2.2 Measures of Central Tendency
LEARNING OBJECTIVES
- Recognize, describe, calculate, and analyze the measures of the centre of data: mean, median, and mode.
The “centre” of a data set is a way of describing location. The two most widely used measures of the “centre” of the data are the mean (average) and the median. To calculate the mean weight of [latex]50[/latex] people, add the [latex]50[/latex] weights together and divide by [latex]50[/latex]. To find the median weight of the [latex]50[/latex] people, order the data, and find the number that splits the data into two equal parts so that half of the numbers are below the median and the other half of the numbers are above the median. 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 centre.
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 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 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 [latex]97[/latex], 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 [latex]100[/latex], then [latex]\displaystyle{\frac{n+1}{2}=\frac{100+1}{2}=50.5}[/latex] and the median occurs midway between the [latex]50[/latex]th and [latex]51[/latex]st 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 |
|---|---|
| Field 1 | A1:A40 |
| Answer | 23.575 months |
For the median:
| Function | median |
|---|---|
| Field 1 | A1:A40 |
| Answer | 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 |
|---|---|
| Field 1 | A1:A39 |
| Answer | 13.949 months |
For the median:
| Function | median |
|---|---|
| Field 1 | A1:A39 |
| Answer | 13 months |
EXAMPLE
Suppose that in a small town of [latex]50[/latex] people, one person earns [latex]\$5,000,000[/latex] per year and the other [latex]49[/latex] each earn [latex]\$30,000[/latex]. Which is the better measure of the “centre”: 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 “centre” 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 [latex]60[/latex] 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 “centre”: the mean or the median?
Click to see Solution
The median is the better measure of the “centre” than the mean because [latex]59[/latex] 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 [latex]20[/latex] 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(…)) |
|---|---|
| Field 1 | A1:A20 |
| Answer | 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 |
|---|---|
| Field 1 | A1:A20 |
| Answer | 72 |
By examining the data, we can see that [latex]72[/latex] is the most frequently occurring value ([latex]5[/latex] times) and that [latex]72[/latex] is the only value that occurs [latex]5[/latex] times.
TRY IT
The number of books checked out from the library from [latex]25[/latex] 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(…)) |
|---|---|
| Field 1 | A1:A25 |
| Answer | 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 |
|---|---|
| Field 1 | A1:A25 |
| Answer | 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(…)) |
|---|---|
| Field 1 | A1:A40 |
| Answer | 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 |
|---|---|
| Field 1 | A1:A40 |
| Answer | 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(…)) |
|---|---|
| Field 1 | A1:A10 |
| Answer | 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 |
|---|---|
| Field 1 | A1:A10 |
| Answer | 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).
Video: “Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy” by Khan Academy [3:55] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.
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 tendency.
- Median is the preferred measure of central tendency when:
- 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).
- There are some missing or undetermined values in the data.
- There is an open-ended distribution (For example, if a data field measuring the number of children has options 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).
- Data that is measured on an ordinal scale.
- Mode is the preferred measure when data are measured on a nominal or ordinal scale.
Exercises
- How much time does it take to travel to work in a particular region? The table below shows the commute time for a sample of workers in the region who are at least [latex]16[/latex] years old and do not work at home.
24.0 24.3 25.9 18.9 27.5 17.9 21.8 20.9 16.7 27.3 18.2 24.7 20.0 22.6 23.9 18.0 31.4 22.3 24.0 25.5 24.7 24.6 28.1 24.9 22.6 23.6 23.4 25.7 24.8 25.5 21.2 25.7 23.1 23.0 23.9 26.0 16.3 23.1 21.4 21.5 27.0 27.0 18.6 31.7 23.3 30.1 22.9 23.3 21.7 18.6 - Find the mean for this data.
- Find the median for this data.
- Find the mode for this data.
Click to see Answer
- [latex]23.462[/latex] minutes
- [latex]23.5[/latex] minutes
- [latex]24[/latex] minutes, [latex]24.7[/latex] minutes, [latex]25.7[/latex] minutes, [latex]23.1[/latex] minutes, [latex]18.6[/latex] minutes, [latex]23.3[/latex] minutes, [latex]22.6[/latex] minutes, [latex]23.9[/latex] minutes, [latex]25.5[/latex] minutes, [latex]27[/latex] minutes
- The following data shows the lengths, in feet, of a sample of boats moored in a marina.
19 35 29 26 21 40 33 33 34 25 20 37 30 26 23 24 29 16 28 25 20 39 32 27 27 27 17 - Calculate the mean.
- Calculate the median.
- Find the mode.
Click to see Answer
- [latex]27.33[/latex] feet
- [latex]27[/latex] feet
- [latex]25[/latex] feet, [latex]27[/latex] feet
- The data below is the weight, in pounds, of all members of a particular NFL team.
177 210 270 275 212 185 200 241 250 220 259 185 210 272 285 212 250 302 205 232 280 285 184 265 215 223 265 260 278 185 228 273 242 185 241 290 210 276 290 206 174 286 247 190 215 245 205 178 290 280 188 230 260 - Find the mean for this data.
- Find the median for this data.
- Find the mode for this data.
Click to see Answer
- [latex]236.25[/latex] pounds
- [latex]241[/latex] pounds
- [latex]185[/latex] pounds
- A sample of 35 post-secondary institutions was taken from across the U.S. The data below shows the number of students enrolled at each institution.
6,414 1,550 2,109 9,350 21,828 4,300 5,944 5,722 2,825 2,044 5,481 5,200 5,853 10,012 6,357 27,000 9,414 7,681 3,200 17,500 9,200 7,380 18,314 6,557 13,713 17,768 7,493 2,771 2,861 1,263 7,285 28,165 5,080 11,622 2,750 - Calculate the mean for this data.
- Calculate the median for this data.
- Find the mode for this data.
- If you were to build a new community college, which piece of information would be more valuable: the mode or the mean? Explain.
Click to see Answer
- [latex]8,628.74[/latex] students
- [latex]6,414[/latex] students
- no mode
- The mean because there is no mode in this data.
- Forty randomly selected students were asked the number of pairs of sneakers they owned. The data is recorded below
1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 7 - Calculate the mean for this data.
- Calculate the median for this data.
- Find the mode for this data.
Click to see Answer
- [latex]3.775[/latex] sneakers
- [latex]4[/latex] sneakers
- [latex]4[/latex] sneakers, [latex]5[/latex] sneakers
- The median age of the U.S. population in 1980 was [latex]30.0[/latex] years. In 1991, the median age was [latex]33.1[/latex] years.
- What does it mean for the median age to rise?
- Give two reasons why the median age could rise.
Click to see Answer
- In 1980, half of the population was younger than [latex]30[/latex] years of age, and the other half of the population was older than 30 years of age. In 1991, half of the population was older than [latex]33.1[/latex] years of age, and the other half of the population was older than [latex]33.1[/latex] years of age. Because the median age rose over that decade, it means that the overall age of the population rose over that time period.
- The median age could rise because fewer children are being born and because people are living longer.
“2.3 Measures of Central Tendency” and “2.7 Exercices” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
