Which measure of central tendency refers to the middle score in the ordered list of values?

Measures of central tendency are numbers that describe what is average or typical within a distribution of data. There are three main measures of central tendency: mean, median, and mode. While they are all measures of central tendency, each is calculated differently and measures something different from the others.

The Mean

The mean is the most common measure of central tendency used by researchers and people in all kinds of professions. It is the measure of central tendency that is also referred to as the average. A researcher can use the mean to describe the data distribution of variables measured as intervals or ratios. These are variables that include numerically corresponding categories or ranges (like race, class, gender, or level of education), as well as variables measured numerically from a scale that begins with zero (like household income or the number of children within a family).

A mean is very easy to calculate. One simply has to add all the data values or "scores" and then divide this sum by the total number of scores in the distribution of data. For example, if five families have 0, 2, 2, 3, and 5 children respectively, the mean number of children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five households have an average of 2.4 children.

The Median

The median is the value at the middle of a distribution of data when those data are organized from the lowest to the highest value. This measure of central tendency can be calculated for variables that are measured with ordinal, interval or ratio scales.

Calculating the median is also rather simple. Let’s suppose we have the following list of numbers: 5, 7, 10, 43, 2, 69, 31, 6, 22. First, we must arrange the numbers in order from lowest to highest. The result is this: 2, 5, 6, 7, 10, 22, 31, 43, 69. The median is 10 because it is the exact middle number. There are four numbers below 10 and four numbers above 10.

If your data distribution has an even number of cases which means that there is no exact middle, you simply adjust the data range slightly in order to calculate the median. For example, if we add the number 87 to the end of our list of numbers above, we have 10 total numbers in our distribution, so there is no single middle number. In this case, one takes the average of the scores for the two middle numbers. In our new list, the two middle numbers are 10 and 22. So, we take the average of those two numbers: (10 + 22) /2 = 16. Our median is now 16.

The Mode

The mode is the measure of central tendency that identifies the category or score that occurs the most frequently within the distribution of data. In other words, it is the most common score or the score that appears the highest number of times in a distribution. The mode can be calculated for any type of data, including those measured as nominal variables, or by name.

For example, let’s say we are looking at pets owned by 100 families and the distribution looks like this:

Animal   Number of families that own it

• Dog: 60
• Cat: 35
• Fish: 17
• Hamster: 13
• Snake: 3

The mode here is "dog" since more families own a dog than any other animal. Note that the mode is always expressed as the category or score, not the frequency of that score. For instance, in the above example, the mode is "dog," not 60, which is the number of times dog appears.

Some distributions do not have a mode at all. This happens when each category has the same frequency. Other distributions might have more than one mode. For example, when a distribution has two scores or categories with the same highest frequency, it is often referred to as "bimodal."

Introduction: Connecting Your Learning

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

Lesson Objectives

By the end of this lesson, you should be able to:

1. Compute the mean, median, and mode of a given set of data.
2. Identify an outlier given a set of data.
3. Identify the mode or modes of a data set for both quantitative and qualitative data.

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.

The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution.

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.

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.

Look at another approach. If you were to take a sample of 3 employees from the group of 8 and calculate the mean age for these 3 workers, would the results change?

Use the ages 55, 29, and 46 for one sample of 3, and the ages 34, 41, and 59 for another sample of 3:

The mean age of the first group of 3 employees is 43.33 years.

The mean age of the second group of 3 employees is 44.66 years.

The mean age for a sample of a population depends upon the values that are included in the sample. From this example, you can see that the mean of a population and that of a sample from the population are not necessarily the same.

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.

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.

Example: Outliers

If you were employed by a company that paid all of its employees a salary between $60,000 and $70,000, you could probably estimate the mean salary to be about $65,000.

However, if you had to add in the $150,000 salary of the CEO when calculating the mean, then the value of the mean would increase greatly. It would, in fact, be the mean of the employees' salaries, but it probably would not be a good measure of the central tendency of the salaries.

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

The Median

What is the Median?

The median is the number that falls in the middle position once the data has been organized. Organized data means the numbers are arranged from smallest to largest or from largest to smallest. The median for an odd number of data values is the value that divides the data into two halves. If n represents the number of data values and n is an odd number, then the median will be found in the

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?

The mode of a set of data is simply the value that appears most frequently in the set.

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.

Remember that the mode can be determined for qualitative data as well as quantitative data, but the mean and the median can only be determined for quantitative data.

Now that you have added to your knowledge by reviewing the lesson and the examples, it is time to watch the following Khan Academy videos. These videos will provide additional explanations and working examples of how to determine the mean, median, and mode to help you gain a better understanding of this new concept.

In this lesson, you have learned how to calculate the mean, median, and mode of a set of data values. In addition, you have been introduced to other key terms such as measures of central tendency, unimodal, bimodal, and outliers. You also learned that the mode is the only measure of central tendency used in both quantitative and qualitative data.

As with every lesson and module, you are encouraged to research how these topics pertain to your particular area of study within the world of information technology. By now you are very aware that not every topic in mathematics will be directly implemented in your future career field. However, do not rule out the possibility that this topic might be an integral part of your future until you do some research.

1) Complete the Statistics: Finding Mean, Median, and Mode.

“Chapter 5: Measures of Central Tendency” by Merry, B. © 2012 retrieved from http://www.ck12.org/flexbook/chapter/9079 and used under a Creative Commons Attribution http://creativecommons.org/licenses/by/3.0/. This is an adaption of the lesson titled, “Measures of Central Tendency: Mean, Median, and Mode” by the National Information Security and Geospatial Technologies Consortium (NISGTC) is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0

Which measure of central tendency refers to the middle value?

Which measure is greatly affected by extreme values?

What measure of central tendency is the middle score in the list after the scores are arranged in decreasing or increasing order?

Which measure of central tendency is the middle value of a rank order distribution?