Measures of Central Tendency

 Measures of Central Tendency

Several different measures of central tendency are defined below.

Arithmetic Mean: The arithmetic mean is the most common measure of central tendency. It simply the sum of the numbers divided by the number of numbers. The symbol 'm' is used for the mean of a population. The symbol M is used for the mean of a sample. The formula for 'm' is shown below: Where ∑X is the sum of all the numbers in the numbers in the sample and N is the number of numbers in the sample. As an example, the mean of the numbers 1 + 2 + 3 + 6 + 8 = 20/5 = 4 regardless of whether the numbers constitute the entire population or just a sample from the population. The table, Number of touchdown passes (Table 1: Number of touchdown passes), shows the number of touchdowns (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season. The mean number of touchdown passes thrown is 20.4516 as shown below. Number of touchdown passes although the arithmetic mean is not the only "mean" (there is also a geometric mean), it is by far the most commonly used.

Therefore, if the term "mean" is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean.

Median: The median is also a frequently used measure of central tendency. The median is the midpoint of a distribution: the same number of scores is above the median as below it. For the data in the table, Number of touchdown passes (Table 1: Number of touchdown passes), there are 31scores. The 16th highest score (which equals 20) is the median because there are 15 scores below the 16th score and 15 scores above the 16th score. The median can also be thought of as the 50th percentile3. Let's return to the made up example of the quiz on which you made a three discussed previously in the module Introduction to Central Tendency4 and shown in Table 2: Three possible datasets for the 5-point make-up quiz. Three possible datasets for the 5-point make-up quiz For Dataset 1, the median is three, the same as your score. For Dataset 2, the median is 4. Therefore, your score is below the median. This means you are in the lower half of the class. Finally for Dataset 3, the median is 2. For this dataset, your score is above the median and therefore in the upper half of the distribution. Computation of the Median: When there is an odd number of numbers, the median is simply the middle number. For example, the median of 2, 4, and 7 is 4. When there is an even number of numbers, the median is the mean of the two middle numbers. Thus, the median of the numbers 2, 4, 7, 12 is 4+7/2 = 5:5.

Modes: The mode is the most frequently occurring value. For the data in the table, Number of touchdown passes (Table 1: Number of touchdown passes), the mode is 18 since more teams (4) had 18 touchdown passes than any other number of touchdown passes. With continuous data such as response time measured to many decimals, the frequency of each value is one since no two scores will be exactly the same (see discussion of continuous variables5). Therefore the mode of continuous data is normally computed from a grouped frequency distribution. The Grouped frequency distribution (Table 3: Grouped frequency distribution) table shows a grouped frequency distribution for the target response time data. Since the interval with the highest frequency is 600-700, the mode is the middle of that interval (650). Grouped frequency distribution

Proportions and Percentages: When the focus is on the degree to which a population possesses a particular attribute, the measure of interest is a percentage or a proportion.

• A proportion refers to the fraction of the total that possesses a certain attribute. For example, we might ask what proportion of women in our sample weigh less than 135pounds. Since 3 women weigh less than 135 pounds, the proportion would be 3/5 or 0.60.

• A percentage is another way of expressing a proportion. A percentage is equal to the proportion times 100. In our example of the five women, the percent of the total who weigh less than 135 pounds would be 100 x (3/5) or 60 percent. Notation of the various measures, the mean and the proportion are most important.

The notation used to describe these measures appears below:

• X: Refers to a population mean.

• x: Refers to a sample mean.

• P: The proportion of elements in the population that has a particular attribute.

• p: The proportion of elements in the sample that has a particular attribute.

• Q: The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P.

• q: The proportion of elements in the sample that does not have a specified attribute. Note that q = 1 - p.