It is interesting to consider the possible effect the shape of a distribution, that is the shape of the curve or frequency polygon of a set of scores, might have on measures of central tendency. This consideration in turn might lead us to decide which measure of central tendency, the mean, the median, or the mode, is the best to use in a given situation.
For a symmetrical situation the scores are evenly distributed about some point. See the frequency polygon below which is your instructors attempt to draw a frequency polygon, illustrating a symmetrical distribution, using a mouse with a graphics draw program.
In a symmetrical distribution, the mean lies along the abscissa at the center of the curve. However, the median, with an equal number of scores above and below it also lies at the same point. The most frequently occurring score, the mode, also lies at this same point. Thus point A in the figure below represents the mean, the median, and the mode.
In general the mean is the best measure of central tendency. In its calculation (the sum of all the scores divided by the number of scores), it does represent all of the scores. If any score changes then the mean will change. This is not true of the median or mode.
This implies that extreme scores in either the high or the low direction will have a much greater effect on the mean than they would have on the median or the mode. For example, take the situation where we have a small factory with 9 employees and a manager. Each of the employees is paid $23,000 per year, while the manager is paid $93,000. If we were to make a frequency distribution of these salaries it would look somewhat like the following:
This frequency distribution is said to be positively skewed, that is the long slope of the curve is in the positive direction. A positively skewed distribution is caused by a relatively few high scores, or in the present example a single high score (the manager's salary). The total for salaries is $300,000 so the mean salary would be $30,000. We can see that the mean has been drastically increased relative to the mode and the median. In the figure below, a positively skewed distribution, the mode is the highest point of the curve and is represented by point A. The mean is the most shifted in the positive direction and is represented by point C. The median (in the typical situation of a positively skewed distribution) lies between the mode and the mean and is represented by point B. In this case of a skewed distribution, the median is probably a better measure of central tendency than the mean.
We can also have a skewed distribution because of a relatively few very low scores. For example, you live in a neighborhood in which there are nine homes valued at $150,000, $140,000, $160,000, $150,000, $160,000, $170,000, $160,000, $150,000, and $160,000. There is one small empty lot in the area and someone builds a very nice garage on it with a valuation of $20,000. If we were to make a frequency polygon of these 10 values it would be a negatively skewed currve and would look something like:
As an activity find the mean, the median, and the mode for these 10 properties. You should have found the mode as $160,000, the median as $155,000, and the mean as $142,000. We can see that the mode is the most frequently occurring value, the mean is much less, that is skewed in the negative or smaller direction, and the median is between the two in value. In the following distribution, point C represents the mode, point B represents the median, and point A represents the mean. This distribution, of course, represents a negatively skewed distribution. In this case the median might be a more representative measure of central tendency than the mean.
Although in general, the mean is usually the best measure of central tendancy, and is the measure we will use to develop further statistics such as the variance and the standard deviation. However, in the case of a badly skewed distribution (either negatively or positively skewed) the median may be a better measure of central tendency than the mean.
As a final thought on this matter, say you lived in one the properties mentioned above and the city tax assessor told you that you could choose to use the either the median, the mean, or the mode of the property values to determine your property tax. Which would you choose? The answer to this question is the mean, as that would give you the lowest tax base.