## Easy Mathematics Step-by-Step (2012)

### Chapter 19. Mean, Median, Mode, and Range

In this chapter, you learn how to calculate the mean, median, mode, and range for a set of numbers. The mean, median, and mode are measures of central tendency. A *measure of central tendency* is a numerical value that describes a set of numbers (such as students’ scores on a test) by attempting to provide a “central” or “typical” value of the numbers. The range is a *measure of variability*. It gives you an idea of the spread of a set of numbers.

**Mean**

The *mean* of a set of numbers is another name for the arithmetic average of the numbers in the set. To calculate the mean: first, sum the numbers; then, divide by how many numbers are in the set. Thus, you have the following formula:

Remember that the fraction bar indicates division.

The braces around a list of numbers mean that the numbers belong together as a set. You read {1,3,5} as “the set consisting of the numbers 1, 3, and 5.”

**Problem** Find the mean for the given set of numbers.

**Solution**

*Step 1*. Sum the numbers.

*Step 2*. Divide by 5.

When computing a mean, don’t forget to divide by how many numbers you have.

If all the numbers are positive, then the mean of the numbers is also positive.

*Step 3*. State the answer.

*Step 1*. Sum the numbers.

*Step 2*. Divide by 6.

*Step 3*. State the answer.

*Step 1*. Sum the numbers.

*Step 2*. Divide by 8.

*Step 3*. State the answer.

*Step 1*. Sum the numbers.

*Step 2*. Divide by 6.

When you compute a mean, you divide by how many numbers you have, *including the number of zeros*, if any. Don’t forget to count the zeros.

*Step 3*. State the answer.

*Step 1*. Sum the numbers.

*Step 2*. Divide by 7.

*Step 3*. State the answer.

As you might expect, if all the numbers are the same, the mean of the numbers is the common number.

*Step 1*. Sum the numbers.

*Step 2*. Divide by 6.

*Step 3*. State the answer.

If all the numbers are negative, then the mean of the numbers is also negative.

**Weighted Average**

A *weighted average* (or *weighted mean*) is an average computed by assigning weights to the numbers (e.g., test scores).

**Problem** A student scores 50, 40, and 96 on three exams. Find the weighted average of the student’s three scores, where the score of 50 counts 20%, the score of 40 counts 30%, and the score of 96 counts 50%.

**Solution**

*Step 1*. Multiply each score by its corresponding “weight” and then add.

*Step 2*. Sum the weights.

*Step 3*. Divide the sum in step 1 by the sum in step 2.

*Step 4*. State the answer.

The student’s weighted average is 70.

**Median**

The *median* is the middle value or the arithmetic average of the two middle values in an *ordered* set of numbers. To find the median, do the following:

1. Put the numbers in order from least to greatest (or greatest to least).

2. Locate the middle number. If there is no single middle number, compute the arithmetic average of the two middle numbers.

**Problem** Find the median.

**Solution**

*Step 1*. Put the numbers in order from least to greatest.

*Step 2*. Locate the middle number.

*Step 3*. State the answer.

*Step 1*. Put the numbers in order from least to greatest.

*Step 2*. Locate the middle number.

There is no single middle number, so compute the arithmetic average of the two middle numbers, 1 and 8.

*Step 3*. State the answer.

*Step 1*. Put the numbers in order from least to greatest.

*Step 2*. Locate the middle number.

There is no single middle number, so compute the arithmetic average of the two middle numbers, 7.5 and 7.6.

*Step 3*. State the answer.

*Step 1*. Put the numbers in order from least to greatest.

*Step 2*. Locate the middle number.

There is no single middle number, so compute the arithmetic average of the two middle numbers, 0 and 100.

*Step 3*. State the answer.

*Step 1*. Put the numbers in order from least to greatest.

*Step 2*. Locate the middle number.

*Step 3*. State the answer.

As with the mean, if all the numbers are the same, the median of the numbers is the common number.

*Step 1*. Put the numbers in order from least to greatest.

*Step 2*. Locate the middle number.

There is no single middle number, so compute the arithmetic average of the two middle numbers, –50 and –50.

If the two middle numbers are the same, then the arithmetic average is the common number.

*Step 3*. State the answer.

As with the mean, if all the numbers are negative, then the median of the numbers is also negative.

**Mode**

In a set of numbers, the *mode* is the number (or numbers) that occurs most often. A set of numbers in which each number occurs the same number of times has *no mode*. If only one number occurs most often, then the set is *unimodal*. If exactly two numbers occur with the same frequency that is more often than any of the other numbers, then the set is *bimodal*. If three or more numbers occur with the same frequency that is more often than any of the other numbers, then the set is *multimodal*.

**Problem** Find the mode, if any. For number sets that have modes, state whether the data set is unimodal, bimodal, or multimodal.

**Solution**

*Step 1*. For each number, determine how often it occurs.

*Step 2*. State the mode, if any.

*Step 1*. For each number, determine how often it occurs.

*Step 2*. State the mode, if any.

*Step 1*. For each number, determine how often it occurs.

*Step 2*. State the mode, if any.

*Step 1*. For each number, determine how often it occurs.

*Step 2*. State the mode, if any.

*Step 1*. For each number, determine how often it occurs.

*Step 2*. State the mode, if any.

*Step 1*. For each number, determine how often it occurs.

*Step 2*. State the mode, if any.

**Range**

The *range* describes the spread of a set of numbers. You find the range by computing the difference between the greatest number and the least number; that is,

Range = greatest number – least number

The range of a set of number is *always* nonnegative.

**Problem** Find the range.

**Solution**

*Step 1*. Identify the greatest number and the least number.

*Step 2*. Compute the range.

*Step 3*. State the answer.

*Step 1*. Identify the greatest number and the least number.

*Step 2*. Compute the range.

*Step 3*. State the answer.

*Step 1*. Identify the greatest number and the least number.

*Step 2*. Compute the range.

*Step 3*. State the answer.

*Step 1*. Identify the greatest number and the least number.

*Step 2*. Compute the range.

*Step 3*. State the answer.

*Step 1*. Identify the greatest number and the least number.

*Step 2*. Compute the range.

*Step 3*. State the answer.

If all the numbers are the same, then the range of the numbers is 0.

*Step 1*. Identify the greatest number and the least number.

*Step 2*. Compute the range.

*Step 3*. State the answer.

**Exercise 19**

*For 1–4, find the mean*.

__1.__ {15, 33, 30, 50, 0}

__2.__ {–4, 25, –4, 11, 19, 4}

__3.__ {4.7, 5.6, 2.5, 4.9, 7.3, 4.7, 5.6, 6.5}

__4.__ {–10, 0, 3, 3, 6, 16}

__5.__ A student scores 80, 90, and 70 on three exams. Find the weighted average of the student’s three scores, where the score of 80 counts 10%, the score of 90 counts 30%, and the score of 70 counts 60%.

*For 6–10, find the median*.

__6.__ {15, 33, 30, 50, 0}

__7.__ {–4, 25, –4, 11, 19, 4}

__8.__ {4.7, 5.6, 2.5, 4.9, 7.3, 4.7, 5.6, 6.5}

__9.__ {–10, 0, 3, 3, 6, 16}

__10.__ {1, 1, 4, 4, 4, 10, 10, 10}

*For 11–15, find the mode, if any. For number sets that have modes, state whether the set is unimodal, bimodal, or multimodal*.

__11.__ {15, 33, 30, 50, 0}

__12.__ {–4, 25, –4, 11, 19, 4}

__13.__ {4.7, 5.6, 2.5, 4.9, 7.3, 4.7, 5.6, 6.5}

__14.__ {–10, 0, 3, 3, 6, 16}

__15.__ {1, 1, 4, 4, 4, 10, 10, 10}

*For 16–20, find the range*.

__16.__ {15, 33, 30, 50, 0}

__17.__ {–4, 25, –4, 11, 19, 4}

__18.__ {4.7, 5.6, 2.5, 4.9, 7.3, 4.7, 5.6, 6.5}

__19.__ {–10, 0, 3, 3, 6, 16}

__20.__ {150, 330, 300, 500, 0}