Measures of Central Tendency
Tools that are called measures of central tendency are designed to give information concerning what the average (or typical) data value is from a large number of data values.
There are three methods for obtaining a measure of the central tendency. When used appropriately, each is designed to give the most accurate estimation possible of the average data value. Depending on how the average data value was found, the interpretation of the data can vary widely.
The Mean
The most widely, though not always correctly, used measure of central tendency is the mean, denoted as M or x bar.
The mean is the arithmetic average of all data values. It is calculated by adding all the data values together and then dividing this sum by the total number of data values involved.
The following is the formula for the mean:
M, or x bar, is the mean, the letter x is the raw data value, or the measure of the trait in question. The sigma Σ is an operational term that indicates the addition of all data values x. It is usually read as, "summation of." The letter n is the total number of observations being dealt with.
The following is the calculation of the mean from a distribution of raw data values:
In this case, the mean is an appropriate measure of central tendency, because the distribution is well balanced; most of the data values occur in the middle range, and there are no extreme data values in one direction. Since the mean is calculated by adding all the data values in the distribution together, it is not usually influenced by the presence of extreme data values, unless the extreme data values are all at one end of the range.
The following is the calculation of the mean from a distribution of raw data values where the result is not a whole number:
Interpreting the Mean
Interpreting the mean correctly can sometimes be challenging, especially when the group or the size of the group changes.
For example, the mean I.Q. of the average freshmen college class is about 115 and the mean I.Q. of the average senior class is 5 points higher. This does not indicate that students increase their I.Q. as they progress through college, but since the size of the senior class is usually always smaller than the size of the freshmen class, the two populations are not the same. Among the freshmen who never become seniors generally have a low I.Q., and their I.Q. scores are not included in the mean I.Q. for seniors, and so that explains why the I.Q. for freshmen is lower.
The Mean of Skewed Distributions
In some situations, the use of the mean can lead to a distorted picture of the average value of a distribution of data values.
For example, consider the following calculation of the mean of a distribution of annual incomes:
Notice that one annual income ($10,000,000.00) is extremely far above the other annual incomes, so the use of the mean annual income as a reflection of average annual income gives a very misleading picture. A distribution which is unbalanced by an extreme data values at or near one end of the range is called a skewed distribution.
The following is a graphic illustration of skewed distributions:
In the distribution on the left, most of the data values fall to the right, or the high end, and there are not many extremely low data values. This is called a negatively skewed, or skewed to the left, distribution. The skew is in the direction of the tail of the data values, not in the direction of the majority of data values.
In the distribution on the right, most of the data values fall to the left, or the low end, and there are not many extremely high data values. This is called a positively skewed, or skewed to the right, distribution. When the tail of the distribution goes to the left, the curve is negatively skewed, or skewed to the left, and when the tail of the distribution goes to the right, the curve is positively skewed, or skewed to the right.
The Median
The median is the exact midpoint of any distribution. It is the point that separates the upper half from the lower half. The median, denoted as Mdn or Me, is a more accurate representation of central tendency for a skewed distribution than using the mean.
For example, recall the distribution of the annual incomes above. The median annual income is $19,400.00, which is a more descriptive reflection of the average annual income for that distribution than using the mean income, which was $786,776.92.
Calculating the Median
To calculate the median, the data must first be arranged in distribution form, which is in order of magnitude.
If the total number of data values is an odd number, then the median will be one of the data values itself. Divide the total number of data values minus 1 by two. This result is a position value of a data value that is beside the median value. Start from the beginning or the end of the distribution and count the data values until it reaches the position value of the median. If you started at the beginning, the median is to the right of this position value. If you started at the end, the median is to the left of this position value.
If the total number is an even number, there will be two median data values, in which the arithmetic average of those two is taken to get the median value. Divide the total number of data values by two. This result is the position of where the median value is. Start from the beginning or the end of the distribution and count the data values until it reaches the position value of the median. Do the same for the opposite side you chose. Take the arithmetic average of the two to get the median.
The following is the calculation of the median with an even number of data values:
The Mean and Median of Skewed Distributions
Unlike the mean, the median is not affected by skewed distributions.
For example, the following shows a similar distribution as the previous one, but there is an extreme data value in the distribution:
The median data value is still the same as the previous distribution, despite having an extreme data value. However, the two means are not the same, because the mean is always pulled towards the extreme data value in a skewed distribution. When the extreme data value is at the high end, the mean is too high to reflect true centrality, and when the extreme data value is at the low end, the mean is too low to reflect true centrality.
The Mode
The third measure of central tendency is called the mode, denoted as Mo. The mode is the most frequently occurring data value in a distribution.
In a histogram, the mode is always located beneath the tallest rectangular bar. In a frequency polygon, the mode is always found directly below the point where the curve is at its highest. This is because the y-axis, or the ordinate, represents the frequency of occurrence.
Finding the Mode
When the distribution is not graphed, the mode of the distribution is the data value that occurs the most times. Consider the following table:
The data value 103 occurs the most, and so the mode is 103.
When the distribution is graphed, to find the mode, determine which data value x has the highest frequency of occurrence f. The following is the location of the mode in a histogram (left) and a frequency polygon (right):
The mode is a useful tool for obtaining some idea of a distribution's centrality.
Unimodal and Bimodal Distributions
A distribution only having one mode is referred to as a unimodal distribution.
However, some distributions have more than one mode. When a distribution has two modes, the distribution is called bimodal. When there are more than two modes, the distribution is called multimodal. Distributions of this type occur when data values cluster together at several points, or if the group being measured represents two or more subgroups.
The following is the graph of a bimodel distribution of running times:
Assume the above distribution represents the running times in a 100-yard dash for a large group of high school seniors. There are two modes: one at 13 seconds and the other at 18 seconds. Since there are two running times that both occur with the same high frequency, it is possible that data about two separate subgroups are being displayed, such as one group is females and the other males.
Interpreting Bimodal Distributions
Whenever a distribution has two modes, neither the mean nor the median can be used, since a bimodal distribution cannot be described with a single value. Bimodal distributions should not be represented by the use of a single average of the data values.
For example, consider the following graph of responses to an attitude questionnaire showing two modes:
The use of either the mean or the median to report the results of this questionnaire would provide a misleading interpretation of the group's performance, for the mean or the median of the group's attitude would be presented as neutral. While no one scored at the neutral point, using either the mean or the median as a description of centrality would imply that the average individual in the group was neutral.
Therefore, when a distribution has more than one mode, the modes themselves should be used to describe the centrality of the distribution.
