Calculating Defects Per Million Using Normal Probability Tables

The following procedure provides step-by-step instructions on how to calculate the defects per million assuming that data follow a normal distribution and the process has a bilateral tolerance (+/- tolerance from a target value such as 25.4 +/- 0.05 cm).

Step 1: Obtain necessary input data information.

Specifications: Target, Upper Specification Limit (USL) and Lower Specification Limit (LSL)

Summary Statistics from Data Set: Estimate of the Sample Mean and Standard Deviation

Example: Suppose you are trying to produce parts for a specification of 25.4 +/- 0.05. You sample 100 parts and obtain a mean = 25.41 and sample standard deviation = 0.02

Target = 25.4; USL = 25.45; LSL = 25.35; Mean = 25.41; Std Dev = 0.02

Step 2: Pictorially show the USL, LSL, Target, Mean, and Std Deviation

Note: identify whether the mean is closer to the USL or the LSL as the defects per million should be greater on the side that is closest to the mean.

Example: graph of the above problem.

Step 3: Calculate the probability of a defect above the USL and below the LSL.

3a. Calculate Pr(Defect > USL). To obtain the probability that a part will be produced greater than the USL, we need to calculate a Z-value for the USL (Zusl) to be looked up in a table. We may also use an Excel built-in function to obtain this probability.

          Compute Zusl = (USL – Mean) / std deviation

          From Zusl, we may determine the Pr (Defect > USL).

          Pr (Defect > USL) = 1 – Pr(Z<Zusl).

NOTE: Normal probability tables are presented as the probability from negative infinity to Z. Thus, for calculating defects greater than the USL, we need to let Pr (Defect > USL) = 1 – Pr (Z < Zusl). Pr(Z < Zusl) is obtained by looking up the value for Zusl in a normal probability table.

Example: Target = 25.4; USL = 25.45; LSL = 25.35; Mean = 25.41; Std Dev = 0.02

Zusl = (25.45 – 25.41) / 0.02 = 2.00

Pr (Z < Zusl) = 0.97725  (based on Normal Table Lookup where Zusl = 2.0)

          Alternatively in Excel:  =normsdist(2.0)    à  0.97725 

Pr (Defect > USL) = 1 – Pr (Z < Zusl) = 1 – 0.97725 = 0.02275

3b. Calculate Pr(Defect < LSL). To obtain the probability that a part will be produced less than the LSL, we need to calculate a Z-value for the LSL (Zlsl) to be looked up in a table. We may also use an Excel built-in function to obtain this probability.

          Compute Zlsl = (LSL – Mean) / std deviation

          From Zlsl, we may determine the Pr (Defect < LSL).

          Pr (Defect < LSL) = Pr(Z<Zlsl).

NOTE: Normal probability tables are presented as the probability from negative infinity to Z. Thus, for calculating defects less than the LSL, we need to let Pr (Defect < LSL) = Pr (Z<Zlsl). Pr(Z < Zlsl) is obtained by looking up the value for Zlsl in a normal probability table.

Example: Target = 25.4; USL = 25.45; LSL = 25.35; Mean = 25.41; Std Dev = 0.02

Zlsl = (25.35 – 25.41) / 0.02 = -3.00

Pr (Z < Zlsl) = 0.00135  (based on Normal Table Lookup where Z = -3.0)

          Alternatively in Excel:  =normsdist(-3.0)    à  0.00135 

Pr (Defect < LSL) = 0.00135

Step 4: Calculate the probability of a defect.

          Pr (Defect) = Pr (Defect > USL) + Pr (Defect < LSL)

          Example: Pr (Defect) = 0.02275 + 0.00135 = 0.02410

Step 5: Calculate the Actual DPM

          Actual DPM = Pr (Defect) * 1,000,000

          Example: Actual DPM = 0.02410 * 1M = 24,100 DPM

Calculating Potential DPM:

We may want to calculate the Potential DPM, which represents the DPM that could be achieved if the process mean is shifted to the target value and the standard deviation does not change. To compute the potential DPM, repeat the above steps but substitute the target value for the mean. Note: Pr (Defect < LSL) should be equal to Pr (Defect > USL) if your target value is at the center of the USL and LSL. Also, your potential DPM should be less than your actual DPM if your current mean is not equal to your target value.

Example:

          Zusl = (25.45 – 25.4) / 0.02 = 2.5  (table lookup à 0.99379)

          Pr (Defect > USL) = 1- 0.99379 = 0.00621

          Zlsl = (25.35 – 25.4) / 0.02 = -2.5  (table lookup à 0.00621)

          Pr (Defect < LSL) = 0.00621

          Pr (Defect) = 0.00621 + 0.00621 = 0.01242

          DPM = 0.01242 * 1 M = 12,420 DPM

COMMENT: For this example, shifting the mean to the target value (given the same standard deviation) would have the effect of reducing the DPM by approx one-half.