Mean

Point of Interest, x

Z Score

0 3-3 1 2-1-2σ 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.

http://www.ztable.org

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.

http://www.ztable.org

μ = 3.4 ampsσ = 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= (5.6-3.4)/0.8 = 2.753. Using the Z table on this site, the

area under the standard normal curve, to the left of Z is 0.997. Therefore, a theoretical 99.7% of the data will fall to the left of the USL

http://www.ztable.org

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 ampsσ = 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

http://www.ztable.org

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= (2.5-3.4)/0.8 = -1.13.

3. Using the Z table on this site, the area under the standard normal curve, to the left of Z = -1.13 is 0.13, or 13%.

4. Therefore, the overall process yield is 100% - 0.3% - 13% = 86.7%