1.8 Distributions

z-Scores

Each variable we might want to measure has its own distribution, or shape.

While the normal distribution is one shape, you could actually think of it as a symmetrical unimodal distribution with zero skew and zero kurtosis.

A simple way to describe data is to explain how the data is distributed. For example, we might expect there to be an average salary with most employees receiving around that amount with a smaller group of new employees who receive less and another small group of long-term employees who receive more, like a classic bell curve. When you think about the data in this way, you can determine what is common and what is rare while looking at what is likely versus what is unlikely. You can see how your outcomes are distributed.

There is no limit to the number of distribution shapes that exist. For example, if everyone had an equal salary, the distribution would be flat when the x-axis represents the years with the company and the y-axis represents the salary earned. If employees started with low salaries and later earned increasingly higher ones, the same axis labels would give us a distribution that would look like an upward sloping line.

At first, the possible number of distributions caused analysts no small worry when they were trying to find a common way to look at information. In the end, there were two concepts that made things much easier for them. The first was the realization that while there are an infinite number of distribution shapes. If you sample from them in groups, you can make them much simpler. For example, imagine a distribution that has two equally sized groups so that it looks like two bell curves next to each other, as in the figure below.

While the population would technically be called a bimodal distribution (bi- meaning 2), analysts realized that if they randomly took 30 people from that distribution, about half would come from the upper and half from the lower "hump" of the distribution. If they averaged their scores together, their average would still be the average of the overall distribution, but if they kept on averaging the scores and measuring how frequently they occurred, the analysts would get a sample distribution that looks like a simple bell curve. Analysts realized that this could be done for any distribution shape.

This idea—known more formally as the central limit theorem—made it possible for any kind of data to be described by the same distribution shape, so long as enough data (at least 30 measurements) is gathered, and the data are randomly chosen. Just remember this bell curve is discussing groups rather than individuals.

Based on the central limit theorem, the second concept that arose among data analysts was the assumption that all grouped data could be mathematically described by the same equation for a normal curve. If all groups could be described by the same kind of distribution and equation, it's simple to discover where the data fall on the distribution. This concept allows analysts to calculate a z-score for the data, which indicates how many standard deviations an element is from the mean and allows analysts to compare data across variables with different units.

To do this, take a score, subtract it from the average of the group, and then divide it by the standard deviation (the measure of how spread out the scores are around the mean of the group)—so long as the population mean and standard deviation—or even just the sample mean and standard deviation—is known. The number that results is part of the distribution that applies universally and can be specified mathematically.

z-Score for a Population

z-Score for a Sample

For example, a z-score of 0 means that 50% of others in the distribution score higher (and lower) than the chosen score—the chosen score is the average. We can see that 68% of scores fall within one z-score (-1 to +1) of the average (0), 95% fall within two z-scores, and 99.7% fall within three z-scores.

To calculate a z-score for a measurement, just take the score and subtract the mean of the group from it. Then divide by the standard deviation. This is a measure of how spread out the scores are.

For example, if one marketing firm charges between $5,000 and $6,000 per advertisement, their prices would have a smaller standard deviation than those of a company whose products range from $2,000 to $9,000, even though the average cost at both companies is $5,500. While a computer program will do the work of calculation in this course, it is important to remember that the mathematics behind the processes shown to you are used by every area of science from physics to psychology and are powerful tools to understand and analyze your business.

Another concept to remember is that every measurement will have a different standard deviation and therefore a different scale, but z-scores can be used to compare these measurements. By enabling the comparison of percentile ranks, z-scores allow the comparison of things with very different scales.

For example, the means and standard deviations for a business might include things like salaries, how many items were sold, and how happy employees and customers are. For salaries, perhaps the mean is $50,000 and the standard deviation is $5,000, while the average score of customer happiness might be an 8 out of 10 with a standard deviation of 0.5. These scores can't be compared directly because the scales are not the same. z-scores help because they put things on the same z-scale. For instance, convert the raw "happiness" scores to z-scores and examine the findings: compared to other employees, one of the manager's scores on both salary and happiness had a z-score of 1.0. This shows that the manager was above 84.1% of other employees in both areas but below about 16% of other employees.