2.3 Measures of Central Tendency

Valerie Watts

2.3 Measures of Central Tendency

LEARNING OBJECTIVES

Recognize, describe, calculate, and analyze the measures of the center of data: mean, median, and mode.

The “center” of a data set is a way of describing location. The two most widely used measures of the “center” of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data, and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

NOTE

The words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term for mean is “arithmetic mean” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”

Mean

The mean is calculated by adding up all of the values in the data and then dividing the sum by the total number of data values.

The letter used to represent the sample mean is [latex]\displaystyle{\overline{x}}[/latex] (read [latex]x[/latex]-bar). The Greek letter [latex]\mu[/latex] (pronounced “mew”) represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

Consider the sample:

1

2

3

4

[latex]\displaystyle{\overline{x} = \frac{1+1+1+2+2+3+4+4+4+4+4}{11} = 2.7}[/latex]

CALCULATING THE MEAN IN EXCEL

To find the mean in Excel, use the average(array) function.

For array, enter the array or cell range containing the data.

The output from the average function is the mean of the entered data.

Visit the Microsoft page for more information about the average function.

Median

The median is the middle value in an ordered set of data. You can quickly find the location of the median by using the expression [latex]\displaystyle{\frac{n+1}{2}}[/latex] where [latex]n[/latex] is the total number of data values in the sample. If [latex]n[/latex] is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If [latex]n[/latex] is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then the median is located in position [latex]\displaystyle{\frac{n+1}{2}=\frac{97+1}{2}=49}[/latex] of the ordered list. If the total number of data values is 100, then [latex]\displaystyle{\frac{n+1}{2}=\frac{100+1}{2}=50.5}[/latex] and the median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The upper case letter [latex]M[/latex] is often used to represent the median.

CALCULATING THE MEDIAN IN EXCEL

To find the median in Excel, use the median(array) function.

For array, enter the array or cell range containing the data.

The output from the median function is the median of the entered data.

Visit the Microsoft page for more information about the median function.

EXAMPLE

AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):

3	4	8	8	10	11	12	13	14	15
15	16	16	17	17	18	21	22	22	24
24	25	26	26	27	27	29	29	31	32
33	33	34	34	35	37	40	44	44	47

Calculate the mean and the median.

Solution:

Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A40.

For the mean:

Function

average

Answer

Field 1

A1:A40

23.575 months

For the median:

Function

median

Answer

Field 1

A1:A40

24 months

TRY IT

The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median.

3	4	5	7	7	7	7	8	8	9
9	10	10	10	10	10	11	12	12	13
14	14	15	15	17	17	18	19	19	19
21	21	22	22	23	24	24	24	24

Click to see Solution

Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A39.

For the mean:

Function

average

Answer

Field 1

A1:A39

13.949 months

For the median:

Function

median

Answer

Field 1

A1:A39

13 months

EXAMPLE

Suppose that in a small town of 50 people, one person earns [latex]\$5,000,000[/latex] per year and the other 49 each earn [latex]\$30,000[/latex]. Which is the better measure of the “center”: the mean or the median?

Solution:

[latex]\displaystyle{\mu=\frac{5,000,000+(49 \times30,000)}{50}=129,400}[/latex]

[latex]M=30,000[/latex]

The median is a better measure of the “center” than the mean because 49 of the values are [latex]\$30,000[/latex] and one is [latex]\$5,000,000[/latex]. The [latex]\$5,000,000[/latex] is an outlier. The median of [latex]\$30,000[/latex] gives us a better sense of the middle of the data.

TRY IT

In a sample of 60 households, one house is worth [latex]\$2,500,000[/latex]. Half of the rest are worth [latex]\$280,000[/latex], and all the others are worth [latex]\$315,000[/latex]. Which is the better measure of the “center”: the mean or the median?

Click to see Solution

The median is the better measure of the “center” than the mean because 59 of the values are either [latex]\$280,000[/latex] or [latex]\$315,000[/latex] and only one is [latex]\$2,500,000[/latex]. The [latex]\$2,500,000[/latex] is an outlier. Either [latex]\$280,000[/latex] or [latex]\$315,000[/latex] gives us a better sense of the middle of the data.

Mode

Another measure of the center of the data is the mode. The mode is the most frequently occurring value in the set of data. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A set of data can also have no mode if all of the observations in the data are unique.

Unlike the mean and the median, the mode can be calculated for both qualitative data and quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red.

CALCULATING THE MODE IN EXCEL

To find the mode in Excel:

Use the count and mode.mult function to determine the number of modes in the data. Enter count(mode.mult(array)) into a cell where array is the array or cell range containing the data. This function will output the number of modes present in the data.
If the output from the count(mode.mult(array)) function is 1, then the data has a single mode. To find the single mode, use the mode.sngl(array) function, where array is the array or cell range containing the data. The output from the mode.sngl function is the value of single mode in the data.
- Visit the Microsoft page for more information about the mode.sngl function.
If the output from the count(mode.mult(array)) function is greater than 1, then the data contains multiple modes. To find the multiple modes:
- Left click on a cell, hold and drag down to highlight a number of vertical cells equal to the number of modes in the data. For example, if there are 4 modes in the data, highlight 4 cells in the vertical array.
- In the highlighted cells, enter the mode.mult(array) function, where array is the array or cell range containing the data.
- After entering the mode.mult function in the vertical array, press CTRL+SHIFT+ENTER. Because the output from this function is an array, we must press CTRL+SHIFT+ENTER (and not ENTER) to produce the array output.
- The output from the mode.mult function are the modes in the data.
- Visit the Microsoft page for more information about the mode.mult function.

EXAMPLE

Statistics exam scores for 20 students are as follows:

50	53	59	59	63	63	72	72	72	72
72	76	78	81	83	84	84	84	90	93

Find the mode.

Solution:

Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A20.

Start by using the count function to count the number of modes in the data:

Function

count(mode.mult(…))

Answer

Field 1

A1:A20

1

Because the output from the count(mode.mult(…)) function is 1, there is only 1 mode in the data. To find the single mode, we use the mode.sngl function:

Function

mode.sngl

Answer

Field 1

A1:A20

72

By examining the data, we can see that 72 is the most frequently occurring value (5 times) and that 72 is the only value that occurs 5 times.

TRY IT

The number of books checked out from the library from 25 students are as follows:

0	0	0	1	2
3	3	4	4	5
5	7	7	7	7
8	8	8	9	10
10	11	11	12	12

Find the mode.

Click to see Solution

Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A25.

Start by using the count function to count the number of modes in the data:

Function

count(mode.mult(…))

Answer

Field 1

A1:A25

1

Because the output from the count(mode.mult(…)) function is 1, there is only 1 mode in the data. To find the single mode, we use the mode.sngl function:

Function

mode.sngl

Answer

Field 1

A1:A25

7

The most frequent number of books is [latex]7[/latex], which occurs four times.

EXAMPLE

AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):

3	4	8	8	10	11	12	13	14	15
15	16	16	17	17	18	21	22	22	24
24	25	26	26	27	27	29	29	31	32
33	33	34	34	35	37	40	44	44	47

Calculate the mode.

Solution:

Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A40.

Start by using the count function to count the number of modes in the data:

Function

count(mode.mult(…))

Answer

Field 1

A1:A40

12

Because the output from the count(mode.mult(…)) function is 12, there are 12 modes in the data. To find the multiple modes, we use the mode.mult function. Left click on a cell, hold and drag down to highlight 12 vertical cells. In the highlighted cells, enter the mode.mult function:

Function

mode.mult

Answer

Field 1

A1:A40

8, 15, 16, 17, 22, 24, 26, 27, 29, 33, 34, 44

Because the output from the mode.mult function is a (vertical) array after entering the function press CTRL+SHIFT+ENTER (not ENTER by itself).

TRY IT

Ten credit scores are

645

680

700

720

517

630

598

739

720

680

Calculate the mode.

Click to see Solution

Enter the data into an Excel spreadsheet. For this example, suppose we entered the data in column A from cell A1 to A10.

Start by using the count function to count the number of modes in the data:

Function

count(mode.mult(…))

Answer

Field 1

A1:A10

2

Because the output from the count(mode.mult(…)) function is 2, there are 2 modes in the data. To find the multiple modes, we use the mode.mult function. Left click on a cell, hold and drag down to highlight 2 vertical cells. In the highlighted cells, enter the mode.mult function:

Function

mode.mult

Answer

Field 1

A1:A10

680, 720

Because the output from the mode.mult function is a (vertical) array after entering the function press CTRL+SHIFT+ENTER (not ENTER by itself).

_{Watch this video: Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy by Khan Academy [3:54] (transcript available).}

Calculating the Mean of Grouped Frequency Tables

When only grouped data is available, we do not know the individual data values (we only know intervals and interval frequencies). Therefore, we cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean:

[latex]\displaystyle{\text{mean}=\frac{\text{data sum}}{\text{number of data values}}}[/latex]

We simply need to modify the definition to fit within the restrictions of a frequency table. Because we do not know the individual data values, we use the midpoint of each interval. The midpoint of an interval is

[latex]\displaystyle{\mbox{midpoint}=\frac{\text{lower boundary } + \text{ upper boundary}}{2}}[/latex]

We can now modify the mean definition to be

[latex]\displaystyle{\text{Mean of Frequency Table} = \frac{\sum (f \times m)}{\sum f}}[/latex]

where [latex]f[/latex] is the frequency of the interval and [latex]m[/latex] is the midpoint of the interval.

EXAMPLE

A frequency table displaying professor Blount’s last statistic test is shown. Find the best estimate of the class mean.

Grade Interval	Number of Students
50–56.5	1
56.5–62.5	0
62.5–68.5	4
68.5–74.5	4
74.5–80.5	2
80.5–86.5	3
86.5–92.5	4
92.5–98.5	1

Solution:

Find the midpoints for all intervals

Grade Interval	Midpoint
50–56.5	53.25
56.5–62.5	59.5
62.5–68.5	65.5
68.5–74.5	71.5
74.5–80.5	77.5
80.5–86.5	83.5
86.5–92.5	89.5
92.5–98.5	95.5

[latex]\begin{eqnarray*} \mbox{Mean} & = & \frac{53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1)}{19} \\ & = & \frac{1460.25}{19} \\& = & 78.86 \end{eqnarray*}[/latex]

TRY IT

Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:

Hours Teenagers Spend on Video Games	Number of Teenagers
0–3.5	3
3.5–7.5	7
7.5–11.5	12
11.5–15.5	7
15.5–19.5	9

What is the best estimate for the mean number of hours spent playing video games?

Click to the Solution

The midpoints are [latex]1.75[/latex], [latex]5.5[/latex], [latex]9.5[/latex], [latex]13.5[/latex], [latex]17.5[/latex].

[latex]\begin{eqnarray*}\mbox{Mean} & = & \frac{(1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9)}{38} \\ & = & \frac{409.75}{38}\\ & = &10.78\end{eqnarray*}[/latex]

When to Use Each Measure of Central Tendency

The measures of central tendency tell us about the center of the data, but often give different answers. So how do we know when to use each? Here are some general rules:

The mean is the most frequently used measure of central tendency and is generally considered the best measure of central location.
Median is the preferred measure of central tendency when:
1. There are a few extreme values or outliers in the distribution of the data. (Note: Remember that a single outlier can have a great effect on the mean).
2. There are some missing or undetermined values in the data
3. There is an open ended distribution (For example, if you have a data field which measures the number of children and your options are 0, 1, 2, 3, 4, 5 or “6 or more,” then the “6 or more field” is open ended and makes calculating the mean impossible because we do not know the exact values for this field).
4. You have data measured on an ordinal scale.
Mode is the preferred measure when data are measured in a nominal or ordinal scale.

Concept Review

The mean, the median, and the mode are measures of the “center” of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode tells us the most frequently occurring datum (or data) in our data set. The mean, median, and mode are extremely helpful when we need to analyze our data, but if the data set consists of ranges which lack specific values, the mean may be impossible to calculate. However, the mean of grouped data can be approximated by multiplying the midpoint of each interval with the frequency, adding up these values and then dividing by the total number of values in the data set.

Attribution

“2.5 Measures of the Center of the Data“ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

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