When a set of data follows a normal distribution pattern, the mean and standard deviation can be used to calculate the percentage of data falling inside a given range. The z table makes this possible.
Simple Definition of a Z Score
Referencing the illustration above, the Z-score is simply how many standard deviations our point of interest is from the mean of our data set (click on the image to see a full view).
To calculate the percentage of data falling inside a given range, the Z score is calculated using the equation: Z= (x-μ)/σ, where x is the value we are interested in, such as an upper or lower specification limit.
Once the z-score is calculated, a z-table can be referenced to determine the area to the left or right of x. This area, when multiplied by 100, represents the percentage of data to the left or right of x, depending on how we are reading the z-table.
A design engineer would like to change the maximum current specification on a DC motor from 6.4 amps to 5.6 amps.
Knowing that the average current is 3.4 amps and the standard deviation is 0.8 amps, estimate the process yield with the 5.6 amp specification.
Average current = 3.4 amps
Standard deviation = 0.8 amps
X = point of interest = 5.6 amps
Z= (x-μ)/σ = (5.6 amps – 3.4 amps)/0.8 amps = 2.75
Using the Z table, the area to the left of Z is 0.997. Since our point of interest is an upper specification limit, we know that this area represents the acceptable product. Therefore, the process yield will be 99.7%, and the theoretical defect rate will be 0.3%, which is 3,000 PPM.