The Mean, average and Mode

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Central tendency: Mean, Median, Mode

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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 illustrated statistic, it is often misused. For example, a buyer is told that the midpoint delivery time is noon, but his actual delivery time turns out to be 3:00 pm. The customer, in this case, does not feel the midpoint and may feel that he has been lied to.

The central tendency of a dataset 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 dataset. Knowing the central tendency of a process' outputs, in mixture with its thorough deviation, will allow the prediction of the process' hereafter performance. The tasteless measures of central tendency are the mean, the median, and the mode.

Mean, Median, Mode

The mean (also called the average) of a dataset 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 comprehension of its limitations. The midpoint is easy to calculate. It is the sum of the magnitudes of all observations divided by the whole of observations.

In a regularly distributed dataset, the midpoint is the statistical tool of selection 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 feebleness of the mean is that it tells nothing about segmentation in the data. Reconsider the batting midpoint of a pro 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 estimate 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 house of distributions instead of a single distribution.

As a result, coaches use specific averages for specific situations. That way they can predict who will best keep the team's offense, given a specific pitcher and game location. This is a tasteless situation with datasets. Many processes furnish data that laid out families of distributions. Knowledge of these data characteristics can tell a lot about how a process behaves.

Another feebleness 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. A histogram of this data from a call center would show the mean is shifted to the right due to the skewedness of the distribution. This happens because we theorize the mean from the magnitudes of the individual observations. Since the data points to the right have a higher magnitude, they bias the calculation, even though they have lower frequencies of occurrence.

What we need in this case is a method that establishes central tendency without "magnitude bias". There are two ways of doing this: the midpoint and the mode. The midpoint is the middle of the dataset, when arranged in order of smallest to largest. If there are nine data points, for example, then five is the midpoint of the set. 1 2 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 dataset below, the central tendency as described by the mode is three. 1 2 3 3 4 5 6 7 8 9

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

Let's suppose that this dataset shows that the mean height would be 5.5 feet. The midpoint would be of similar magnitude and both would be worthless in predicting the height of the next man to be measured. Knowing the gender of the next person, on the other hand, would allow for a best prediction of the next persons height. This is because there would be a mode for males and a mode for females. The mode in this case would be a good predictor.

In other words, the thorough method of calculating central tendency is dependent upon the nature of the data. In a nonskewed distribution of data, the mean, median, and mode are equally great 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 exiguous value. The midpoint and the mode would best predictors, but each tells a dissimilar story. Which is best depends upon why the data is skewed and how the corollary will be used.

In a skewed dataset, the midpoint may be the best indication of central tendency for hypothesis testing (Non-Parametric Tests), but the mode may be a best predictor of the next observation. Only a through knowledge of the data will show what method to use.

A shift in the process' production can make an otherwise general dataset seem skewed. In that case, the new data is evidence of special cause variation. It means that the dataset may be on the way to becoming bimodal, not skewed. For example, Reconsider measuring the height of all thirty-year-old-people in a town as above. If females are measured first, there will be a regularly distributed dataset centered around 5 feet. As the men begin to be measured, the date set will begin to take on a skewed look. Eventually, the dataset will become bimodal. This phenomenon can make statistical decision manufacture difficult. The key is to understand the theorize for the dataset's skewedness.

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