# Distributions

Tools

Intervals & Tests

Hypothesis Test Of Sample Mean Example

Hypothesis Test Of Two Sample Variances Example

Hypothesis Test Of A Standard Deviation Compared To A Standard Value Example

Distributions

Area Under the Standard Normal Curve

Non-Normal Distributions in the Real World

Rayleigh Distribution for True Position

Normal distributions are perhaps the most widely known distribution: the familiar bell-shaped curve. While some statisticians would have you believe they are the most widely occurring distribution in nature, others would suggest you take a good look at one in a textbook, since you are not likely to see one occur in the "real world."

Some years ago, some statisticians held the belief that when processes were not Normally distributed, there was something "wrong" with the process, or even that the process was "out of control." In their view, the purpose of the control chart was to determine when processes were non-normal so they could be "corrected," and returned to Normality. Most statisticians and quality practitioners today would recognize that there is nothing inherently "normal" (pun intended) about the Normal distribution, and its use in statistics is only due to its simplicity. It is well defined, so it is convenient to assume Normality when errors associated with that assumption would be minor. In fact, most of the efforts done in the interest of quality improvement lead to non-normal processes, since they try to narrow the distribution using process stops. Similarly, nature itself can impose stops to a process, such as a service process whose waiting time is physically bounded at the lower end by zero. The design of a waiting process would move the process as close as economically possible to zero, causing the process mode, median and average to move towards zero. This process would tend towards non-normality, regardless of whether it is stable or non-stable.

Any distribution can be characterized by four parameters, whose calculations are the same for any distribution:

Average: For symmetrical distributions (see Skewness, below) the average (or Mean) provides a good description of the central tendency or location, of the process. For very skewed distributions, the median is a much better indicator of location (or central tendency).

Standard Deviation: Denoted with the Greek symbol Sigma, the standard deviation provides an estimate of variation. In mathematical terms, it is the second moment about the mean. In simpler terms, you might say it is how far the observations vary from the mean.

Skewness: provides a measure of the location of the mode (or high point in the distribution) relative to the average. In mathematical terms, it is the third moment about the mean. Symmetrical distributions, such as the Normal distribution, have a skewness of zero. When the mode is to the left of the average, the skewness is negative; to the right it is positive.

Kurtosis: provides a measure of the "peaked-ness" of a distribution. In mathematical terms, it is the fourth moment about the mean. The Normal distribution has a kurtosis of one. Distributions that are more peaked have higher kurtosis.

Distributions can be fit to data using Curve Fitting techniques.

See also:

Non-Normal Distributions in the Real World

Rayleigh Distribution for True Position

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