Normal DistributionKarl Gauss is generally given credit for his recognition of the normal curve of errors and also Pierre-Simon de LaPlace for discovering the normal distribution. The normal curve is also referred to as the Gaussian Distribution. Manufacturing processes and occurrences in nature frequently create a normal distribution, a unimodal bell curve. The data or performance is spread symmetrically around the central location. Most Six Sigma projects will involve analyzing normal sets of data or assuming normality. Some data will require a transformation to provide an accurate capability analysis. It is necessary that a Black Belt or Green Belt understands when and how to conduct a data transformation. When the distribution is not normally distributed, the Central Limit Theorem usually applies or a transformation of the data, such as a Box-Cox transformation applies. This determination MUST be done prior to using hypothesis testing tools. A normal distribution exhibits the following: 68.3% of the population is contained within 1 standard deviations from the mean. 95% of the population is contained within 2 standard deviations from the mean. 99.7% of the population is contained within 3 standard deviations from the mean.
The mean is used to define the central location in a set of normally distributed data. In a normal data set the median, mode, and mean are near equal. The area under the curve equals all of the observations or measurements.
Throughout this site the following assumptions apply: P-Value < alpha risk set at 0.05 indicates non-normal distribution although normality assumptions may apply. The level of confidence assumed throughout is 95%. P-Value > alpha risk set at 0.05 indicates a normal distribution. The Z-statistic can be derived from any variable point of interest (X) with the mean and standard deviation. The Z-statistic can be referenced to a table that will estimate a proportion of the population that applies to the point of interest. Recall, one of the two important implications of the Central Limit Theorem states that regardless of the type of distribution (unimodal, bi-modal, skewed, symmetric) the distribution of the sample means will take the shape of a normal distribution as the sample size increases. So the higher the sample size the more normality can be assumed.
Some tables and software programs compute the Z-statistic differently but will all get the correct results if interpreted correctly. Some tables incorporate single-tail probability and another table may incorporate double-tail probability. Examine each table carefully to make the correct reference to your problem. The bell curve theoretically spreads from negative infinity to positive infinity and approaches the x-axis without ever touching it - asymptotic to the x-axis. The area under the curve represents the probabilities and the whole area is estimated to be equal to 1.0 or 100%. The normal distribution is described by the mean and the standard deviation. The formula for the normal distribution density function is shown below (e = constant = 2.71828):
With prepopulated values based on a given value for "x", the probabilities can be assessed using a conversion formula (shown below) from the z-distribution, also known as the standardized normal curve.
A z-score is the number of standard deviations that a given value "x" is above or below the mean of the normal distribution.
Normal Distribution Tables
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