Introduction: Connecting Your Learning

In real-world applications, you can use tables and graphs of various kinds to show information and to extract information from data that can lead to analyses and predictions. Graphs allow you to communicate a message from data.

Measures of central tendency are a key way to discuss and communicate with graphs. The term central tendency refers to the middle, or typical, value of a set of data, which is most commonly measured by using the three m's: mean, median, and mode. The mean, median, and mode are known as the measures of central tendency. In this lesson, you will explore these three concepts.

Focusing Your Learning

Key Terms

Presentation

The Mean, Median, and Mode

Mean, median, and mode are three basic ways to look at the value of a set of numbers. You will start by learning about the mean.

For instance, take a look at the following example. Use the formula to calculate the mean number of hours that Stephen worked each month based on the example below.

Example
Problem	Stephen has been working on programing and updating a Web site for his company for the past 15 months. The following numbers represent the number of hours Stephen has worked on this Web site for each of the past 7 months: 24, 25, 31, 50, 53, 66, 78 What is the mean (average) number of hours that Stephen worked on this Web site each month?
	Step 1: Add the numbers to determine the total number of hours he worked. 24 + 25 + 33 + 50 + 53 + 66 + 78 = 329
	Step 2: Divide the total by the number of months.
Answer Move Close Print The mean number of hours that Stephen worked each month was 47.

The calculations for the mean of a sample and the total population are done in the same way. However, the mean of a population is constant, while the mean of a sample varies from sample to sample.

Example
Problem	Mark operates Technology Titans, a Web site service that employs 8 people. Find the mean age of his workers if the ages of the employees are as follows: 55, 63, 34, 59, 29, 46, 51, 41
	Step 1: Add the numbers to determine the total age of the workers. 55 + 63 + 34 + 59 + 29 + 46 + 51 + 41 = 378 Step 2: Divide the total by the number of months.
Answer Move Close Print The mean age of all 8 employees is 47.25 years, or 47 years and 3 months.

In addition to calculating the mean for a given set of data values, you can apply your understanding of the mean to determine other information that may be asked for in everyday problems.

Example
Problem	Two weeks before Mark opened Technology Titans, he launched his company Web site. During those 14 days, Mark had an average of 24.5 hits on his Web site per day. In the first two days that Technology Titans was open for business, the Web site received 42 and 53 hits respectively. Determine the new average for hits on the Web site.
	Step 1: Multiply the given average by 14 to determine the total number of hits on Mark's Web site. 24.5 x 14 = 343 Step 2: Add the hits for the first two days his business was open. 343 + 42 + 53 = 438 Step 3: Divide this new total by 16 to determine the new average.
Answer Move Close Print The average number of hits Mark's Web site has received per day since it was launched is 27.375.

All values for the means you have calculated so far have been for ungrouped, or listed, data. A mean can also be determined for data that is grouped, or placed in intervals. Unlike listed data, the individual values for grouped data are not available, and you are not able to calculate their sum. To calculate the mean of grouped data, the first step is to determine the midpoint of each interval or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.

The following example will show how the mean value for grouped data can be calculated.

Example
Problem	In Tim's office, there are 25 employees. Each employee travels to work every morning in his or her own car. The distribution of the driving times (in minutes) from home to work for the employees is shown in the table below.
Driving Times (minutes) 0 to less than 10 10 to less than 20 20 to less than 30 30 to less than 40 40 to less than 50 Number of Employees 3 10 6 4 2
	Calculate the mean of the driving times.
	Step 1: Determine the midpoint for each interval. For 0 to less than 10, the midpoint is 5. For 10 to less than 20, the midpoint is 15. For 20 to less than 30, the midpoint is 25. For 30 to less than 40, the midpoint is 35. For 40 to less than 50, the midpoint is 45. Step 2: Multiply each midpoint by the frequency for the class. For 0 to less than 10, (5)(3) = 15 For 10 to less than 20, (15)(10) = 150 For 20 to less than 30, (25)(6) = 150 For 30 to less than 40, (35)(4) = 140 For 40 to less than 50, (45)(2) = 90 Step 3: Add the results from Step 2 and divide the sum by 25. 15 + 150 + 150 + 140 + 90 = 545
Answer Move Close Print Each employee spends an average (mean) time of 21.8 minutes driving from home to work each morning.

The mean is often used as a summary statistic. However, it is affected by extreme values (outliers): either an unusually high or low number. When you have extreme values at one end of a data set, the mean is not a very good summary statistic.

The Median

What is the Median?

This measure of central tendency is typically used when the mean value is affected by an unusually low number or an unusually high number in the data set (outliers). Outliers distort the mean value to the extent that the mean value no longer accurately depicts the set of data.

For example: If one of the houses in your neighborhood was broken down and maintained a low property value, then you would not want to include this property when determining the value of your own home. However, if you are purchasing a home in that neighborhood, you may want to include the outlier since it would drive down the price you would have to pay.

Try a few examples to follow the steps needed to calculate the median.

Example
Problem	Find the median of the following data: 12, 2, 16, 8, 14, 10, 6
	Step 1: Organize the data, or arrange the numbers from smallest to largest. 2, 6, 8, 10, 12, 14, 16 Step 2: Since the number of data values is odd, the median will be found in the position. Step 3: In this case, the median is the value that is found in the fourth position of the organized data. 2, 6, 8, 10, 12, 14, 16
Answer Move Close Print The median is 10.

Another way to look at the example is to narrow the data down to find the middle number.

2, 6, 8, 10, 12, 14, 16

Χ, 6, 8, 10, 12, 14, Χ

Χ, Χ, 8, 10, 12, Χ, Χ

Χ, Χ, Χ, 10, Χ, Χ, Χ

Here is another example of how to calculate the median of a set of numbers.

Example
Problem	Find the median of the following data: 7, 9, 3, 4, 11, 1, 8, 6, 1, 4
	Step 1: Organize the data, or arrange the numbers from smallest to largest. 1, 1, 3, 4, 4, 6, 7, 8, 9, 11 Step 2: Since the number of data values is even, the median will be the mean value of the numbers found before and after the position. Step 3: The number found before the 5.5 position is 4 and the number found after the 5.5 position is 6. Now, you need to find the mean value. 1, 1, 3, 4, 4, 6, 7, 8, 9, 11
Answer Move Close Print The median is 5.

The Mode

What is the Mode?

If two or more values appear with the same frequency, each is a mode. The downside to using the mode as a measure of central tendency is that a set of data may have no mode, or it may have more than one mode. However, the same set of data will have only one mean and only one median.

• The word modal is often used when referring to the mode of a data set.
• If a data set has only one value that occurs most often, the set is called unimodal.
• A data set that has two values that occur with the same greatest frequency is referred to as bimodal.
• When a set of data has more than two values that occur with the same greatest frequency, the set is called multimodal.

When determining the mode of a data set, calculations are not required, but keen observation is a must. The mode is a measure of central tendency that is simple to locate, but it is not used much in practical applications.

Example
Problem	Find the mode of the following data: 76, 81, 79, 80, 78, 83, 77, 79, 82, 75
	There is no need to organize the data, unless you think that it would be easier to locate the mode if the numbers were arranged from least to greatest. In the above data set, the number 79 appears twice, but all the other numbers appear only once. Since 79 appears with the greatest frequency, it is the mode of the data values.
Answer Move Close Print The mode is 79.

Example
Problem	The ages of 12 randomly selected customers at a local Best Buy are listed below: 23, 21, 29, 24, 31, 21, 27, 23, 24, 32, 33, 19 What is the mode of the above ages?
	The above data set has three values that each occur with a frequency of 2. These values are 21, 23, and 24. All other values occur only once. Therefore, this set of data has three modes.
Answer Move Close Print The modes are 21, 23, and 24.