Mean, Median, and Mode Study Guide (page 3)
Introduction to Mean, Median, and Mode
Poor teaching leads to the inevitable idea that the subject (mathematics) is only adapted to peculiar minds, when it is the one universal science and the one whose… ground rules are taught us almost in infancy and reappear in the motions of the universe.
—HENRY J. S. SMITH, Irish mathematician (1826–1883)
This lesson focuses on three numbers that researchers often use to represent their data. These numbers are sometimes referred to as measures of central tendency. Translated to English, that simply means that these numbers are averages. This lesson defines mean, median, and mode; explains the differences among them; and shows you how to use them.
An average is a number that typifies or represents a group of numbers. You come into contact with averages on a regular basis—your batting average, the average number of pizza slices you can eat at one sitting, the average number of miles you drive each month, the average income for entry-level programmers, the average number of students in a classroom, and so forth.
There are actually three different numbers that typify a group of numbers:
- the mean
- the median
- the mode
Most of the time, however, when you hear people mention the average, they are probably referring to the mean. In fact, whenever this book uses the word average, it refers to the mean.
Let's look at a group of numbers, such as the number of students in a classroom at the Chancellor School, and find these three measures of central tendency.
The mean (average) of a group of numbers is the sum of the numbers divided by the number of numbers:
|Example:||Find the average number of students in a classroom at the Chancellor School.|
|Average = 17|
The average (mean) number of students in a classroom at the Chancellor School is 17. Do you find it curious that only two classrooms have more students than the average or that the average isn't right smack in the middle of things? Read on to find out about a measure that is right in the middle of things.
The median of a group of numbers is the number in the middle when the numbers are arranged in order. When there is an even number of numbers, the median is the average of the two middle numbers.
|Example:||Find the median number of students in a classroom at the Chancellor School.|
|Solution:||Simply list the numbers in order (from low to high or from high to low) and identify the number in the middle:|
Did you know that median is the most common type of average used to measure the price of homes in the real estate market? The median is a helpful tool because it is protected against skewed data that is very far from the true center number. (Like a neighborhood where most of the houses cost around $200,000, but there's one house for $750,000 because it has a horse stable.)
Had there been an even number of classrooms, then there would have been two middle numbers:
With ten classrooms instead of nine, the median is the average of 15 and 16, or , which is also halfway between the two middle numbers.
If a number above the median is increased significantly or if a number below the median is decreased significantly, the median is not affected. On the other hand, such a change can have a dramatic impact on the mean—as did the one classroom with 30 students in the previous example. Because the median is less affected by quirks in the data than the mean, the median tends to be a better measure of central tendency for such data.
Consider the annual income of the residents of a major metropolitan area. A few multimillionaires could substantially raise the average annual income, but they would have no more impact on the median annual income than a few above-average wage earners. Thus, the median annual income is more representative of the residents than the mean annual income. In fact, you can conclude that the annual income for half the residents is greater than or equal to the median, while the annual income for the other half is less than or equal to the median. The same cannot be said for the average annual income.
The mode of a group of numbers is the number that appears most often.
Example: Find the mode the most common classroom size, at the Chancellor School.
Solution: Scanning the data reveals that there are more classrooms with 15 students than any other size, making 15 the mode:
Had there also been three classrooms of, say, 16 students, the data would be bimodal—both 15 and 16 are the modes for this group:
On the other hand, had there been an equal number of classrooms of each size, the group would NOT have a mode—no classroom size appears more frequently than any other:
|Hook:||Here's an easy way to remember the definitions of median and mode.|
|Median: Picture a divided highway with a median running right down the middle of the road.|
|Mode: The MOde is the MOst popular member of the group.|
Use these tricks to remember what each term means:
If there's an even-spacing pattern in the group of numbers being averaged, you can determine the average without doing any arithmetic! For example, the following group of numbers has an even-spacing "3" pattern: Each number is 3 greater than the previous number:
The average is 18, the number in the middle. When there is an even number of evenly spaced numbers in the group, there are two middle numbers, and the average is halfway between them:
This shortcut works even if each number appears more than once in the group, as long as each number appears the same number of times, for example:
You could have used this method to figure out sample question 1. Order Barbara's grades from low to high to see that they form an evenly spaced "2" pattern:
Thus, Barbara's average grade is the number in the middle, 94.
In a weighted average, some or all of the numbers to be averaged have a weight associated with them.
Example: Don averaged 50 miles per hour for the first three hours of his drive to Seattle. When it started raining, his average fell to 40 miles per hour for the next two hours. What was his average speed?
You cannot simply compute the average of the two speeds as , because Don spent more time driving 50 mph than he did driving 40 mph. In fact, Don's average speed is closer to 50 mph than it is to 40 mph precisely because he spent more time driving 50 mph. To correctly calculate Don's average speed, we have to take into consideration the number of hours at each speed: 3 hours at an average of 50 mph and 2 hours at an average of 40 mph, for a total of 5 hours.
Or, take advantage of the weights, 3 hours at 50 mph and 2 hours at 40 mph:
Write down your age on a piece of paper. Next to that number, write down the ages of five of your friends or family members. Find the mean, median, and mode of the ages you've written down. Remember that some groups of numbers do not have a mode. Does your group have a mode?
- This term, Barbara's test scores are 88, 96, 92, 98, 94, 100, and 90. What is her average test score?
Find the average test score, the median test score, and the mode of the test scores for the 30 students represented in the following table.
Number of Students Test Score 1 100 3 95 6 90 8 85 5 80 4 75 2 70 1 0
Solutions to Sample Questions
Calculate the average by adding the grades together and dividing by 7, the number of tests:
Average = 94
Use the number of students achieving each score as a weight:
Even though one of the scores is 0, it must still be accounted for in the calculation of the average.
Since the table is already arranged from high to low, we can determine the median merely by locating the middle score. Since there are 30 scores represented in the table, the median is the average of the 15th and 16th scores, which are both 85. Thus, the median is 85. Even if the bottom score, 0, were significantly higher, say 80, the median would still be 85. However, the mean would be increased to 84.1. The single, peculiar score, 0, makes the median a better measure of central tendency than the mean.
Just by scanning the table, we can see that more students scored an 85 than any other score. Thus, 85 is the mode. It is purely coincidental that the median and mode are the same.
Find practice problems and solutions for these concepts at Mean, Median, and Mode Practice Questions.