# 5.1. Find the percentage above the upper specification

The first step in determining the percentage above the upper specification is to calculate the Z value for the upper specification. This is found by subtracting the overall average from the upper specification, and then dividing by the estimated standard deviation. The Z value for the upper specification is denoted as Z_{upper}. The upper specification for the example is 14, the overall average is 10.00, and the estimated standard deviation is 2.00. Thus, the value of Z_{upper} for the example is:

This means that the upper specification is located 2.00 estimated standard deviations away from the overall average. Look up the Z value in the standard normal distribution table to find the estimated proportion of output that is outside the upper specification.

Z values are listed along the left and top of the table. The whole number (number to the left of the decimal) and the tenths digit (first number to the right of the decimal) are listed on the left hand side of the table, and the hundredths digit (second number to the right of the decimal) is along the top. The table shows Z values only up to 4. If the Z value is greater than 4, the proportion outside the specification is virtually 0. In the example, the Z value is 2.00. To find the percentage outside the specification, go down the left hand side of the table to 2.0 and then across to the column marked x.x0. The number is 0.0228, which is the proportion outside the specification. To convert the proportion to a percentage, multiply it by 100. The percentage outside the upper specification is 2.28 percent. Place this percentage on the diagram.

The above article is an excerpt from the "Operational definition" chapter of *Practical Tools for Continuous Improvement Volume 2 Statistical Tools*.