Six Sigma and Beyond: Statistical Process Control, Volume IV

STANDARDIZED VALUES ( Z SCORES)

The normal distribution may be used to estimate percentages for zones that do not match whole number standard deviations. To do this, define the zone of interest by standard deviations or fractions of a standard deviation. Convert the actual measurement scale (in millimeters) to a standard scale ( Z ). Standardized values ( Z ) make the normal distribution useful for any distribution that is similar in shape to the normal curve. The units of measurement (inches, millimeters, seconds, pounds , grams, volts , ohms, etc.) will not influence the calculations once the measurement scale is standardized. The standardized value ( Z ) formula for sample data is listed below:

where

The results of the Z formula produce a scale that is equal to that at the bottom of Figure 5.2. The size of the Z value indicates how many standard deviations the value is from the distribution mean. The sign of the answer (positive or negative) indicates whether the value is above or below the mean.

The standardized value ( Z ) needed to answer the item height question is calculated below:

• Gather the needed information and illustrate the problem (see Figure 5.5). X = value to be standardized (e.g., largest part that will not jam = 7.035 mm), Xbar = distribution mean (7.008 mm), s = distribution standard deviation (0.17 mm).

  Figure 5.5: Distribution of the height items.

• Calculate the Z value.

• Round the calculated Z value to two (2) decimal places (1.5882353 = 1.59). This is shown in Figure 5.6.

  Figure 5.6: The relationship of the z value to the distribution.

• Use Table 5.1 to identify the area of probability.

   

  Table 5.1: The Area under the Normal Curve

  1. The left-hand column of Table 5.1 is labeled Z. The numbers in the column are the units and tenths digits of the standardized Z value. For this example, the unit is 1 and the tenths digit is 5. The appropriate row is highlighted in Table 5.1.

  2. The top row of Table 5.1 has labels that include x 's. The x 's stand for the units and tenths digits of the standardized Z value.

     The vertical column of this table stands for the hundredths digit. For this example, the hundredths digit is 9. The appropriate column is highlighted in Table 5.1.

• The answer is found where the horizontal row 1.5 and the vertical column x.x9 meet. The answer is 0.559. The values that are contained in the body of Table 5.1 are proportions or percentages in decimal form. These numbers may be made into whole-number percentages by multiplying by 100.

  0.0559 — 100 = 5.59%

  • This means that approximately 5.59% of the parts made by Station No. 9 will become jammed in Station 19.