Measures of Central Tendency for Numerical Data Study Guide (page 2)

Introduction to Measures of Central Tendency for Numerical Data

We need a greater statistical vocabulary if we are to describe the distributions of dotplots and stem-and-leaf plots. In this lesson, we will begin to think about measures of the middle of the distribution.

Population Measures of Central Tendency

Measures of central tendency attempt to quantify the middle of the distribution. If we are working with the population, these measures are parameters. If we have a sample, the measures are statistics, which are estimates of the population parameters. There are many ways to measure the center of a distribution, and we will learn about the three most common: mean, median, and mode.

Mean

The mean, denoted by μ, is the most common measure of central tendency. The mean is the average of all population values. If a population has N members, the mean is

,

where x_i is the value of the variable associated with the ith unit in the population. We have used two different notations to symbolize the mean. The first is the Greek letter μ, which has become a conventional representation of the mean. The other is E(X), representing the "expected value of X." The average of a random variable across the whole population is the mean or the expected value of that variable. Notice that the symbol for a capital sigma (∑), is a shorthand way of saying to add a set of numbers. The terms to be added are those beginning with i = 1 (because "i = 1" is below the sigma) to i = N (because "N" is at the top of the sigma). The index, here i, is incremented by one in each subsequent term.

Example

Find the mean height of the 62 high school orchestra members given in Dotplots and Stem-and-Leaf Plots: Study Guide.

Solution

The mean height for this population is = 66.4.

Although we have presented a formal equation for the mean, it is important to remember that the population mean is simply the average of all population values.

Median

The median, another measure of central tendency, is the middle value of the population distribution. To find the median, order all of the values in the population from largest to smallest and find the middle value. For example, suppose the following five values constituted the population distribution:

The middle value is the 7. Now suppose the following four values represent the population distribution:

Here, the middle value is somewhere between the 6 and the 9. The median is any value between 6 and 9; however, usually, the average of the two values, = 7.5, is taken as the median value. Through these two illustrations, we can see that we find the median in a slightly different manner when there is an even number of observations in the distribution than when there is an odd number of observations. This can be written generally as follows.

If N, the number of values in the population, is odd, the median is the st value in the ordered list of population values. If N is even, the median is any value between the th and the st values in the ordered list of population values; usually, the average of the two values is taken as the median. Note that this definition of population median is appropriate only if the number of population units is finite. For a continuous random variable, the median is still the middle value in the population, but we must use other methods to define it.

Example

Find the sample mean, sample median, and sample mode for the difference in the number of blinks while playing video games and the number of blinks during normal conversation. Which one do you think is the best measure of central tendency for these data?

Solution

The sample mean is the average of the sample values, that is, the average of the sample differences:

To find the median, we first order the sample values from smallest to largest:

13   13   17   17   18   19

21   24   26   26   27   29

32   34   40

Because there are 15 values in the sample, the middle value is the = 8th value in the list; that is, the sample median is 24.

The sample mode is the most frequently occurring value in the sample. Here, the values of 13, 17, and 26 were each observed twice. In these cases, the sample mode is of little value in measuring central tendency.

The mean and median provide good measures of the center of distribution, but the mode does not.

Measures of Central Tendency for Numerical Data In Short

Mean, median, and mode are three measures of central tendency. Each provides a measure of the middle value in the population. The mean is the average of all population values. The median is the middle of the population values. The mode is the most frequently occurring population values. The sample mean, sample median, and sample mode are sample estimates of the corresponding population parameters.

Find practice problems and solutions for these concepts at Measures of Central Tendency for Numerical Data Practice Exercises.