TMGT 361
Assignment V Instructions
Lecture/Essay
Statistics 001
Though you might have forgot most of it, you have already had course work on most of the math and statistics required in this course. There was a prerequisite math quiz to review some of this math. Statistics is merely math (mostly algebra) aimed at (a) summarizing data (descriptive statistics) or (b) judging how well sample data fits a population of data (inferential statistics). Statistics as a term refers to doing a or b, the results of a or b, or the profession or field of study of the math to do a or b.
What makes statistics difficult or scary is not the math (software does that in a second) but the qualitative knowledge you need to do the statistics correctly (especially when software will do the number crunching) and interpret the results.
Terminology
It is helpful to understand the interrelationship of the following.
Population. A population is made up of things (or units or pieces, subjects, or test blanks, dogs, test tubes, persons, molecules, or other things). The population is the big set of things we are really interested in. Most hypotheses have to do with a population. Knowledge is most useful and generalizable when it pertains to a population. Usually, we do not have access to a population (because of time, money, availability, or other reasons).
Sample. A sample is a subset of the population. We can look at (test, measure, observe, experiment with) a sample much easier than we can the population. We can make decisions about the population based on the results of the sample (inferential statistics).
Sampling unit. The sampling unit is often called the unit of observation. The population and sample must have the same types of units or things because the sample is a subset of the things/units in the population. It is easy to understand a unit when it is common, discreet, individual thing, e.g., a person or a car. However, the unit can be a foot of rope, or a mile of rope, one marble, a gram of marble dust, a bag of marbles. The unit is often called the unit of observation due to the traditional reminder that for it to be measurable it must be observable. If I was measuring empathy or love or other emotion or desire, the same is true. I have to define those qualities (those variables, those characteristics) but I also need to know what they are qualities of. Aristotle’s explanation is still used and valid. There are objects (things) like an apple. The object has qualities (like color or sugar content or weight or number of worms).
Characteristic or quality. A unit/object has any number of characteristics or qualities. The important characteristics are often redundantly called quality characteristics (meaning that those are the important characteristics). Characteristics are also called variables (when they can vary; they are not constant), factors, inputs, descriptors, signals, attributes, and many others depending on the situation or profession.
Variable value. Variables have values (or levels, amounts, settings, labels, quantities, and many other synonymous terms, depending on the situation or profession). For example, the variable could be color; the value could be red. We make a lot of distinctions (accompanied by different or modified terminology) about variables depending on whether they are an input or an output, if they are the cause or the effect, how much influence they have, if we care about them or not, and other distinctions. Variables that we can’t control and might mess up our experiments can be called noise, environmental, extraneous, or confounding. There are too many variations of variables to discuss all of them here. The thing to remember is that things (units of observation) have variables and those variables have values, regardless of the specific terminology used. Quality as a discipline gets its name from its focus on keeping the values of the important variables at the levels the should be.
Measurement. Measurement can be a noun or a verb; it might mean the thermometer (the instrument or tool), the process (instructions, protocol, method) of using a thermometer to find someone’s temperature, the act of finding that temperature (performing the process), or the data resulting from finding the temperature, i.e., the temperature. In metrology (the science of measurement), measurement usually means the result, e.g., the temperature, but always be careful (written or oral communication) to be clear about the meaning of all terms. Measurement requires and instrument or tool, which might be a classroom quiz, a questionnaire, a micrometer, a bath scale, your eyes, … We generically call this instrument a test. Be aware that we also use the word test as a verb synonymous with measure used as a verb (English, gotta love it!). We often call the instrument by its name or its process name, e.g., the SAT, quiz 1, tryouts, ruler, bath scale, gauge, inspection, etc. The best measurement situations use standardized instruments and procedures (to use the instrument, produce a result-the measurement, and analyze the results). The analysis of measurement results is called an evaluation or a validation (and other things). The point is that evaluation places a value judgement on the measurement, e.g., my temperature is too high, the cake tastes good, the part is too small, etc.
Data scale levels. The following table summarizes the scale levels. I have chosen to call the lowest level—1.

Level

Name

Characteristics

Appropriate measures of central tendency

Appropriate measures of dispersion

Examples

1

Nominal, categorical

Data are labels (though the label may be a number)

Mode (largest category)

Frequency (number or percent/proportion) of each category

Color, gender, part numbers

2

Ordinal, rank

The labels are in a meaningful order

Median

Range (difference between high and low values)

Education degrees, your favorite foods

3

Interval

The distance between each of the things labeled is uniform.

Mean

Variance, Standard deviation (the square root of the variance)

Temperature in C or F. A scaled, standardized psychometric instrument, e.g., IQ, test or SAT exam

4

Ratio

Interval with an absolute zero

Mean

Variance, Standard deviation (the square root of the variance)

Length, weight, temperature in Kelvin

Note that each level has all the characteristics of all lower levels and that all the appropriate measures for each lower level are interpretable.

One of the biggest problems in using data is using the wrong type of data and/or measure of central tendency or variation. Various mathematical procedures require data that are categorical, rank, or interval/ratio. For a certain technique or situation, some data have to be continuous or dichotomous. Data might need a minimum discrimination or range. There is nothing you can do to make categorical or rank data into interval/ratio data, therefore, the mean or standard deviations of those data types will never be interpretable (although you can calculate a number). To find the average part number, to rank employee numbers, to find the standard deviation of region of the country (labeled region 1, region 2, etc.), or the range of game show guests (labeled contestant no. 1, no. 2, etc.) is meaningless. Remember that software doesn’t know what a number represents; you have to be smarter than the software.
We often have to break things down (cross tabulate) into subcategories to compare counts, percentages, and other information. There is never a need to show all descriptive statistics of any variable or subcategories or combinations of variables. The general rule always applies: Show what is necessary to accurately describe what needs to be described.
The nature of the variables is important, e.g., categorical, interval, etc. The purpose of the data is important, e.g., to describe (to teach, to inform, to convince, etc.) to show a relationship (or not), to test a hypothesis, etc. The audience and what they want to know and/or can understand is important, e.g., can they understand statistical conclusions, are you presenting to clients or your boss, do you need to get the idea across in 30 seconds or do you have an hour. The type of summary statistic is important, e.g., do you need a count, a percentage, a row total, an average, etc. Remember that just because something can be calculated (assuming the math is done correctly) doesn’t mean that it is interpretable. Even if something is calculated correctly and it is a meaningful statistic (e.g., the mean of salary, instead of the mean of eye color), doesn’t mean that it will be useful. You have to know when the frequency, the relative frequency, the cumulative frequency, the proportion, the ratio, the average, etc. is important. Likewise, you need to know what combinations and subcategories are important. If I tell you that the average person lives to be 75 years, this tells you nothing about females compared to males, smokers compared to non-smokers, etc.
Distributions. Strictly speaking, a distribution only pertains to interval or ratio data. However, the term is often used to describe the pattern or results of summarizing nominal or ordinal data, e.g., in a frequency table or bar chart. You are going to learn much more about distributions later in the course. For this assignment, simply go by eye and make a judgement if your data looks like it has a certain distribution, e.g., normal, exponential, etc.
There are many other related terms, e.g., accuracy and precision, that you will learn about in future assignments.
Reliability
Reliability is the broad (and mathematically intensive) area of quality that is concerned with things (products, machines, processes, etc.) working as they should (cost, quality, safety, etc.) for as long as they should with minimal maintenance or repair and if preventative or corrective maintenance is needed is it quick and easy. Reliability includes (or is greatly related to the following).
· Designing things to be more reliable.
· Testing things to see how reliable they are.
· Predicting the reliability or uptime of things (and conversely the downtime for maintenance, repair, changeovers, coffee breaks, etc.).
· Figuring out the best way to use things, maintain things, and service things to increase the reliability.
· To do all the above with the most quality, safety, economy (best use of money and resources), and other important concerns.
We do know general rules of thumb for making something more reliable, e.g., make it simple, build in parallel redundancies or backups (as compared to series elements, where if one part quits the entire thing quits). The simplest type of reliability test is to calculate the mean time until a unit fails. As an example, you turn on 10 light bulbs; you record how long each one works until it burns out. Then you state that average. Of course it usually is much more complicated. The average is often an exponential mean (not the mathematical average) because the failures usually are not normally distributed but some sort of exponential distribution. What if you don’t have time to test every light bulb to failure? Are you going to time-terminate the test (e.g., quit testing after 100 hours)? There is math to do this and estimate the failure rate even if no bulb burnt out in the 100 hours. During the testing, are you going to replace failures?
Initial Post
See the general assignment instructions for information about the quality and quantity expectations and evaluation criteria.
V. Data collection and analysis.
a. Decide on something you are going to measure. In your initial post write up, include the following. Keep an industrial focus if possible but you can measure anything. The measurement instrument can be anything, e.g., your eyes, a bath scale, a micrometer, etc.
i. Define the population measured.
ii. Define the unit of observation (e.g., case, subject).
iii. Define the sample measured and state the sample size (must be at least n=10).
iv. Describe the variable measured. Include scale level, i.e., nominal, ordinal, interval, or rank.
v. Describe the measurement tool and measurement method.
vi. Define the variable values measured and/or that could have been measured, e.g., there might have been green apples but none in your sample were.
vii. Calculate appropriate measures of central tendency and dispersion.
viii. Create an appropriate chart to display the data. Does the data look like it fits a particular distribution; should it, i.e., does the data fit the definition of a particular distribution?
b. Similar to example 5.8 in the Primer, calculate the mean and 3 sigma limits for 10 coin tosses. Toss an actual coin 10 times and compare the theoretical and actual results*. Are the theoretical and actual results different? If so, why? How do you interpret the control limits, e.g., what do they tell us?
c. Create your own probability example of compound events* (similar to examples 5.20 – 5.23 in the Primer).
d. Discuss the general principles of how to make something more reliable.
e. Do your own failure experiment.* Calculate Lambda and Theta.
 
Obviously, for a you have to measure something yourself and cannot get data from a source such as the internet, book, or company data.
For b, follow the example but use the values you need to for 10 coin tosses. You have to actually toss the coin to compare the theoretical population results with your actual sample results.
For c, come up with your own example. The goal is to make you practice with something more real than the text book. Do not get an example from any source except your life or your imagination.
For e, you could test anything to failure. I would pick something that fails in short order under testing, e.g., flexing something until it breaks (the bends can be the unit instead of time, but time would work too if you keep your flex rate constant). You could consider a basketball shot the event and a miss as a failure. You could bother your roommate until he or she gets mad (the failure) and see how long that takes (maybe that isn’t a good idea)! The point is to really do this, with something simple if need be (and not use an internet or book example and merely change some words around).
Page 4 of 5