-year period was $550,000, what was the company’s total revenue during that period?
SOLUTION: Since you want to solve for the sum, manipulate the average formula to isolate the sum:

Substitute the given values for A and N: 550,000 × 10 = $5,500,000.
Sometimes you will have to use the average formula multiple times in a question:

Example: The average height of eight students is 64 inches. If the average height of seven of the students is 62, how tall is the eighth student?
SOLUTION: Use the average formula to determine the sum of the heights of all eight students:

Next, use the average formula to determine the sum of the heights of seven of the students:

Let the height of the eighth student = h. You know that h + (the sum of the heights of the seven other students) = (the sum of the heights of the eight students). Thus h + 432 = 512 → h = 80.

An even more difficult example will use three averages in the question:
 

Example: A company’s average daily revenue over a 10-day period was $40,000. If the average daily revenue for the first 4 days was $25,000, what was the average daily revenue for the last 6 days?
SOLUTION: revenue for the first 4 days + revenue for the last 6 days = total revenue
Total revenue = A × N = $40,000 × 10 = $400,000
Revenue for the first four days = A × N = $25,000 × 4 = $100,000
Revenue for the last 6 days = A × 6 = 6A, where A is the average daily revenue for the last 6 days.

Thus:


Weighted Averages
When determining the average of two data points, the average will always fall in the middle of the two data points.

For example, the average of 30 and .


But what if one of the data points appears more often than the other data point?

For example: What is the average of 30, 30, and 40? In this case, the average .

Now that you have added an additional data point of 30 to the set, the average is weighted closer to 30 than it is to 40.


The preceding example represents a weighted average. You will have a weighted average any time the frequency of a data point pulls the average closer to that data point than to the other data point. In the preceding example, the fact that there were more 30s than 40s meant that the average was skewed more toward 30 than toward 40.

Quantitative Comparison Strategy
Weighted averages are tested most frequently in Quantitative Comparison questions.

For these questions, it is important to keep in mind the mandate that you must minimize calculations! Oftentimes, one of the quantities will be the average of the set if the number of data points were equal. Think about which data point the average is skewed toward, and you will cut back on time spent calculating.

Example: A student took 10 exams for his biology course. His average on 6 of the exams was 80. His average on the other 4 exams was 90.

Solution: Look at the columns! Notice that 85 is the average of 80 and 90. If 80 and 90 appeared with equal frequency, then the student’s average for the course would be 85. However, six of the data points correspond with the average of 80 and four of the data points correspond with the average of 90. The average for the course will thus be closer to 80 than to 90 and therefore less than 85. The correct answer is B.

Though it is tested less commonly, you should also know how to calculate a weighted average.

Let’s use the preceding example:
A student took 10 exams for his biology course. His average on 6 of the exams was 80. His average on the other 4 exams was 90. What was his average for the 10 exams?
- Approach 1: Use the average formula. For the 4 courses in which he averaged 90, the sum is 90(4). For the 6 courses in which he averaged 80, the sum is 80(6). The average for all 10 courses is thus .

- Approach 2: Use the weights.

Note that . The fractions represent the weight of each data point as a piece of the total weights of the set.

From this, you can extrapolate the following formula:

Note that this approach works when the number of data points is represented as a ratio or percent as well!

Example: For the first 10 days of a 30-day period, a stock’s average return was $6.00. For the last 20 days of the 30-day period, the stock’s average return was $12.00. What was the stock’s average return for the 30-day period?
SOLUTION: The two data points are $6.00 and $12.00.

The frequency of the $6.00 data point is .

The frequency of the $12.00 data point is .

The weighted average is thus .
 

More on the Median
Earlier, the section defined the median as the middle data point when the data points are in increasing order. Though calculating the median for small sets simply requires that you put the data points in increasing order and find the middle value, it becomes trickier when the set has a large number of data points.

For example, if you want the median of 51 data points, it would be too time-consuming to list all of them.

Instead, you should recognize that the median will be the 26th data point. Why? Because there will be 25 data points below this value and 25 data points above this value.

Example: A certain test was administered to 29 students. 6 students scored 60, 8 students scored 73, and 15 students scored 79. What was the median score for the 29 students?
SOLUTION: The median is the 15th-largest (or smallest) score. The smallest 6 data points have a value of 60. The next 8 data points have a value of 73. These two groups account for the first 14 data points. The next data point will thus equal the median. That data point occurs in the last set. Since all the data points in the last set are 79, the median is 79.

Standard Deviation
Standard deviation refers to how far from the mean the numbers in a set typically fall. Two sets can have the same mean and the same number of items, but completely different spreads from the mean. The concept of standard deviation is employed to represent this spread. The greater the average spread of the data, the greater the standard deviation.

Look at the following three sets, all of which have the same mean:


In the first set, all the data points are equal, so there is no spread from the mean. The standard deviation is thus zero. In the second set, the distance between successive numbers is one, so the standard deviation is greater than in the first set. In the third set, the distance between successive numbers is two, so the standard deviation is even greater than in the second set.

You might be wondering what the standard deviations of the first and second sets are. Fortunately, the GRE almost never tests the exact formula for standard deviation. It does, however, expect you to understand the concept and to be able to compare the standard deviation of different sets of numbers.

Example: Which of the following sets has the greatest standard deviation?
{4, 4, 4, 4, 4}
{3, 4, 4, 4, 5}
{3, 4, 4, 4, 8}
{0, 2, 4, 4, 12}
{1, 4, 4, 4, 7}

SOLUTION: Look at how the choices compare to A, which has a spread of zero. The further the values in a set move away from 4, the greater the standard deviation. The values in Choice D spread the most from 4. Thus the correct answer is D.