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What Is Normal Probability Distribution?

By Franklin Jeffrey
Updated Feb 12, 2024
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The principles of statistics hold that, given a sufficient sample size, it is possible to predict the normal probability distribution of a greater population. Most people associate distribution probability with the shape resulting when the data is graphed, which will form a bell curve. The normal curve will show a greater concentration near the mean, or the point at which half of the sample lies on either side. There are fewer elements of the sample as one moves away from the mean point.

It is easy to picture the bell curve representing the normal probability distribution if one imagines what happens when flour is sifted onto a plate. Most of the flour lands in a heap directly underneath the sifter. Moving away from the top of the mound, the flour becomes less deep, and by the edge of the plate, little or no flour can be found.

To quantify the way that the sample, such as the flour, is dispersed, it is necessary to explain standard deviations. In simplest terms, the standard deviation indicates how widely spread each piece of data is from other data points and the mean. If the points are clustered together closely, the standard deviation will be less than if they are widely dispersed. For example, if the average temperature in a city varies dramatically by season, it will have a greater standard deviation than the normal probability distribution of a city on the equator where the temperature remains relatively constant all year.

As an example, consider that in the US, 27.8 percent of the women's shoes sold are in sizes 8 and 8.5, 23.7 percent are sizes 7 and 7.5 and 17.5 percent are sizes 9 or 9.5. Based on this information, shoe manufactures have established the average shoe size as an 8 to 8.5; using 27.8 as the mean and assigning a standard deviation of one shoe size should prove that approximately 68 percent of all women wear between a 7 and a 9.5 shoe. Adding the numbers yields 69 percent, well within the normal probability distribution.

Moving outward from the mean, the numbers should indicate that approximately 99 percent wear between a size 5 and a size 11. Given manufacturers' reports that 4.8 percent of all sales are a size 5 or 5.5, 11.7 percent are a size 6 or 6.5, 10 percent are a size 10 or 10.5 and 3 percent are a size 11, one can see that 98.5 percent of all sales follow the principle of normal probability distribution. Only 1.5 percent of all shoes sold fall beyond three standard deviations of mean.

The principles of normal probability distribution are used for many different applications. Pollsters sometimes use distribution probability to predict the accuracy of the data they collect. The normal curve may also be used in financial applications, such as to analyze the performance of a particular stock. Educators may apply the laws of normal probability distribution to predict future test scores or to grade papers on a curve.

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