Measures of Center: Means and Medians

Learning Outcomes

  • Calculate means, medians, and modes for a set of data
  • Determine if a mean or median is a better representation for the center of a set of data

The “center” of a data set is also 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 5050 people, add the 5050 weights together and divide by 5050. To find the median weight of the 5050 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 is “arithmetic mean” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”

When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an xx with a bar over it (read “xx bar”): ¯x¯¯¯x.

The Greek letter μμ (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.

To see that both ways of calculating the mean are the same, consider this sample:

11; 11; 11; 22; 22; 33; 44; 44; 44; 44; 44

¯x=1+1+1+2+2+3+4+4+4+4+411=2.7¯¯¯x=1+1+1+2+2+3+4+4+4+4+411=2.7
¯x=3(1)+2(2)+1(3)+5(4)11=2.7¯¯¯x=3(1)+2(2)+1(3)+5(4)11=2.7

In the second example, the frequencies are 33, 22, 11, and 55.

Recall: Evaluating Algebraic Expression

To evaluate an algebraic expression, you replace the variable in the expression with a value.

Example: Evaluate the following algebraic expression where n=99n=99.

n+12=99+12=1002=50n+12=99+12=1002=50

Recall: Summation Notation

Summation notation is used when multiple numbers in a set need to be added together. The Greek letter capital sigma, σσ, is used to represent the addition of the numbers in a set.

You can quickly find the location of the median by using the expression n+12n+12.

The letter nn is the total number of data values in the sample. If nn is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If nn 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 9797, then n+12=97+12=49n+12=97+12=49. The median is the 4949th value in the ordered data. If the total number of data values is 100100, then n+12=100+12n+12=100+12 = 50.550.5. The median occurs midway between the 5050th and 5151st values. The location of the median and the value of the median are not the same. The upper case letter MM is often used to represent the median. The next example illustrates the location of the median and the value of the median.

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):

33; 44; 88; 88; 1010; 1111; 1212; 1313; 1414; 1515; 1515; 1616; 1616; 1717; 1717; 1818; 2121; 2222; 2222; 2424; 2424; 2525; 2626; 2626; 2727; 2727; 2929; 2929; 3131; 3232; 3333; 3333; 3434; 3434; 3535; 3737; 4040; 4444; 4444; 4747

Calculate the mean and the median.

Finding the Mean and the Median Using the TI-83, 83+, 84, 84+ Calculator

Clear list L1. Pres STAT 4:ClrList. Enter 2nd 1 for list L1. Press ENTER.

Enter data into the list editor. Press STAT 1:EDIT.

Put the data values into list L1.

Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and then ENTER.

Press the down and up arrow keys to scroll.

¯x¯¯¯x= 23.623.6, MM = 2424

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.

33; 44; 55; 77; 77; 77; 77; 88; 88; 99; 99; 1010; 1010; 1010;1010; 1010; 1111; 1212; 1212; 1313; 1414; 1414; 1515; 1515; 1717; 1717; 1818; 1919;1919; 1919; 2121; 2121; 2222; 2222; 2323; 2424; 2424; 2424; 2424

example

Suppose that in a small town of 5050 people, one person earns $5,000,0005,000,000 per year and the other 4949 each earn $30,00030,000. Which is the better measure of the “center”: the mean or the median?

Try It

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

Another measure of the center is the mode. The mode is the most frequent value. 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 data set with two modes is called bimodal.

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.

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.

Example

Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.

When is the mode the best measure of the “center”? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.

Note

The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red.

Statistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software.

Try It

Five credit scores are 680, 680, 700, 720, 720. The data set is bimodal because the scores 680 and 720 each occur twice. Consider the annual earnings of workers at a factory. The mode is $25,000 and occurs 150 times out of 301. The median is $50,000 and the mean is $47,500. What would be the best measure of the “center”?

Watch the following video from Khan Academy on finding the mean, median, and mode of a set of data.