The normal probability distribution can be predicted with a sufficient sample size, forming a bell curve with higher concentration near the mean. Standard deviations explain the spread of data points from the mean. The principles are used in various applications, such as predicting test scores or analyzing financial trends.
The principles of statistics hold that, given a sufficient sample size, it is possible to predict the normal probability distribution of a larger population. Most people associate the distribution probability with the resulting shape when the data is graphed, which will form a bell curve. The normal curve will show a higher concentration near the mean, or the point where half of the sample is on either side. There are fewer elements in the sample the further away from the midpoint.
It’s easy to imagine the bell curve representing the normal probability distribution if you imagine what happens when flour is sifted onto a plate. Most of the flour ends up in a pile directly under the sieve. Moving away from the top of the mound, the flour becomes shallower and little or no flour is found at the edge of the dish.
To quantify how the sample, such as flour, is dispersed, the standard deviations need to be explained. In simpler terms, the standard deviation means the spread of each data point from other data points and the mean. If the points are clustered very close together, the standard deviation will be lower than if they are widely dispersed. For example, if the average temperature in a city varies greatly by season, it will have a larger standard deviation than the normal probability distribution for a city on the equator where the temperature remains relatively constant throughout the year.
For example, consider that in the United States, 27.8 percent of women’s shoes sold are in sizes 8 and 8.5, 23.7 percent are in sizes 7 and 7.5, and 17.5 percent are in sizes 9 or 9.5. Based on this information, shoe manufacturers have determined that the average shoe size is 8 to 8.5; using 27.8 as the mean and assigning one standard deviation of one shoe size should show that approximately 68 percent of all women wear a size 7 to 9.5 shoe. Adding the numbers together gives 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, the 11.7 percent are a size 6 or 6.5, 10% are a size 10 or 10.5 and 3% are a size 11, it can be seen that 98.5% of all sales follow the principle of normal probability distribution. Only 1.5% of all shoes sold exceed three standard deviations of the 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 can also be used in financial applications, for example to analyze the trend of a given security. Educators can apply the laws of normal probability distribution to predict future test scores or to grade papers on a curve.
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