What are the characteristics of a normal distribution and wh

What are the characteristics of a normal distribution, and why are these important for statistical reasoning?

Solution

Jane is about to take an SAT. The school she is applying for gives priority to candidates whose SAT scores are in the 84th percentile or above. Jane wonders what she should score on the test to achieve this.

Sam is designing an electric car. To design it properly, he needs to know how long 95% of the lithium ion batteries will last.

What do these questions have in common? They can be solved with a greater understanding of the normal distribution. The normal distribution is a continuous distribution of data that has the shape of a symmetrical bell curve. It\'s also known as the Bell Curve. It is also called the Gaussian Distribution, after Carl Gauss who created a mathematical formula for the curve.

So, what\'s so special about this curve? A lot of data in nature have this shape when compiled and graphed. For example, heights and weights of men and women have this distribution. Standardized test scores are normally distributed. Sometimes lifespans of manufactured parts or equipment form a normal distribution.

By compiling the data into a frequency table and graphing in a histogram, we can often see this phenomena. Notice that the normal distribution, or curve, has a bell shape and is symmetrical. This is a property of the normal distribution. Another property is that \'mean = median = mode.\' This is because the shape of the data is symmetrical with one peak.

And, since the curve is symmetrical, the mean or median or mode (which are all the same number for this distribution) divide the data in half. From now on, we will just refer to this value in the middle as the mean. However, note that the symbol Mu represents a population mean, and x bar represents a sample mean.

he normal distribution is the most important and most widely used distribution in statistics. It is sometimes called the \"bell curve,\" although the tonal qualities of such a bell would be less than pleasing. It is also called the \"Gaussian curve\" after the mathematician Karl Friedrich Gauss. As you will see in the section on thehistory of the normal distribution, although Gauss played an important role in its history, Abraham de Moivre first discovered the normal distribution.

Strictly speaking, it is not correct to talk about \"the normal distribution\" since there are many normal distributions. Normal distributions can differ in their means and in their standard deviations. Figure 1 shows three normal distributions. The green (left-most) distribution has a mean of -3 and a standard deviation of 0.5, the distribution in red (the middle distribution) has a mean of 0 and a standard deviation of 1, and the distribution in black (right-most) has a mean of 2 and a standard deviation of 3. These as well as all other normal distributions are symmetric with relatively more values at the center of the distribution and relatively few in the tails.

he normal distribution is important because