3.1: Measures of Center

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Learning Objectives

• Describe measures of center to identify the typical or central value in a data set.
• Understand and calculate the mean, median, mode, and midrange.
• Explore variations such as the grouped mean for frequency tables and the weighted mean for data with varying importance.

Mean, Median, Mode, and Midrange for Raw Data

This section focuses on measures of central tendency. Many times you are asking what to expect on average. Such as when you pick a major, you would probably ask how much you expect to earn in that field. If you are thinking of relocating to a new town, you might ask how much you can expect to pay for housing. If you are planting vegetables in the spring, you might want to know how long it will be until you can harvest. These questions, and many more, can be answered by knowing the center of the data set. There are three measures of the “center” of the data. They are the mode, median, and mean. Any of the values can be referred to as the “average.”

• The mean is the arithmetic average of the numbers. This is the center that most people call the average, though all three – mean, median, and mode – really are averages.
• The median is the data value in the middle of a sorted list of data. To find it, you put the data in order, and then determine which data value is in the middle of the data set.
• The mode is the data value that occurs the most frequently in the data. To find it, you count how often each data value occurs, and then determine which data value occurs most often.
• The midrange is the sum of the least and the greatest value in the data set, divided by 2.

There are no symbols for the mode and the median, but the mean is used a great deal, and statisticians gave it a symbol. There are two symbols, one for the population parameter and one for the sample statistic. In most cases, you cannot find the population parameter, so you use the sample statistic to estimate the population parameter.

Definition \(\PageIndex{1}\): Population Mean

The population mean is given by

\(\mu=\dfrac{\sum x}{N}\), pronounced mu

where

• \(N\) is the size of the population.
• \(x\) represents a data value.
• \(\sum x\) means to add up all of the data values.

Definition \(\PageIndex{2}\): Sample Mean

Sample Mean:

\(\overline{x}=\dfrac{\sum x}{n}\), pronounced x bar, where

• \(n\) is the size of the sample.
• \(x\) represents a data value.
• \(\sum x\) means to add up all of the data values.

The value for \(\overline{x}\) is used to estimate \(\mu\) since \(\mu\) can't be calculated in most situations.

Example \(\PageIndex{1}\) Finding the mean, median, mode, and midrange

Suppose a vet wants to find the average weight of cats. The weights (in pounds) of five cats are in Example \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Finding the Mean, Median, Mode, and Midrange

6.8	8.2	7.5	9.4	8.2

Find the mean, median, mode, and midrange of the weight of a cat.

Solution

Before starting any mathematics problem, it is always a good idea to define the unknown in the problem. In this case, you want to define the variable. The symbol for the variable is \(x\).

The variable is \(x =\) weight of a cat

Mean:

\(\overline{x}=\dfrac{6.8+8.2+7.5+9.4+8.2}{5}=\dfrac{40.1}{5}=8.02\) pounds

Median:

You need to sort the list for both the median and the mode. The sorted list is in Example \(\PageIndex{2}\).

Table \(\PageIndex{2}\): Sorted List of Cat's Weights

6.8	7.5	8.2	8.2	9.4

There are 5 data points so the middle of the list would be the 3rd number. (Just put a finger at each end of the list and move them toward the center one number at a time. Where your fingers meet is the median.)

Table \(\PageIndex{3}\): Sorted List of Cats' Weights with Median Marked

6.8	7.5	8.2	8.2	9.4

The median is therefore 8.2 pounds.

Mode:

This is easiest to do from the sorted list that is in Example \(\PageIndex{2}\). Which value appears the most number of times? The number 8.2 appears twice, while all other numbers appear once.

Mode = 8.2 pounds.

Midrange:

To find the midrange we first need to know the lowest value, which is 6.8, and the highest value, which is 9.4. We add them and divide by 2.

Midrange = MR \(=\dfrac{Lowest + Highest}{2}=\dfrac{6.8+9.4}{2}=\dfrac{16.2}{2}=8.1\)

A data set can have more than one mode. If there is a tie between two values for the most number of times, then both values are the mode, and the data is called bimodal (two modes). If every data point occurs the same number of times, there is no mode. If there are more than two numbers that appear the most times, then usually there is no mode.

In Example \(\PageIndex{1}\), there were an odd number of data points. In that case, the median was just the middle number. What happens if there is an even number of data points? What would you do?

Example \(\PageIndex{2}\) finding the median with an even number of data points

Suppose a vet wants to find the median weight of cats. The weights (in pounds) of six cats are in Example \(\PageIndex{4}\). Find the median.

Table \(\PageIndex{4}\): Weights of Six Cats

6.8	8.2	7.5	9.4	8.2	6.3

Solution

Variable: \(x =\) weight of a cat

First, sort the list if it is not already sorted.

There are 6 numbers in the list so the number in the middle is between the 3rd and 4th numbers. Use your fingers starting at each end of the list in Example \(\PageIndex{5}\) and move toward the center until they meet. There are two numbers there.

Table \(\PageIndex{5}\): Sorted List of Weights of Six Cats

6.3	6.8	7.5	8.2	8.2	9.4

To find the median, just average the two numbers.

Median \(=MD=\dfrac{7.5+8.2}{2}=7.85\) pounds

The median is 7.85 pounds.

Example \(\PageIndex{3}\) finding mean and median using technology

Suppose a vet wants to find the median weight of cats. The weights (in pounds) of six cats are in Example \(\PageIndex{4}\). Find the median

Solution

Variable: \(x=\) weight of a cat

You can do the calculations for the mean and median using the technology.

The procedure for calculating the sample mean ( \(\overline{x}\) ) and the sample median (Med) on the TI-83/84 is in Figures 3.1.1 through 3.1.4. First, you need to go into the STAT menu and then Edit. This will allow you to type in your data (see Figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): TI-83/84 Calculator Edit Setup

Once you have the data into the calculator, you then go back to the STAT menu, move over to CALC, and then choose 1-Var Stats (see Figure \(\PageIndex{2}\)). The calculator will now put 1-Var Stats on the main screen. Now type in L1 (2nd button and 1) and then press ENTER. (Note if you have the newer operating system on the TI-84, then the procedure is slightly different.) If you press the down arrow, you will see the rest of the output from the calculator. The results from the calculator are in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{2}\): TI-83/84 Calculator CALC Menu

Figure \(\PageIndex{3}\): TI-83/84 Calculator Input for Example \(\PageIndex{3}\) Variable

Figure \(\PageIndex{4}\): TI-83/84 Calculator Results for Example \(\PageIndex{3}\) Variable

Example \(\PageIndex{4}\) affect of extreme values on mean and median

Suppose you have the same set of cats from Example \(\PageIndex{1}\), but one additional cat was added to the data set. Example \(\PageIndex{6}\) contains the six cats’ weights, in pounds.

Table \(\PageIndex{6}\): Weights of Six Cats

6.8	7.5	8.2	8.2	9.4	22.1

Find the mean and the median.

Solution

Variable: \(x=\) weight of a cat

Mean \(=\overline{x}=\dfrac{6.8+7.5+8.2+8.2+9.4+22.1}{6}=10.37\) pounds

The data is already in order, thus the median is between 8.2 and 8.2.

Median \(=MD=\dfrac{8.2+8.2}{2}=8.2\) pounds

The mean is much higher than the median. Why is this? Notice that when the value of 22.1 was added, the mean went from 8.02 to 10.37, but the median did not change at all. This is because the mean is affected by extreme values, while the median is not. The hefty cat brought the mean weight up. In this case, the median is a much better measure of the center.

An outlier is a data value that is very different from the rest of the data. It can be really high or really low. Extreme values may be outliers if the extreme value is far enough from the center. In the example above, the data value 22.1 pounds is an extreme value, and it may be an outlier.

If there are extreme values in the data, the median is a better measure of the center than the mean. If there are no extreme values, the mean and the median will be similar, so most people use the mean.

The mean is not a resistant measure because it is affected by extreme values. The median and mode are resistant measures because extreme values do not affect them.

Determine Best Representation for Center: Mean or Median

As a consumer, you need to be aware that people choose the measure of center that best supports their claim. When you read an article in the newspaper that talks about the “average,” it usually means the mea,n but sometimes it refers to the median. Some articles will use the word “median” instead of “average” to be more specific. If you need to make an important decision and the information says “average”, it would be wise to ask if the “average” is the mean or the median before you decide.

As an example, suppose that a company wants to use the mean salary as the average salary for the company. This is because the high salaries of the administration will pull the mean higher. The company can say that the employees are paid well because the average is high. However, the employees want to use the median since it discounts the extreme values of the administration and will give a lower value of the average. This will make the salaries seem lower, and that a raise is in order.

Why use the mean instead of the median? The reason is that when multiple samples are taken from the same population, the sample means tend to be more consistent than other measures of the center. The sample mean is the more reliable measure of center.

To understand how the different measures of center relate to skewed or symmetric distributions, see the figure below. As you can see, sometimes the mean is smaller than the median and mode, sometimes the mean is larger than the median and mode, and sometimes they are the same value.

Figure \(\PageIndex{5}\): Mean, Median, Mode as Related to a Distribution

One last type of average is a weighted average. Weighted averages are used quite often in real life. Some teachers use them in calculating your grade in the course or your grade on a project. Some employers use them in employee evaluations. The idea is that some activities are more important than others. As an example, a full-time teacher at a community college may be evaluated on their service to the college, their service to the community, whether their paperwork is turned in on time, and their teaching. However, teaching is much more important than whether their paperwork is turned in on time. When the evaluation is completed, more weight needs to be given to the teaching and less to the paperwork. This is a weighted average.

Weighted Mean

Definition \(\PageIndex{3}\)

Weighted Mean

\(\overline{x}=\dfrac{\sum x \cdot w}{\sum w}\) where \(w\) is the weight of the data value, \(x\).

Example \(\PageIndex{5}\) weighted mean

In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. The lab score is worth 15% of the course, the two exams are worth 25% of the course each, and the final exam is worth 35% of the course. Suppose you earned scores of 95 on the labs, 83 and 76 on the two exams, and 84 on the final exam. Compute your weighted mean for the course.

Solution

Variable: \(x=\) score

(Weighted Mean) \(\overline{x} = \dfrac{\sum x \cdot w}{\sum w} = \dfrac{\text{sum of the scores times their weights}}{\text{sum of all the weights}}\)

\(\overline{x}=\dfrac{95(0.15)+83(0.25)+76(0.25)+84(0.35)}{0.15+0.25+0.25+0.35}=\dfrac{83.4}{1.00}=83.4 \%\)

A weighted mean can be found using technology.

The procedure for calculating the weighted average on the TI-83/84 is in Figures 3.1.6 through 3.1.9. First, you need to go into the [STAT] menu, and then [EDIT]. This will allow you to type in the scores into L1 and the weights into L2 (see Figure \(\PageIndex{6}\)).

Figure \(\PageIndex{6}\): TI-3/84 Calculator Edit Setup

Once you have the data in the calculator, you then go back to the [STAT] menu, move over to [CALC], and then choose [1-Var Stats] (see Figure \(\PageIndex{7}\)). The calculator will now put [1-Var Stats] on the main screen. Now type in L1 ([2ND] button and [1]), then a comma (button above the 7 button), and then L2 ([2ND] button and [2]), and then press [ENTER]. (Note: if you have the newer operating system on the TI-84, then the procedure is slightly different.) The results from the calculator are in Figure \(\PageIndex{9}\). The \(\overline{x}\) is the weighted mean.

Figure \(\PageIndex{7}\): TI-83/84 Calculator CALC Menu

Figure \(\PageIndex{8}\): TI-83/84 Calculator Input for Weighted Mean

Figure \(\PageIndex{9}\): TI-83/84 Calculator Results for Weighted Mean

Example \(\PageIndex{6}\) weighted mean

The faculty evaluation process at John Jingle University rates a faculty member on the following activities: teaching, publishing, committee service, community service, and submitting paperwork promptly. The process involves reviewing student evaluations, peer evaluations, and supervisor evaluations for each teacher and awarding them a score on a scale from 1 to 10 (with 10 being the best). The weights for each activity are 20 for teaching, 18 for publishing, 6 for committee service, 4 for community service, and 2 for paperwork.

1. One faculty member had the following ratings: 8 for teaching, 9 for publishing, 2 for committee work, 1 for community service, and 8 for paperwork. Compute the weighted mean.
2. Another faculty member had ratings of 6 for teaching, 8 for publishing, 9 for committee work, 10 for community service, and 10 for paperwork. Compute the weighted mean.
3. Which faculty member had the higher mean?

Solution

a. Variable: \(x=\) rating

(Weighted Mean) \(\overline{x} = \dfrac{\sum x \cdot w}{\sum w} = \dfrac{\text{sum of the scores times their weights}}{\text{sum of all the weights}}\)

\(\overline{x}=\dfrac{8(20)+9(18)+2(6)+1(4)+8(2)}{20+18+6+4+2}=\dfrac{354}{50}=7.08\)

b. \(\overline{x}=\dfrac{6(20)+8(18)+9(6)+10(4)+10(2)}{20+18+6+4+2}=\dfrac{378}{50}=7.56\)

c. The second faculty member has a higher mean.

You can find a weighted mean using technology. The last thing to mention is which average is used on which type of data.

The mode can be found on nominal, ordinal, interval, and ratio data since the mode is just the data value that occurs most often. You are just counting the data values. Median can be found on ordinal, interval, and ratio data since you need to put the data in order. As long as there is order to the data, you can find the median. Mean can be found on interval and ratio data since you must have numbers to add together.

Grouped Data

Sometimes the mean is computed from grouped data rather than the raw data set. The mean for grouped data is computed using the following steps. Also, the modal class is the class with the highest frequency.

Perform the following steps:

1. Add two columns to the table with headings "Midpoint (\(X_M\))" and "f \(\cdot X_M\)."
2. Fill in the columns. The first of these columns consists of midpoints and the second column is each frequency multiplied with the corresponding midpoint.
3. Find the sum of the frequency columns. This sum is n = ∑ f.
4. Find the sum of the last column, which is denoted as ∑ f • X_M.
5. Compute the mean.

\(\overline{x}=\dfrac{\sum_{}^{}f\cdot X_M}{\sum_{}^{}f}\)

Example \(\PageIndex{7}\) Grouped Mean and Modal Class

Find the grouped mean and modal class from the grouped data.

Grouped Frequency Distribution

Class Limits	Frequency
5 - 11	5
12 - 18	9
19 - 25	18
26 - 32	6
33 - 39	8

Table \(\PageIndex{7}\): Grouped Frequency Distribution

Solution

Grouped Frequency Distribution

Class Limits	Frequency	\(X_M\)	\(f\cdot X_M\)
5 - 11	5	8	40
12 - 18	9	15	135
19 - 25	18	22	396
26 - 32	6	29	174
33 - 39	8	36	288

Table \(\PageIndex{8}\): Grouped Frequency Distribution With Midpoint \(X_M\) and \(f\cdot X_M\) Columns

n = Σf = 46

Σf \(\cdot \) X_M = 1033

\(\overline{x}=\dfrac{1033}{46}\approx22.46\)

Modal class: 19 - 25

Authors

"3.1: Measures of Center" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY-SA 4.0

Attributions

"3.1: Measures of Center" by Kathryn Kozak is licensed CC BY-SA 4.0