The Mea sur e P has e 203

NOT Easy to Contact BBTM

SUPPLIERS INPUTS

• Phone company • Customer doesn't

• Product know where to look

• Internal computer for number

systems • Number in manual is

• Internal phone incorrect

systems • Dial phone

incorrectly

• Customer makes wrongVRU selections

• CSA sees no customer info

• Customer presents incorrect question or issue

• Customer won't give CC info

• Customer won 't listen to upsell

CSA selects wrong transfer option

• Phone system error

FIGURE 7.4 SIPOC for undesirable outcomes.

CSAobtains credit card

information. Upsell.

CSA transfers call to

technician

OUTPUTS CUSTOMERS

• Customer can't find • User number • Technician

• Customer gets 'wrong number ' message

• No callerID

• VRU drops call

• VRU routes call incorrectly

• VRU loses call

• Wrong person answers call

• Call queued

• Customer annoyed by upsell

• Call goes to wrong person

• Call is lost

• Call goes to queue

and the results of the manipulation are studied to help the analyst better understand the empirical system.

In reality, measurement is problematic: the analyst can never know the "true" value of the element being measured. The numbers provide information on a cer­tain scale and they represent measurements of some unobservable variable of inter­est. Some measurements are richer than others, that is, some measurements provide more information than other measurements. The information content of a number is dependent on the scale of measurement used. This scale determines the types of statistical analyses that can be properly employed in studying the numbers. Until one has determined the scale of measurement, one cannot know if a given method of analysis is valid.

Measurement Scales The four measurement scales are: nominal, ordinal, interval, and ratio. Harrington (1992) summarizes the properties of each scale in Table 7.1.

Numbers on a nominal scale aren't measurements at all; they are merely category labels in numerical form. Nominal measurements might indicate membership in a group (1 = male, 2 = female) or simply represent a designation (John Doe is 43 on the team). Nominal scales represent the simplest and weakest form of measurement. Nominal variables are perhaps best viewed as a form of classification rather than as a measurement scale.

204 C hap te r S eye n

Scale Definition Example Statistics

Nominal Only the presence/absence of an Go/no go; Percent; attribute; can only count items success/fail; proportion;

accept/reject chi-square tests

Ordinal Can say that one item has more or Taste; Rank-order less of an attribute than another item; attractiveness correlation can order a set of items

Interval Difference between any two successive Calendar time; Correlations; points is equal; often treated as a temperature t-tests; F-tests; ratio scale even if assumption of multiple regression equal intervals is incorrect; can add, subtract, order objects

Ratio True zero point indicates absence Elapsed time; t-test; F-test; of an attribute; can add, subtract, distance; weight correlations; multiply and divide multiple regression

From Harrington (1992). P516. Copyright © 1992. Used by permission of the publisher, ASQ Quality Press, Milwaukee, Wisconsin.

TABLE 7.1 Types of Measurement Scales and Permissible Statistics

Ideally, categories on the nominal scale are constructed in such a way that all objects in the universe are members of one and only one class. Data collected on a nominal scale are called attribute data . The only mathematical operations permitted on nominal scales are = (equality, which shows that an object possesses the attribute of concern) or "* (inequality).

An ordinal variable is one that has a natural ordering of its possible values, but for which the distances between the values are undefined. An example is product prefer­ence rankings such as good, better, or best. Ordinal data can be analyzed with the math­ematical operators, = (equality), "* (inequality), > (greater than), and < (less than). There are a wide variety of statistical techniques which can be applied to ordinal data includ­ing the Pearson correlation, discussed in Chap. 10. Other ordinal models include odds­ratio measures, log-linear models and logit models, both of which are used to analyze cross-classifications of ordinal data presented in contingency tables. In quality manage­ment, ordinal data are commonly converted into nominal data and analyzed using binomial or Poisson models. For example, if parts were classified using a poor-good­excellent ordering, the quality analyst might plot a p chart of the proportion of items in the poor category.

Interval scales consist of measurements where the ratios of differences are invari­ant. For example, 90°C = 194°F, 180°C = 356°F, 270°C = 518°F, 360°C = 680°F. Now 194°F /90°C"* 356°F /180°C, but

356°F -194°F 680°F - 518°F

180°C - 90°C 360°C - 270°C

The Mea sur e P has e 205

Conversion between two interval scales is accomplished by the transformation

y =ax+ b a> 0

For example,

where a = 9/5, b = 32. As with ratio scales, when permissible transformations are made statistical, results are unaffected by the interval scale used. Also, 0° (on either scale) is arbitrary. In this example, zero does not indicate an absence of heat.

Ratio scale measurements are so called because measurements of an object in two different metrics are related to one another by an invariant ratio. For example, if an object's mass was measured in pounds (x) and kilograms (y), then x/y = 2.2 for all values of x and y. This implies that a change from one ratio measurement scale to another is performed by a transformation of the form y = ax, a > 0; for example, pounds = 2.2 x kilograms. When permissible transformations are used, statistical results based on the data are identical regardless of the ratio scale used. Zero has an inherent meaning: in this example it signifies an absence of mass.

Discrete and Continuous Data A more general classification of measurements may also be made, which is also use­ful in defining suitable probability distributions and analysis tools discussed later in this chapter. Data are said to be discrete when they take on only a finite number of points that can be represented by the nonnegative integers. An example of discrete data is the number of defects in a sample. Data are said to be continuous when they exist on an interval, or on several intervals. An example of continuous data is the measurement of pH.

For most purposes, nominal and ordinal data are considered discrete, while inter­vals and ratios possess qualities of continuous data. While discrete data may take on integer form only, a continuous data value may be defined theoretically to an infinite number of decimal places, assuming it could be measured as such. In the real world, even though length or time, for example, could theoretically be measured to an infinite number of decimal places, we are be limited by our measurement system. If the varia­tion in length or time is smaller than the smallest unit of measure, the resulting data is essentially discrete, since the same data value is recorded for the majority of the data. Similar information content could be obtained by counting the number of items with that dimension.

Fundamentally, any item measure should meet two tests:

1. The item measures what it is intended to measure (Le., it is valid).

2. A remeasurement would order individual responses in the same way (Le., it is reliable) .

The remainder of this chapter describes techniques and procedures designed to ensure that measurement systems produce numbers with these properties.

CHAPTER 9 Measurement Systems

Evaluation

A gOOd measurement system possesses certain properties. First, it should pro­duce a number that is "close" to the actual property being measured, that is, it should be accurate. Second, if the measurement system is applied repeatedly to

the same object, the measurements produced should be close to one another, that is, it should be repeatable. Third, the measurement system should be able to produce accurate and consistent results over the entire range of concern, that is, it should be linear. Fourth, the measurement system should produce the same results when used by any properly trained individual, that is, the results should be reproducible. Finally, when applied to the same items the measurement system should produce the same results in the future as it did in the past, that is, it should be stable. The remainder of this section is devoted to discussing ways to ascertain these properties for particular measurement systems. In general, the methods and definitions presented here are consistent with those described by the Automotive Industry Action Group (AIAG) MSA Reference Manual (3rd ed.).

Definitions Bias: The difference between the average measured value and a reference value is referred to as bias . The reference value is an agreed-upon standard, such as a standard traceable to a national standards body (see below). When applied to attribute inspec­tion, bias refers to the ability of the attribute inspection system to produce agreement on inspection standards. Bias is controlled by calibration, which is the process of comparing measurements to standards. The concept of bias is illustrated in Fig. 9.1.

Repeatability: AIAG defines repeatability as the variation in measurements obtained with one measurement instrument when used several times by one appraiser, while measuring the identical characteristic on the same part. Variation obtained when the measurement system is applied repeatedly under the same conditions is usually caused by conditions inherent in the measurement system.

ASQ defines precision as "The closeness of agreement between randomly selected individual measurements or test results. NOTE: The standard deviation of the error of measurement is sometimes called 'imprecision"' . This is similar to what we are calling repeatability. Repeatability is illustrated in Fig. 9.2.

Reproducibility: Reproducibility is the variation in the average of the measurements made by different appraisers using the same measuring instrument when measuring the identical characteristic on the same part. Reproducibility is illustrated in Fig. 9.3.

289

290 C hap te r N i n e

Reference value

FIGURE 9.1 Bias illustrated.

Bias

Average measurement

1 ..... ------ Repeatability -------1

FIGURE 9.2 Repeatability illustrated.

Mea sur e men t S y stem s E y a I u a t ion 291

Frank

FIGURE 9.3 Reproducibility illustrated.

Stability: Stability is the total variation in the measurements obtained with a mea­surement system on the same master or parts when measuring a single characteristic over an extended time period. A system is said to be stable if the results are the same at different points in time. Stability is illustrated in Fig. 9.4.

Linearity: the difference in the bias values through the expected operating range of the gage. Linearity is illustrated in Fig. 9.5.

Historically, calibration has been the standard approach to limit the effects of bias, long considered the fundamental source of measurement error. Modern measurement system analysis goes well beyond calibration. A gage can be perfectly accurate when checking a standard and still be entirely unacceptable for measuring a product or controlling a process. This section illustrates techniques for quantifying discrimina­tion, stability, bias, repeatability, reproducibility and variation for a measurement sys­tem. Control charts are used to provide graphical portrayals of the measurement processes, enabling the analyst to detect special causes that numerical methods alone would not detect.

Measurement System Discrimination Discrimination, sometimes called resolution, refers to the ability of the measurement system to divide measurements into "data categories." All parts within a particular data category will measure the same. For example, if a measurement system has a

292 C hap te r N i n e

FIGURE 9.4 Stability illustrated.

I I I I I I

Friday

1--.. I-----------1~ Stability

resolution of 0.001 inch, then items measuring 1.0002, 1.0003, 0.9997 would all be placed in the data category 1.000, that is, they would all measure 1.000 inch with this particular measurement system. A measurement system's discrimination should enable it to divide the region of interest into many data categories. In Six Sigma, the region of inter­est is the smaller of the tolerance (the high specification minus the low specification) or six standard deviations. A measurement system should be able to divide the region of interest into at least five data categories. For example, if a process was capable (Le., Six Sigma is less than the tolerance) and cr = 0.0005, then a gage with a discrimination of 0.0005 would be acceptable (six data categories), but one with a discrimination of 0.001 would not (three data categories). When unacceptable discrimination exists, the range chart shows discrete "jumps" or "steps." This situation is illustrated in Fig. 9.6.

Note that on the control charts shown in Fig. 9.6, the data plotted are the same, except that the data on the bottom two charts were rounded to the nearest 25. The effect is most easily seen on the R chart, which appears highly stratified. As sometimes hap­pens (but not always), the result is to make the X-bar chart go out of control, even though the process is in control, as shown by the control charts with unrounded data. The remedy is to use a measurement system capable of additional discrimination, that is, add more significant digits. If this cannot be done, it is possible to adjust the control limits for the round-off error by using a more involved method of computing the con­trollimits, see Pyzdek (1992a, pp. 37-42) for details.

Part size near high

end of range

Measurements of a part checked

repeatedly

FIGURE 9.5 Linearity illustrated.

Stability

Mea sur e men t S y stem s E y a I u a t ion 293

Reference value

Larger bias near small

end of range

Part size near small

end of range

Measurement system stability is the change in bias over time when using a measurement system to measure a given master part or standard. Statistical stability is a broader term that refers to the overall consistency of measurements over time, including variation from all causes, including bias, repeatability, reproducibility, etc. A system's statistical stability is determined through the use of control charts. Averages and range charts are typically plotted on measurements of a standard or a master part. The standard is mea­sured repeatedly over a short time, sayan hour; then the measurements are repeated at predetermined intervals, say weekly. Subject matter expertise is needed to determine the subgroup size, sampling intervals and measurement procedures to be followed. Control charts are then constructed and evaluated. A (statistically) stable system will show no out-of-control signals on an X-control chart of the averages' readings. No "sta­bility number" is calculated for statistical stability; the system either is or is not statisti­cally stable.