Therefore, we’ll be able to compare relative positions of data values within their own distribution to determine which data values are closer to or farther from the mean. In other words, the z-score allows us to standardizetwo or more normal distributions, or more appropriately, to put them on the same scale. Z-scores measure the distance of any data point from the mean in units of standard deviations and are useful because they allow us to compare the relative positions of data values in different samples. The distribution on the right is a standard normal distribution with a standard score of z = −0.60 indicated. The distribution on the left is a normal distribution with a mean of 48 and a standard deviation of 5. Now draw each of the distributions, marking a standard score of z = −0.60 on the standard normal distribution. Given a normal distribution with μ = 48 and s = 5, convert an x-value of 45 to a z-value and indicate where this z-value would be on the standard normal distribution.īegin by finding the z-score for x = 45 as follows. Obviously a z-score will be positive if the data value lies above (to the right) of the mean, and negative if the data value lies below (to the left) of the mean.Įxample 6.1: Calculating and Graphing z-Values The z-score for any single data value can be found by the formula (in English): But what we’d really like to know is, relative to the spread of our data set, how far is x from μ? Remember that the standard deviation σ gives us a measure of how spread out our entire set of individual data values is. This value will be positive if your data value lies above (to the right) of the mean, and negative if it lies below (to the left) of the mean. In other words, the main objective of your quality management and controls should be to have your production process outcome as close to the normal distribution as possible.īecause of the six sigma methodology, in the last three decades the normal distribution has been used to enhance processes from manufacturing to transactions, both in factories and offices.Given any data value, we can identify how far that data value is away from the mean, simply by doing a subtraction x – μ. The basic notion is that a process requires a serious correction when it deviates more than three sigma from its mean. There are five main elements to this process: a) define, b) measure, c) analyze, d) improve, and e) control. Engineered at Motorola in the 1980s the system uses statistical analysis to measure end eliminate errors. This is the reason behind the quality control system based on the standard normal distribution, called the six sigma. If you perform a repetitive task that can be described by the normal distribution (such as a production of a standardized good), in the long run you may expect serious errors to happen so rarely that they become negligible.
Such events may be considered as very unlikely: accidents and mishaps, on the one hand, and streaks of luck, on the other.
If this principle is successfully applied you can expect to have 3.4 defects for every one million realizations of a process.
If you try to expand this interval and go six sigmas to left and right, you will find out that 99.9999998027% of your data points fall into this principles. Hence, only 0.03% of all the possible realizations of this process will lay outside of the three sigma interval. 99.7% of observation of a process that follows the normal distribution can be found either to the right or to the left from the distribution mean.