Module 1.1 Notes
"Why Statistics for Management"

 

"If you know a thing only qualitatively, you know it no more than vaguely…If you know it quantitatively--grasping some numerical measure that distinguishes it from an infinite number of other possibilities--you are beginning to know it deeply…You comprehend some of its beauty and you gain access to its power and the understanding it provides…"

Carl Sagan (1997), "Billions and Billions: Thoughts on Life and Death At the Brink of the Millennium."

 

Let's move this thought into a business world example to introduce statistics for managers. Don't worry about how to do the computations - we will cover that later. For now, just focus on the concepts illustrated in this introduction.

... If you don't know a process quantitatively, you run the risk of making at least two errors (D. Wheeler, 1993).

The first error is to interpret noise as if it were a signal.

Figure 1.1.1 provides an example of a run chart tracking average monthly cycle times for materials coming into a manufacturing firm (this example is based on an actual experience I had in doing a logistics system analysis and design project for Johnson & Johnson's Sterile Design Company several years ago under a grant at the University of South Florida.) Cycle time is the elapsed time from when materials are ordered until they arrive. The average cycle time would be an average for all materials ordered in a particular month. So in April, 1998 (Month 16), the average cycle time was 20 days, or almost 3 weeks for all of the items ordered that month. Note the "Boss says" that no average monthly cycle time should be over 24 days, so that the April 1999 experience of 25 days produces an "Oh No" from the procurement office. At the minimum, these "Oh No's" require exception reports and create stress. In the worst cases, these "Oh No's" may lead to high turnover, a lot of sick leave, or "cover-up" responses, especially when team members believe the process is behaving as expected.

Figure 1.1.1

If the organization defines cycle time as an important output of the supply chain, then that variable should be measured. Such measurement enables the process to be analyzed so it can be improved, managed and controlled. Verifiable measurements, now often referred to as metrics (S. Melnyk, 1999), put data in its context and gives it meaning (D. Wheeler, 1993).

As we will learn in Modules 1.2 and 1.3 Notes, measurement of continuous numerical data such as time and dollars, involves measuring the center and spread or variability of the data. Once this is done, we can understand the "Voice of the Process," a term coined in 1924 by Walter Shewhart, one of the founding fathers of Statistical Process Control. So, rather than just looking at one data point in comparison to the bosses target, or just comparing a current data point to the same point a year ago, we analyze all 30 months of current cycle time data to discover the Voice of the Process in a Process Control Chart such as Figure 1.1.2.

Figure 1.1.2.

In Figure 1.1.2, the Voice of the Process suggests the process center or mean is 21 days over the past 30 months. Note that some of the monthly average observations are above the mean, and some are below - that is, there is deviation around the process mean. The maximum deviation for this process is computed as 30 days, and the minimum is 12. These are called the upper bound or upper control limit (UCL) of 30 and a lower control limit (LCL) of 12. These upper and lower limits are 9 days above and below the mean (we will see how to compute the mean and spread in Module 1.3.) For now, just accept the fact that any observation within the upper and lower control limit is called "noise." Every process generates noise when it is in control. Processes that are in control have the good properties of being stable and predictable. We will see how to construct these process control charts and control limits in Module 1.3 Notes.

Processes that are in control should not generate signals, or observations outside the control limits. Signals are also known as outliers. Whenever signals are encountered, such as a cycle time of 33 days, there should be an investigation to determine and correct the cause of the out-of-control problem.

The Voice of the Process helps us avoid the error of interpreting noise as if it were a signal - the first error in the interpretation of data. It also helps in avoiding the second error: failing to detect a signal when it is present (D. Wheeler, 1993). Note carefully that Figure 1.1.2 shows a process that is in control. Also note that the boss's specification limit of 24 days (referred to as "Boss Says" in Figure 1.1.1), is formally called an upper or lower specification limit, in the process control chart. As the chart shows, this process is not able to satisfy the boss 100 percent of the time as the boss's upper specification limit (USL) is within the upper control limit. Specification limits express the Voice of the Customer.

This is an important point. The proper translation of data into information gives us vital knowledge about the process. Even though the process is in control, the boss is unsatisfied. That is, being in control is not necessarily good or bad. "Goodness" or "badness" of a process depends on the targets set by the customer (here the boss would be an internal customer of the process - there are obviously both internal customers such as bosses and team workers, and external customers such as buyers of products and services.) A process that is in control is a good process if the customer's target specification limits are being met by the process. A process that is in control is a bad process if the customer's target specification limits are not being met by the process.

When the Voice of the Customer, expressed as target specification limits on the process, is not being met by the Voice of the Process, there is conflict. To resolve the conflict and satisfy the boss, we could reduce the variation of the process (make the upper and lower control limits "tighter"). For example, if the variation of the process were reduced to an UCL of 23 and a lower control limit of 19 around the current mean of 21, we would have an effective process with respect to the Voice of the Customer, since the boss's upper specification limit of 24 would be outside the upper control limit. This would mean there should never be an observation above 23. How can we reduce variation? One way would be to shift from common to dedicated carrier - the transportation cost is higher but the inventory carrying cost due to reduction in safety stock is much lower.

Another option is to shift the mean of the process, keeping the variation constant at plus or minus a total of 9 days. In this example, we would want to shift the mean from 21 to perhaps 14, which results in a new upper control limit of 23 days (14 plus 9). Here again, the upper control limit is lower than the customer's upper specification limit, so this process would be considered capable of meeting customer expectations. The mean could be shifted by switching to quicker, more costly modes of transportation. That cost was an accepted cost of the late 1980's and early 1990's as most companies began competing by both quality and cycle time reduction - getting product to market sooner, without defect.

A side note: please understand in this introductory example that our attention has been on the upper control limit and upper specification limit. This is common when cycle time is the variable of interest, since concern is usually with longer rather than shorter cycle times. Sometimes, we are interested in moving up lower control limits, such as when we are measuring profit contribution or revenue growth. Sometimes we are interested in both upper and lower control and specification limits such as in monitoring the tolerance of manufactured parts.

A third option is to reduce the variability and shift the mean - a combination approach. Here it is customary to reduce the variation before shifting the mean, since processes with little variation are much more stable. If the mean is shifted in a process with great variation, the shift may be undetectable.

Please note carefully that I omitted two other approaches that are, sadly, taken in many organizations. I say "sadly" since neither results in continuous improvement. One is for the "boss" (internal customer) to get angry with the team running the process and identify someone/some unit for punishment. In this example, this is virtually guaranteed since there are occasions in Figure 1.1.2 when the target specification limit is not met (there are observations above the upper specification limit of 24). Rather than getting angry with the team running the process, the boss should work with the team to improve the process (reduce variation or shift the mean), and provide the needed resources for the improvement.

Did you think of another approach that I omitted.....telling the customer ("boss" in this case) that he or she is wrong - the specification limit is too tight.....Hello.....!! In one of the first quality improvement seminars I was conducting for GE Client Business Services several years ago, I said; "to satisfy the customer when specification limits are within process control limits, reduce the variation, shift the mean, do both, or ask the customer to reset their specification limits." Well, there were about 40 students in the class and a strange hush fell upon the room.... the class leader finally said, "what, tell the customer they are wrong... not at GE!"

I agree with the GE philosophy, the customer is right. However, in my defense, I stated that the customer may have set unreasonable targets, at least in the short run. The Japanese auto manufacturers understood this many years ago when they started dealing with US vendors for auto parts. To get US vendors to deliver "zero-defect" parts and subassemblies would take a series of small continuous improvement steps (gradually setting tighter and tighter specification limits) to migrate from high percent defects to zero-defects. Of course the US vendors rapidly caught on when they realized that competition required high quality just for market entry - it wasn't a product differentiator any more.


I chose this example to illustrate that business statistics isn't about formulas to crank to turn in homework in college classes. Rather, it's about putting data in its context through appropriate measurement to understand and improve process capability and performance, and to make inferences and predictions. This course will hopefully expose you to that branch of decision sciences called statistics that enables us to put data in its context, and to transform that data into information and knowledge. Statistics enables managers to know how to (Leven, Berenson and Stephan, 1999):

Properly present and describe information (descriptive statistics)

Draw conclusions about populations based on information obtained from samples (inferential statistics)

Improve processes (Continuous Improvement)

Obtain reliable forecasts and predictions

The remainder of Module 1 will take us through descriptive and inferential statistics for continuous numerical variables such as time, whose data elements are measured in, for example, minutes or fractions of minutes. Other continuous variables include weight, measured in pounds or tons; height, measured in inches or feet; and revenue, variable cost or profit contribution measured in dollars or thousands of dollars. Discrete numerical data is distinguished from continuous numerical data in that the discrete number scale contains only discrete integers such as 1, 2, 3, as would be found in counting. For example, suppose there are 5 firms in a small data sample, and these firms make an average of $5,000 profit contribution. Here, the "5 firms" represent discrete data, and the "$5,000 profit" represents continuous data. Discrete data will be discussed in Module 5.

Modules 2 and 3 focus on regression and correlation analysis, or the study of the strength, form and direction of relationships between variables. In addition to understanding the relationship between variables, regression analysis is the tool managers use to make reliable predictions and forecasts. You may have read about female faculty members at the University of South Florida claiming gender bias in faculty salaries. A regression analysis of male versus female faculty salaries over time (the diamonds represent actual salaries and years of experience for a sample of male and female faculty members) quickly demonstrates the value of quantitative measurement in supporting a claim. Figure 1.1.3, although based on fictitious data, is representative of that analysis. This figure reflects that while both male (top curved prediction line of squares) and female (bottom curved prediction line of squares) faculty salaries increase at a decreasing rate (a common phenomenon at a university), males make more on average than females. In studies such as this, analysts need to take care that subjects are selected from similar disciplines and colleges to reduce the impact of confounding variables.

Figure 1.1.3.

 In Module 5 we introduce categorical data-type variables that are measured by counting observations within a category level. For example, note in Figure 1.1.2 that four observations were above the upper specification limit. The boss would say we have four defects. In this scenario, the cycle time variable is considered a categorical variable with two values: late (cycle times above 24) and not late (cycle times at or below 24 days). The focus here is on the number of shipments that were defective (late), not the length of time they were late as in the continuous numerical variable example. Categorical variables then have named categories such as defective/not defective, in season/not in season, poor/satisfactory/good, and so forth. Much more on this subject when we get to Module 5.

Sampling Concepts & Procedures

 

In business, data usually arises from accounting transactions or management processes (i.e., inventory, sales, and payroll). Much of the data we analyze were recorded without explicit consideration, yet many decisions may depend on the data.  Let's go over some important definitions before you collect the data for your assignments.

 

A subject or individual is a single member of a collection of items that we want to study.  A variable is a characteristic of a subject or individual such as employee’s income or invoice amount.  A data set consists of all the values of all the variables for all the individuals we have chosen to observe, that is, a collection of observations. Table 1.1 relates the type of dataset with the amount of variables you may want to study/present. In Asgn 1, for example, will conduct a univariate study; that is, we are interested in describing a process/construct through only one variable or characteristic process.  In my example for Module 1, I am interested in measuring customer service which of course is composed of many variables/characteristics (reliability, on time delivery, etc).  However, I elected to present/describe it only in terms of cycle time, which is viewed by management as very important to please our customers. Therefore, in Module 1, we'll deal with univariate study.

 

            Table 1.1

 

Modules 2 will study Bivariate data through simple regression analysis. Modules 3, 4, and 5 will require the use of multivariate dataset.  Figure 1.1.4 illustrate a multivariate data set.

 

            Figure 1.1.4

 

Sample or Census?

 

A sample involves looking only at some items selected from the population.  A census is an examination of all items in a defined population.  Why can’t the United States Census survey every person in the population?

 

Situations Where a Sample May Be Preferred …

 

·         Infinite Population:  No census is possible if the population is infinite or of indefinite size (an assembly line can keep producing bolts, a doctor can keep seeing more patients).

 

·         Destructive Testing:  The act of sampling may destroy or devalue the item (measuring battery life, testing auto crashworthiness, or testing aircraft turbofan engine life).

 

·         Timely Results:  Sampling may yield more timely results than a census (checking wheat samples for moisture and protein content, checking peanut butter for aflatoxin contamination).

 

·         Accuracy:  Sample estimates can be more accurate than a census.  Instead of spreading limited resources thinly to attempt a census, our budget of time and money might be better spent to hire experienced staff, improve training of field interviewers, and improve data safeguards.

 

·         Cost: Even if it is feasible to take a census, the cost, either in time or money, may exceed our budget.

 

·         Sensitive Information: Some kinds of information are better captured by a well-designed sample, rather than attempting a census. Confidentiality may also be improved in a carefully-done sample.

 

Situations Where a Census May Be Preferred …

 

·         Small Population: If the population is small, there is little reason to sample, for the effort of data collection may be only a small part of the total cost.

 

·         Large Sample Size: If the required sample size approaches the population size, we might as well go ahead and take a census.

 

·         Database Exists: If the data are on disk we can examine 100% of the cases.  But auditing or validating data against physical records may raise the cost.

 

·         Legal Requirements: Banks must count all the cash in bank teller drawers at the end of each business day.  The U.S. Congress forbade sampling in the 2000 decennial population census.

 

Pause and reflect: A parameter is any measurement that describes an entire population.  Usually, the parameter value is unknown since we rarely can observe the entire population.  Parameters are often (but not always) represented by Greek letters

 

Figure 1.1.5

 

Statistics are any measurement computed from a sample.  Usually, the statistic is regarded as an estimate of a population parameter.  Sample statistics are often (but not always) represented by Roman letters.

 

Let's review two situations in which samples provide estimates of population parameters.

 

1. a tire manufacturer developed a new tire designed to provide an increase in mileage over the firm's current line of tires. To estimate the mean number of miles provided by the new tire, the manufacturer selected a sample of 120 new tires for testing. The test provided a sample mean of 36,500 miles. Hence, an estimate of the mean tire mileage for the population of new tires was 36,500 miles

2. Members of a political party were considering supporting a particular candidate for election to the U.S. Senate, and party leaders wanted an estimate of the proportion of registered voters supporting the candidate. The time and cost associated with contacting every potential voter were prohibitive. Hence, a sample of 400 registered voters was selected and 160 of the 400 voters indicated a preference for the candidate.  An estimaete of the proportion of the population of registered voters supporting the candidate was 160/400 = 0.40.

 

These two examples illustrate why samples are used. It is important to realize that sample results provide only estimates of the value of the population characteristics. We do not expect the mean mileage for all tires in the population to be exactly 36,500 miles, nor do we expect exactly 0.40, or 40%, of the population of registered voters to support the candidate. However, proper sampling methods, the sample results will provide 'good' estimates of the population parameters.  Let's briefly see some types of sampling procedures.

 

Sampling procedures:

 

·         Simple Random Sample:  Use random numbers to select items from a list (e.g., VISA cardholders).

·         Systematic Sample: Select every kth item from a list or sequence (e.g., restaurant customers).

·         Stratified Sample: Select randomly within defined strata (e.g., by age, occupation, gender).

·         Cluster Sample: Like stratified sampling except strata are geographical areas (e.g., zip codes).

·         Judgment Sample : Use expert knowledge to choose “typical” items (e.g., which employees to interview).

·         Convenience Sample : Use a sample that happens to be available (e.g., ask co-worker opinions at lunch).

·         Focus Groups : In-depth dialog with a representative panel of individuals (e.g. iPod users).

 

 

A very important sampling procedure is called Simple Random Sampling: Every item in the population of N items has the same chance of being chosen in the sample of n items. We rely on random numbers to select a name.

 

Example:

Figure 1.1.6

 

There are 48 names in the list presented in figure 1.1.6 and we need to select one at random.  There are random tables we could use for this task but in this course we'll use Excel random number generator.  In Excel, type in a cell, =RANDBETWEEN(1,48).  The formula returns a random number between 1 and 48.  In the example above, the return is 44, so we can select Stephanie from the list. When the data is arranged in a rectangular array (or Table), an item can be chosen at random by randomly selecting a row and column.

 

 

Use =RANDBETWEEN(1,3) function to randomly choose a column and =RANDBEWTEEN(1,4)  to choose a row. This way, each item has an equal chance of being selected.

 

Often, we have the need to randomize a LIST:  In Excel, use function =RAND() beside each row to create a column of random numbers between 0 and 1. Let's see an example.

 

Name

Major

Gender

Claudia

Accounting

F

Dan

Economics

M

Dave

Human Res

M

Kalisha

MIS

F

LaDonna

Finance

F

Marcia

Accounting

F

Matt

Undecided

M

Moira

Accounting

F

Rachel

Oper Mgt

F

Ryan

MIS

M

Tammy

Marketing

F

Victor

Marketing

M

Rand

Name

Major

Gender

0,382091

Claudia

Accounting

F

0,730061

Dan

Economics

M

0,143539

Dave

Human Res

M

0,906060

Kalisha

MIS

F

0,624378

LaDonna

Finance

F

0,229854

Marcia

Accounting

F

0,604377

Matt

Undecided

M

0,798923

Moira

Accounting

F

0,431740

Rachel

Oper Mgt

F

0,334449

Ryan

MIS

M

0,836594

Tammy

Marketing

F

0,402726

Victor

Marketing

M

 

Now you sort from in ascending order.  The first n items are a random sample of the entire list (they are as likely as any others).

 

Rand

Name

Major

Gender

0,143539

Dave

Human Res

M

0,229854

Marcia

Accounting

F

0,334449

Ryan

MIS

M

0,382091

Claudia

Accounting

F

0,402726

Victor

Marketing

M

0,431740

Rachel

Oper Mgt

F

0,604377

Matt

Undecided

M

0,624378

LaDonna

Finance

F

0,730061

Dan

Economics

M

0,798923

Moira

Accounting

F

0,836594

Tammy

Marketing

F

0,906060

Kalisha

MIS

F

 

Systematic Sampling:  Sample by choosing every kth item from a list, starting from a randomly chosen entry on the list.  Systematic sampling should yield acceptable results unless patterns in the population happen to recur at periodicity k.  It can be used with unlistable or infinite populations.  Systematic samples are well-suited to linearly organized physical populations.

 

A systematic sample of n items from a population of N items requires that periodicity k be approximately N/n.  For example, out of 501 companies, we want to obtain a sample of 25. What should the periodicity k be?  k = N/n ,  501/25 » 20. So, we should choose every 20th company from a random starting point.

 

For example, starting at item 2, we sample every k = 4 items to obtain a sample of n = 20 items from a list of N = 78 items. 

Note that N/n = 78/20 » 4.

 

That is all for now folks.  Try to apply some of these concepts when considering your data for the assignments in this course.  Remember that a good sampling procedure helps eliminate bias and increase the chance of a good estimate of the population parameter. We'll see more about sampling when we get to module 5.

 

-x-x-x-x-x-x-

 

 

 

 

 

 



References:

D. Anderson, D. Sweeney, and T. Williams, “Essentials of Modern Business Statistics with MS Excel”.  3rd Edition, Thompson/Southwestern. Chapter 1.

D. Groebner, P. Shannon, P. Fry & K. Smith.  Business Statistics: A Decision Making Approach, Seventh Edition, Prentice Hall, Chapter 1, 18

Ken Black. Business Statistics for Contemporary Decision Making. Fourth Edition, Wiley. Chapter 18

Leven, D., Berenson, M. & Stephan, D. (1999). Statistics for Managers Using Microsoft Excel (2nd ed.). Upper Saddle River, NJ: Prentice-Hall. Chapter 1.

Melnyk S. (March, 1999). "Metrics - The Missing Piece in Operations Management Research," Decision Line, Vol. 30, No. 2

Wheeler, D. (1993). Understanding Variation: The Key to Managing Chaos. Knoxville, TN: SPC Press, Inc.

 

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