1.17: Measures of Central Tendency
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. 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."
Another measure of the center is the mode. The mode is the most frequent value. If a data set has two values that occur the same number of times, then the set is bimodal.
Calculating the Mean and 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 (previously discussed under box plots in this chapter). 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.
The mean can also 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 x with a bar over it (pronounced "x bar").
The Greek letter μ (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.
You can quickly find the location of the median by using the expression (n+1)/2. The letter n is the total number of data values in the sample. If n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by 2 after the data has been ordered. For example, if the total number of data values is 97, then (n+1)/2 = (97+1)/2 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100, then (n+1)/2 = (100+1)/2 = 50.5. 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.
This text is adapted from Barbara Illowsky, Ph.D., Susan Dean, Collaborative Statistics. OpenStax CNX.