## Mean Median and Mode in Central Tendency by Walter McIntyre

Before discussing measures of central tendency, a word of caution is necessary. Customers do not feel averages. They feel their specific experience. As a result, while central tendency is an important descriptive statistic, it is often misused. For example, a customer is told that the average delivery time is noon, but his actual delivery time turns out to be 3:00 PM. The customer, in this case, does not experience the average and may feel that he has been lied to.

The central tendency of a data set is a measure of the predictable center of a distribution of data. Stated another way, it is the location of the bulk of the observations in a data set. Knowing the central tendency of a process’ outputs, in combination with its standard deviation, will allow the prediction of the process’ future performance. The common measures of central tendency are the mean, the median, and the mode. Which of these descriptive statistics you need to use depends on the characteristics of the data set.

Mean, Median, Mode

The mean (also called the average) of a data set is one of the most used and abused statistical tools for determining central tendency. It is the most used because it is the easiest to apply. It is the most abused because of a lack of understanding of its limitations.

In a normally distributed data set, the mean (average) is the statistical tool of choice for determining central tendency. We use averages every day to make comparisons of all kinds such as batting averages, gas mileage, and school grades.

One weakness of the mean is that it tells nothing about segmentation in the data. Consider the batting average of a professional baseball player. It might be said that he bats .300 (Meaning a 30 percent success rate), but this does not mean that on a given night he will bat .300. In fact, this rarely happens. A closer evaluation reveals that he bats .200 against left-handed pitchers and .350 against right-handed pitchers. He also bats close to .400 at home and .250 on the road. What results is a family of distributions

As can be seen, the overall batting average of this baseball player does not do a good job of predicting the actual ability of this athlete on a given night. Instead, coaches use specific averages for specific situations. That way they can predict who will best support the team’s offense, given a specific pitcher and game location. This is a common situation with data sets. Many processes produce data that represent families of distributions, like those in the diagram above. Knowledge of these data characteristics can tell a lot about how a process behaves.

Another weakness of the mean is that it does not give the true central tendency of skewed distributions. An example would be a call center’s cycle time for handling calls.

If you were to diagram call center cycle time data, you would see how the mean is shifted to the right due to the skewedness of the distribution. This happens because we calculate the mean from the magnitudes of the individual observations. The data points to the right have a higher magnitude and bias the calculation, even though they have lower frequencies. What we need in this case is a method that establishes central tendency without “magnitude bias”. There are two ways of doing this: the median and the mode.

The median is the middle of the data set, when arranged in order of smallest to largest. If there are nine data points, as in the number set below, then five is the median of the set. If another three is added to the number set, the median would be 4.5 (the mid-point of the data set residing between 4 and 5).

1 2 3 4 5 6 7 8 9 1 2 3 3 4 5 6 7 8 9

The mode, on the other hand, is a measure of central tendency that represents the most frequently observed value or range of values. In the data set below, the central tendency as described by the mode, is three. Note that the median is 4.5 and the mean is 4.8, indicating that the distribution is skewed to the right.

1 2 3 3 4 5 6 7 8 9

The mode is most useful when the data set has more than one segment, is badly skewed, or it is necessary to eliminate the effect of extreme values. An example of a segmented data set would the observed height of all thirty-year-old people in a town. This data set would have two peaks, because it is made up of two segments. The male and female data points would form two separate distributions, and as a result, the combined distribution would have two modes.

In this data set, the mean would be 5.5 and the median would be of similar magnitude. Using the mean or median to predict the next person’s height would not be of value. Instead, knowing the gender of the next person would allow the use of the appropriate mode. This would result in a better predictor of the next person’s height.

In other words, the appropriate method of calculating central tendency is dependent upon the nature of the data. In a non-skewed distribution of data, the mean, median, and mode are equally suited to define central tendency. They are, in fact right on top of each other.

In a skewed distribution, like that of the call center mentioned earlier, the mean, median, and mode are all different. For prediction purposes, with a skewed distribution, the mean is of little value. The median and the mode would better predictors, but each tells a different story. Which is best depends upon why the data is skewed and how the result will be used.

A shift in the process’ output can make a data set seem skewed. In that case, the recent data is evidence of special cause variation. It means that the data set is on the way to becoming bi-modal, not skewed. For example, consider measuring the height of all thirty-year-old-people in a town as above. If females are measured first, there will be a normally distributed data set centered around 5 feet. As the men begin to be measured, the date set will begin to take on a skewed look. Eventually, the data set will become bi-modal. This phenomenon can make statistical decision making difficult. The key is to understand the reason for the data set’s skewedness.

The lesson to be learned here is that things are not always what they seem to be. You have to know what is happening behind the numbers to make the correct decision about how to calculate central tendency.

Understanding the nature of the data is also critical to making good choices about which statistical tools to use. Many poor conclusions find their origin in a lack of data intelligence.

In summary, as a rule, the mean is most useful when the data set is not skewed or multi-modal. Either the median or mode is useful when the data set is skewed, depending upon why it is skewed. The mode is most useful when the data set is multi-modal. Under all circumstances, the nature of the data will dictate which measure of central tendency will be best.

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