Math Portal
Introductory Statistics
Section 1.4 - Measures of Central Tendency
Population and Sample
Before we start discussing measures of central tendency, it is important to understand the difference between two important concepts: Population and Sample. A population is a collection of people, items, or events about which you want to make inferences. It is not always convenient or possible to examine every member of an entire population.
For example, it is not practical to count the bruises on all apples picked at an orchard. It is possible, however, to count the bruises on a set of apples taken from that population. This subset of the population is called a sample.
A sample a subset of people, items, or events from a larger population that you collect and analyze to make inferences. To represent the population well, a sample should be selected randomly collected and should be sufficiently large enough.
A measure of central tendency is a numerical value that describes a data set by attempting to provide a central or typical value of the data set. Measures of central tendency should have the same units as those of the data values from which they are determined. If no units are specified for the data values, no units are specified for the measures of central tendency. This section describes three common measures of central tendency: mean, median, and mode. Each of these measures is a way to classify a central value of a set of data.
Mean
There is a distinction made when the mean is computed from a whole group, or population, as opposed to when it is computed from a subset, or sample, from a population. The formula for computing the mean of a population is exactly the same as the formula for computing the mean of a sample, but the symbolism for each is different. The symbol for the mean of a population is the Greek letter μ(pronounced mew), while the symbol for the mean of a sample is x̅ (read x bar).
The mean of a data set is found by taking the sum of the data values and dividing by the number of data values. For example the mean of the following set of data can be computed:
25, 43, 40, 60, 12
by taking the sum of the numbers (180) and dividing it by the number of data values (5). 180 divided by 5 equals 36 (the mean of the sample).
Median
The median is the middle value or the arithmetic average of the two middle values in an ordered set of data. You find the median of a data set using a two-step process:
Mode
The mode is the data value (or values) that occurs with the greatest frequency in a data set. A data set in which each data value occurs the same number of times has no mode. If only one data value occurs with the greatest frequency, the data set is unimodal; that is, it has one mode. If exactly two data values occur with the same frequency that is greater than any of the other frequencies, the data set is bimodal; that is, it has two modes. If more than two data values occur with the same frequency that is greater than any of the other frequencies, the data set is multimodal; that is, it has more than two modes.
Range
The range for a data set is the difference between the maximum value (greatest value) and the minimum value (least value) in the data set; that is, range = maximum value - minimum value. The range should have the same units as those of the data values from which it is computed. If no units are specified, the range will not specify units.
Population and Sample
Before we start discussing measures of central tendency, it is important to understand the difference between two important concepts: Population and Sample. A population is a collection of people, items, or events about which you want to make inferences. It is not always convenient or possible to examine every member of an entire population.
For example, it is not practical to count the bruises on all apples picked at an orchard. It is possible, however, to count the bruises on a set of apples taken from that population. This subset of the population is called a sample.
A sample a subset of people, items, or events from a larger population that you collect and analyze to make inferences. To represent the population well, a sample should be selected randomly collected and should be sufficiently large enough.
A measure of central tendency is a numerical value that describes a data set by attempting to provide a central or typical value of the data set. Measures of central tendency should have the same units as those of the data values from which they are determined. If no units are specified for the data values, no units are specified for the measures of central tendency. This section describes three common measures of central tendency: mean, median, and mode. Each of these measures is a way to classify a central value of a set of data.
Mean
There is a distinction made when the mean is computed from a whole group, or population, as opposed to when it is computed from a subset, or sample, from a population. The formula for computing the mean of a population is exactly the same as the formula for computing the mean of a sample, but the symbolism for each is different. The symbol for the mean of a population is the Greek letter μ(pronounced mew), while the symbol for the mean of a sample is x̅ (read x bar).
The mean of a data set is found by taking the sum of the data values and dividing by the number of data values. For example the mean of the following set of data can be computed:
25, 43, 40, 60, 12
by taking the sum of the numbers (180) and dividing it by the number of data values (5). 180 divided by 5 equals 36 (the mean of the sample).
Median
The median is the middle value or the arithmetic average of the two middle values in an ordered set of data. You find the median of a data set using a two-step process:
- Put the data values in order from least to greatest (or greatest to least).
- Locate the middle data value. If there is no single middle data value, compute the arithmetic average of the two middle data values.
Mode
The mode is the data value (or values) that occurs with the greatest frequency in a data set. A data set in which each data value occurs the same number of times has no mode. If only one data value occurs with the greatest frequency, the data set is unimodal; that is, it has one mode. If exactly two data values occur with the same frequency that is greater than any of the other frequencies, the data set is bimodal; that is, it has two modes. If more than two data values occur with the same frequency that is greater than any of the other frequencies, the data set is multimodal; that is, it has more than two modes.
Range
The range for a data set is the difference between the maximum value (greatest value) and the minimum value (least value) in the data set; that is, range = maximum value - minimum value. The range should have the same units as those of the data values from which it is computed. If no units are specified, the range will not specify units.