Mean Point of Interest, x Z Score σ σ σ σ σ σ The Z Table Estimate Area “A” Area “B” Background on the Z Estimate Using the Z score, the Z table estimates the percentage of data that will fall from -∞ to a given point of interest, x. This area is represented by the cross-hatched area, A. Using the calculation Z= (x-μ)/σ, the Z score transforms any data set into the standard normal distribution, which has a total area (under the curve) of 1, therefore A + B = 1. In this case our Z score is 1.6, and the Z table estimates that 94.5% of the population will fall to the left of x. Therefore, the remaining 5.5% of the population will fall to the right of x. Background on the Z Estimate Using the Z score, the Z table estimates the percentage of data that will fall from -∞ to a given point of interest, x. This area is represented by the cross-hatched area, A. Using the calculation Z= (x-μ)/σ, the Z score transforms any data set into the standard normal distribution, which has a total area (under the curve) of 1, therefore A + B = 1. In this case our Z score is 1.6, and the Z table estimates that 94.5% of the population will fall to the left of x. Therefore, the remaining 5.5% of the population will fall to the right of x.

Mean Point of Interest, x Steps in Using a Z Table Area “A” Area “B” Finding the Area Under the Curve As a preliminary step, plot the data in a histogram and make sure it is normally distributed. If the data is not normal, then predictions made with the Z table will be invalid. 1.Calculate the mean and standard deviation of the data set. 2.Using the mean, standard deviation, and the point of interest x, calculate Z using the formula Z= (x-μ)/σ. 3.Using the Z table, find the area under the standard normal curve. Important: look at the diagram on the Z table and understand exactly which area is being provided. Not all Z tables start and end at the same reference points. Finding the Area Under the Curve As a preliminary step, plot the data in a histogram and make sure it is normally distributed. If the data is not normal, then predictions made with the Z table will be invalid. 1.Calculate the mean and standard deviation of the data set. 2.Using the mean, standard deviation, and the point of interest x, calculate Z using the formula Z= (x-μ)/σ. 3.Using the Z table, find the area under the standard normal curve. Important: look at the diagram on the Z table and understand exactly which area is being provided. Not all Z tables start and end at the same reference points.

μ = 3.4 amps σ = 0.8 amps Example #1 – One-Sided Specification Limit Process Yield = 99.7% (Z table value of 0.997) USL = 5.6 amps Product Exceeding the 5.6 amp USL = 0.3% (100% – 99.7%) Calculations 1.Mean = 3.4 amps, Standard Deviation = 0.8 amps 2.Z= ( )/0.8 = Using the Z table on this site, the area under the standard normal curve, to the left of Z is Therefore, a theoretical 99.7% of the data will fall to the left of the USL Calculations 1.Mean = 3.4 amps, Standard Deviation = 0.8 amps 2.Z= ( )/0.8 = Using the Z table on this site, the area under the standard normal curve, to the left of Z is Therefore, a theoretical 99.7% of the data will fall to the left of the USL Problem statement: The design team is proposing a 5.6 amp upper specification limit (USL) for motor manufacturing process. If ongoing production audit results show a mean current of 3.4 amps and a standard deviation of 0.8 amps, what is the theoretical process yield?

μ = 3.4 amps σ = 0.8 amps Example #2 – Upper and Lower Specification Limits USL = 5.6 amps Product Exceeding the 5.6 amp USL = 0.3% (100% – 99.7%) LSL = 2.5 amps Problem statement: The design team is proposing a 5.6 amp upper specification limit (USL) and a 2.5 amp lower specification limit (LSL) for a motor manufacturing process. If ongoing production audit results show a mean current of 3.4 amps and a standard deviation of 0.8 amps, what is the theoretical process yield? Product Falling Below the 2.5 amp LSL = 13% Calculations 1.From Example #1, we know that the area under the standard normal curve exceeding the 5.6 amp USL is 0.3%. 2.To calculate the area to the left of the LSL, we use Z= ( )/0.8 = Using the Z table on this site, the area under the standard normal curve, to the left of Z = is 0.13, or 13%. 4.Therefore, the overall process yield is 100% - 0.3% - 13% = 86.7% Calculations 1.From Example #1, we know that the area under the standard normal curve exceeding the 5.6 amp USL is 0.3%. 2.To calculate the area to the left of the LSL, we use Z= ( )/0.8 = Using the Z table on this site, the area under the standard normal curve, to the left of Z = is 0.13, or 13%. 4.Therefore, the overall process yield is 100% - 0.3% - 13% = 86.7%