1. The Z Table Estimate Area “A” Area “B” Point of Interest, x Mean
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
Area “A”
Area “B”
Z Score -3 σ -2 σ -1 σ 1 σ 2 σ 3 σ

2. Steps in Using a Z Table Area “A” Area “B” Point of Interest, x Mean
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.
Calculate the mean and standard deviation of the data set.
Using the mean, standard deviation, and the point of interest x, calculate Z using the formula Z= (x-μ)/σ.
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.
Point of Interest, x
Mean
Area “A”
Area “B”

3. Example #1 – One-Sided Specification Limit
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?
Calculations
Mean = 3.4 amps, Standard Deviation = 0.8 amps
Z= ( )/0.8 = 2.75
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
USL = 5.6 amps
Process Yield = 99.7%
(Z table value of 0.997)
Product Exceeding the 5.6 amp USL = 0.3%
(100% – 99.7%)
μ = 3.4 amps
σ = 0.8 amps

4. Example #2 – Upper and Lower Specification Limits
Calculations
From Example #1, we know that the area under the standard normal curve exceeding the 5.6 amp USL is 0.3%.
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%.
Therefore, the overall process yield is 100% - 0.3% - 13% = 86.7%
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?
LSL = 2.5 amps
USL = 5.6 amps
Product Exceeding the 5.6 amp USL = 0.3%
(100% – 99.7%)
μ = 3.4 amps
σ = 0.8 amps
Product Falling Below the 2.5 amp LSL = 13%