class: center, middle, inverse layout: yes name: inverse ## STAT 305: Chapter 1 ##Part II ### Amin Shirazi .footnote[Course page: [ashirazist.github.io/stat305_s2020.github.io](https://ashirazist.github.io/stat305_s2020.github.io/)] --- class: center, middle, inverse layout: yes name: inverse ## Why Engineers Study Statistics ### Chapter 1: Introduction, Continued ### Chapter 2: Data Collection --- class: center, middle, inverse layout: yes name: inverse # Section 1.2 ## Basic Terminology, Continued --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures The most basic way to think about data is to imagine how the the raw observations could be organized once collected. Collected data can be referred to as a **data set**. If the data set is simple enough, we can store it in a **data table** or **flat file**. Traditional data tables store values relating to a single observation/unit/individual as a row of the table. Each column in the table represents a value for some observed characterstic observed. **Example**: Failure time of lightbulbs A single brand and model of lightbulb is being examined for average failure time. Five bulbs were run until they burned out and their lifetime was recorded in hours. The first bulb lasted 521.4 hours, the second bulb lasted 501.2 hours, the third bulb lasted 541.8 hours, the fourth bulb lasted 498.1 hours, and the fifth bulb lasted 528.2 hours. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures **Example**: Failure time of lightbulbs, continued Assembling the results in a data table could look like this: ```text Bulb Number Failure Time (hours) 1 521.4 2 501.2 3 541.8 4 498.1 5 528.2 ``` Each bulb tested gets its own row - which row is attached to which bulb is identified by the first column. The only feature being observed is failure time - so only one column of observations are recorded for each bulb. Notice: - Failure Time is a **quantitative continuous** variable. - This is a **univariate data set**. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures **Example**: Type of bill, date of payment, and payment amount for Mediacom ```text Customer Type Date Amount John Doe Internet 01-05-2015 110.00 John Doe Phone 01-15-2015 10.00 John Doe Internet 02-05-2015 110.00 John Doe Phone 02-15-2015 10.00 John Doe Internet 03-05-2015 110.00 John Doe Phone 03-15-2015 10.00 ... ... ... ... John Doe Internet 01-05-2016 110.00 John Doe Phone 01-15-2016 10.00 Jane Doe Internet 04-12-2015 90.00 Jane Doe Internet 05-12-2015 90.00 ... ... ... ... Jane Doe Internet 01-12-2016 90.00 ``` Notice: - Type of bill is is a **Qualitative** variable. - Amount paid is **quantitative discrete**. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures **Example**: Machine Parts > Suppose we get a shipment of 5000 machine parts and would like to verify that the shipment meets the standards the machinist agreed to. We take out 100 parts and examine them carefully. To verify that the parts are as strong as we anticipated, we measure the "Rockwell hardness" with a machine that is accurate to the first decimal place. We also examine each part for scratches and record it weight. Further, we run the part in a test machine to determine if it works correctly. In this case, we are gathering **4** values on each part. So for instance, the first of the 100 parts we examine could have a measured Rockwell hardness of 3.2, no scratches, a weight of 1.7562 g, and it works correctly. The second of the 100 parts we examine could have a measured Rockwell hardness of 3.1, no scratches, a weight of 1.7901 g, and does not work correctly. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures The data as recorded by the researcher might look like this ```text Part identifier: 1/100 Rockwell Hardness: 3.2 scratches: no weight (g): 1.7562 functioning: yes Part identifier: 2/100 Rockwell Hardness: 3.1 scratches: no weight (g): 1.7901 functioning: no ... Part identifier: 100/100 Rockwell Hardness: 3.4 scratches: no weight (g): 1.7651 functioning: yes ``` ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures Which we could turn into structured data table like this: The data as recorded by the researcher might look like this ```text part rockwell_hardness weight scratches functioning 1 3.2 1.7562 no yes 2 3.1 1.7901 no no . . . . . . . . . . . . . . . 100 3.4 1.7651 no yes ``` When data is arranged like this, with each sampling unit on its own row, the data is said to be in **wide format**. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Types of Data Structures However, we could also structure a data table like this: ```text part measurement value 1 Rockwell 3.2 1 weight 1.7562 1 scratches no 1 functioning yes 2 Rockwell 3.1 2 weight 1.7901 2 scratches no 2 functioning no . . . . . . . . . 100 functioning yes ``` When data is arranged like this, with each sampling unit on its own row, the data is said to be in **long format**. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Factorial Studies > **Factorial Studies** involve scenarios in which several process variables are indentified as being of interest and data are collected under different settings of these process variables. > We call the process variables **factors** and the possible settings for a process variable its **levels** > **Complete Factorial Studies** are factorial studies where data is collected from each possible combination of the levels of the factors (also known as **Full Factorial Studies**). > **Partial(Fractional) Factorial Studies** are factorial studies where data is collected from some (but not all) possible combinations of the levels of the factors. ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ## Factorial Studies Example > A pair of chemists, Walter and Jessie, are attempting to synthesize a chemical product and consider purity to be the most important quality. There are three environments available to them ( RV, a basement, and a laboratory) and two precursors (Chemical compound) (pseudoephedrine/methylamine). They are both willing to try all their options in order to get the best results. - What parts of this synthesis are being treated as variables which can be controlled at the start of the experiment? - What are the possible values for each of these variables? - How many ways can the variables be combined? ] ??? lead cook - Walter, Jessie environment - RV, basement, lab precursor - pseudo, methylamine 2 x 3 x 2 = 12 WRITE OUT THE LEVELS AFTER SOME TIME FOR THEM TO WRITE THEM OUT --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ <center> <img src="figs/being_chemists_trim.jpg" alt="dmc logo" height="125"> </center> Here are all the possible combinations of the factors: $$ \scriptsize{ (\text{# of Cooks}) \cdot (\text{# of Environments}) \cdot (\text{# of Precursors}) = 2 \cdot 3 \cdot 2 = 12} $$ ```text cook environment precursor walter RV psuedoephedrine walter RV methylamine walter basement psuedoephedrine walter basement methylamine walter lab psuedoephedrine walter lab methylamine jessie RV psuedoephedrine jessie RV methylamine jessie basement psuedoephedrine jessie basement methylamine jessie lab psuedoephedrine jessie lab methylamine ``` If we collect data from each of these combinations, we have performed a **A Complete Factorial Study** ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ### Factorial Studies Example, cont <center> <img src="figs/new_chemists_trim.jpg" alt="dmc logo" height="125"> </center> After testing each scenario, Walter and Jessie decide that the best combination to use is Walt as cook in the lab with methylamine. However, a new "chemist" Victor has joined the group and is going to try to be the cook and "follow the recipe" in the lab. Jessie also tries a new environment, South America. - If we consider the all the past combinations to be part of this new study, how many combinations of factor levels are now possible? ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ - Victor never works in the RV, the basement, or South America. - Walter never works in South America.  ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ### Factorial Studies Example, cont <center> <img src="figs/still_chemists_trim.jpg" alt="dmc logo" height="95"> </center> ``` cook env precursor 1. walt RV pseudo 2. walt RV methylamine 3. walt basement pseudo 4. walt basement methylamine 5. walt lab pseudo 6. walt lab methylamine 7. jessie RV pseudo 8. jessie RV methylamine 9. jessie basement pseudo 10. jessie basement methylamine 11. jessie lab pseudo 12. jessie lab methylamine 13. jessie so. am. methylamine 14. jessie so. am. pseudo 15. victor lab methylamine 16. victor lab pseudo ``` ] --- layout:false .left-column[ ## What and Why ## Terms ## Data Structures ] .right-column[ ### Factorial Studies Example, cont <center> <img src="figs/still_chemists_trim.jpg" alt="dmc logo" height="125"> </center> In this case, we would have a **Fractional Factorial Study** - a factorial study in which no data is collected for some possible combinations. ] ??? --- name: inverse layout: true class: center, middle, inverse --- # Section 1.3 ## Measurement: It's Importance and Difficulty --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Or Engineering For That Matter - Success in statistical engineering studies requires the ability to measure - Methods of measurements are available for some physical properties (length, mass, temperature, ...) - Often, the behavior of anengineering system can be adequately characterized in terms of such properties - If it cannot, engineers must carefully define what is about the system that needs observing and then create a suitable method of measurement. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Example: Two students wanted to conduct a factorial study comparing joint strengths for combinations of three different woods and three glues. - Didn't know how to have access to strength-testing equipment, so invented their own. - Suspend a large container from one of the pieces of wood involved and poured water into it until the weight was sufficient to break the joint. - Knowing the volume of the water poured into the container and the density of the water, they could determine the force required to break the joint. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics <center> <img src="figs/funny1.jpg" alt="dmc logo" height="200"> <img src="figs/funny2.jpg" alt="dmc logo" height="200"> <img src="figs/funny3.jpg" alt="dmc logo" height="200"> </center> ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Measurement and its importance and difficulty - **Validity**: appropriately represent the feature of interest. Variation is always present in collecting data. - Some come from the the objects under study as they are never alike(that might be of interest to see if the variation is due to the object) - Some of it is due to the fact that the measurement processes have their own inherent variablity. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Measurement and its importance and difficulty - **Precision**: the amount of variation in repeated measurement of the same object A measurement system is called **precise** if it produces small variation in repreated measurement of the same object - Precision is the internal consistency of a measurement system: typically, it can be improved only with basic changes in the configuration of the system. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Measurement and its importance and difficulty Precision of a measurement is important, but for many purposes it alone is not adequate. - **Accuracy**: or **Unbiasedness**; how close a measurement is to the true value "on average". Accuracy is the agreement of a measuring system with some external standard. It can be changed without extensive physical change in a measurement method. So, we **calibrate** to improve accuracy. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Measurement and its importance and difficulty - **Calibration** of a system against the standard (bringing the measurement system in line with the standard) can be - As simple as comparing the measurement system to a standard - Developing an appropriate conversion scheme and then using converted values in place of recording observed measurements. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Accuracy VS. Precision - **Accuracy**: how close a measurement is to the true value "on average" - **Precision**: the amount of variation in repeated measurement of the same object - Comparing measurement to target shooting <center> <img src="figs/shooting.png" alt="dmc logo" height="250" > </center> ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ### Key Words ] .right-column[ ### If You Can't Measure, You Can't Do Statistics ### Validity, Accuracy and Precision  ] --- name: inverse layout: true class: center, middle, inverse --- # Section 1.4 ## Mathematical Models --- layout:false .left-column[ ## What and Why ## Terms ## Measure ## Math Models ] .right-column[ ## Mathematical Models and Data Analysis A discussion on the relationships of mathematics to the physical words and to engineering statistics. > **Mathematical Model**: A description of a physical system using mathematical concepts and language (in terms of symbols, equations, numbers, and the like) - Identifying mathematical relationships between parts of a system allows us to describe complexity in simple terms. - An effective mathematical model is the one which is **simple** and has **predictive ability**. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ## Math Models ] .right-column[ ## Mathematical Models and Data Analysis **Example**: Height of an Object in Projectile Motion We can describe the relationship between height of a projectile \\(h\\) and time \\(t\\) as \\[ h = h_0 + v_h \cdot t - \frac{1}{2} g t^2, \text{ } t \ge 0, \\] where - \\(h_0\\) is the initial height, - \\(v_h\\) is the initial vertical velocity, and - \\(g\\) is the (constant) acceleration due to gravity ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ## Math Models ] .right-column[ ## Mathematical Models and Data Analysis **Example**: Height of an Object in Projectile Motion <center> <h2></h2> <img src="figs/projectile.png" alt="dmc logo" height="200"> </center> ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ## Math Models ] .right-column[ **Example**: Height of an Object in Projectile Motion, cont. However, this is not what we see in real life for a variety of reasons. This model assumes 1. \\(g\\) is constant as the ball falls, while \\(g\\) actually depends on the distance between the object and earth, 2. \\(g\\) is a known to infinite accuracy, while we would be using a value that is estimated, 3. Gravity is the only force acting on the object, ignoring drag force, electrical attractions, etc. 4. There are no other changes in the system (for instance, changes in air pressure) We can fix these by writing a better relationship *or* we can accept that some things won't be known and use a **stochastic model** - a mathematical model that specifically allows for variation (or "randomness"). Understanding how these **stochastic models** work is a major focus of this course. ] --- layout:false .left-column[ ## What and Why ## Terms ## Measure ## Math Models ] .right-column[ ## What's my point - We cannot say there is no variation in the measurement or the relation under study is just affected by the components defined in the mathematical model. - We can control some parts of the variation by planning the data collection process - There is always some error out of control which are stochastic (random) - Statistical methods help to deal with this randomness ]