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Central Tendency

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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.

Organizations and the Laws of Physics

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I am writing this article to create an imbalance in the world of those who read my posts. I am not in any way attempting to consider all of the options, or to be fair.  I just want to Step on your t pets a little. If it makes you uncomfortable, that is a good thing.  It is what this article is meant to do.

Newton’s Second Law of motion, in paraphrase, states that to change the state of motion of an object, a force must act on it to create an imbalance in forces. The object will then move to establish a new state of equilibrium.

The second law of thermodynamics, in paraphrase, states that systems always move toward a state of equilibrium. This movement will persist until the system reaches absolute zero (system death) or equilibrium is reached.

These concepts taken from physics also apply to human endeavors at the individual and organizational levels. In the human experience we call equilibrium the “status quo”.  I personally find that the status quo is a place for those who need rest or are not motivated to move forward. I am not against rest, but if you are resting and your competition isn’t, you’re losing ground. In other words, the status quo for me is good only when the status quo is to avoid the status quo.  Chew on that one for a while.

The status quo mentality usually forms in organizations and individuals who are internally focused. Being internally focused will isolate you from your external operating environment. You do not feel, or you fail to recognize, external forces that create imbalances in your external operating environment. The result is that you become out of alignment with the world around you. You fail to benefit from changes in the environment or maybe you even fall victim to them. The ostrich may have protected his head, but his rear end is more than a little exposed.

I know that some will say that organizations and individuals must isolate themselves from destructive forces in their operating environment in order to protect their assets.  I will answer that I disagree. Individuals or organizations that do not try to manage within the environment they operate in are simply exchanging one master (the larger outside world) for another (isolation). We do not have to be mastered by either. We control our choices and we become stronger and more robust as we exercise our ability to choose.

Let me give you examples. Governments and businesses isolate themselves from the governed and customers with bureaucratic layers of management. Religions do this by operating on a paradigm of exclusion (us, them) instead of a paradigm of inclusion. The result is that some governments, businesses and religions become more and more isolated, lose connection with their sense of purpose and eventually fail.

So what do you do? First understand that nothing stays the same in our world.  We age, tastes change and the people around us change. There is an interesting story line in the movie “The Time Machine”. The time traveler sits in his time machine and watches the world change around him.  He is isolated from the effects of the change and when he arrives in the future he is out of place and out of sync with the world around him.  The world experienced the changes first hand and has adapted, he did not experience the changes and finds himself in danger without a full understanding of how to cope. In the movie the good guys win, but in real life it probably would not have turned out that way.

We don’t have to agree with, or placidly accept, the changes around us. We can push back, adjust our strategy, etc. What we cannot do is ignore what is happening. The wise person evaluates these changes against reality and avoids letting others interpret their meanings for them. In sports we call this “keeping on your toes” or “keeping you eye on the ball.” In life it is simply a matter of paying attention to what is happening around us and keeping the main thing, the main thing.

In short we must embrace change. The world is moving onward with a great deal of inertia and it doesn’t care if you get left behind. The days of large stable bureaucratically ran organizations are coming to an end.  These are the days of smaller, fast and flexible, organizations that can move quickly to take care of customers, no matter how the environment changes. What customers, and people in general, want are solution providers, not protestors or clingers on to the old paradigm.

One way to manage this is to balance long term projects, goals and rewards with short term projects, goals and rewards. The long term perspective tends to add stability to an organization’s progress over time.  The short term perspective creates more employee engagement and a degree of instability, which is also good. Short term projects, goals and rewards operate in the current reality and force us to see what is actually happening right now. Long term projects, goals and rewards keep us focused on our mission and vision, which may be based in another reality. Short and long term efforts tend to modify each other in a healthy way when managed properly.

The balance point is always shifting.  Don’t let it become a tripping point.

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 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 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 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.

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 understanding of its limitations.

In a normally distributed dataset, the 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 datasets. 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 dataset, 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 dataset 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 dataset 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 dataset has more than one segment, is badly skewed, or it is necessary to eliminate the effect of extreme 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 separate distributions, and as a result, the combined distribution would have two modes.

In this dataset, 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. In a skewed dataset, the median may be the best indication of central tendency for hypothesis testing (see “Non-Parametric Tests”). The mode may be a better predictor of the next observation.

A shift in the process’ output can make a dataset seem skewed. In that case, the recent data is evidence of special cause variation. It means that the dataset is on the way to becoming bimodal, 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 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 making difficult. The key is to understand the reason for the dataset’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 dataset is not skewed or multi-modal. Either the median or mode is useful when the dataset is skewed, depending upon why it is skewed. The mode is most useful when the dataset is multi-modal. Under all circumstances, the nature of the data will dictate which measure of central tendency will be best.