STAT 305: Lecture 2
Why Engineers Study Statistics
Chapter 1: Introduction, Continued
Chapter 2: Data Collection
.footnote[Course page: ashirazist.github.io/stat305.github.io]
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 bult 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:
Bulb Number Failure Time (hours)
1 521.4
2 501.2
3 541.8
4 498.1
5 528.2Each 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
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 90.00Notice:
- Type of bill is is a Qualitative variable.
- Amount paid is quantitative discrete.
- Date is … ] — 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
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
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 yesWhen 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:
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 yesWhen data is arranged like this, with each sampling unit on its own row, the data is said to be in long format. Long format matches each recorded value to a unique set of identifiers called keys - in this case, for example, the first row matches the recorded value 3.2 uniquely to the measurement Rockwell hardness and the first part in our sample. ]
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.
Partial Factorial Studies are factorial studies where data is collected from some (but not all) possible combinations of the levels of the factors.
]
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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 (a winnebago, a basement, and a laboratory) and two precursors (pseudoephedrine/methylamine). They are both willing to take the role of “lead cook” and will 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? ]
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lead cook - Walter, Jessie environment - winnebago, basement, lab precursor - pseudo, methylamine
2 x 3 x 2 = 12
WRITE OUT THE LEVELS AFTER SOME TIME FOR THEM TO WRITE THEM OUT
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Factorial Studies Example, cont
Here are all the possible combinations of the factors:
\[ \scriptsize{ (\text{# of Cooks}) \cdot (\text{# of Environments}) \cdot (\text{# of Precursors}) = 2 \cdot 3 \cdot 2 = 12} \]
cook environment precursor
walter winnebago psuedoephedrine
walter winnebago methylamine
walter basement psuedoephedrine
walter basement methylamine
walter lab psuedoephedrine
walter lab methylamine
jessie winnebago psuedoephedrine
jessie winnebago methylamine
jessie basement psuedoephedrine
jessie basement methylamine
jessie lab psuedoephedrine
jessie lab methylamineIf we collect data from each of these combinations, we have performed a A Complete Factorial Study ]
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Factorial Studies Example, cont
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, where only methylamine is available.
If we consider the all the past combinations to be part of this new study, how many combinations of factor levels are now possible?
Victor never works in the Winnebago, 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
cook env precursor
1. walt winne pseudo
2. walt winne methylamine
3. walt basement pseudo
4. walt basement methylamine
5. walt lab pseudo
6. walt lab methylamine
7. jessie winne pseudo
8. jessie winne methylamine
9. jessie basement pseudo
10. jessie basement methylamine
11. jessie lab pseudo
12. jessie lab methylamine
13. jessie so. am. methylamine
14. victor lab methylamineIn this case, we would have a Fractional Factorial Study - a factorial study in which no data is collected for some possible combinations. ]
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Section 1.3
## Measurement: It’s Importance and Difficulty
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If You Can’t Measure, You Can’t Do Statistics
Or Engineering For That Matter
- Validity: faithfully representing the aspect of interest
- Precision: the amount of variation in repeated measures
- Accuracy: aka “unbiasedness”; how close a measurement is to the true value “on average”
We calibrate to improve accuracy ]
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Section 1.4
## Mathematical Models
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Mathematical Models and Data Analysis
Mathematical Model: A description of a physical system using mathematical concepts and language.
Identifying mathematical relationships between parts of a system allows us to describe complexity in simple terms.
Example: Height of an Object in Projectile Motion
We can describe the relationship between height of a projectile \(y\) and time \(t\) as \[ y = h_0 + v_h t - g t^2, t 0, \] where - \(h_0\) is the initial height, - \(v_h\) is the initial vertical velocity, and - \(g\) is the (constant) acceleration due to gravity ]
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Example: Height of an Object in Projectile Motion, cont.
\[ y = h_0 + v_h t - g t^2, t 0, \]
However, this is not what we see in real life for a variety of reasons. This model assumes
\(g\) is constant as the ball falls, while \(g\) actually depends on the distance between the object and earth,
\(g\) is a known to infinite accuracy, while we would be using a value that is estimated,
Gravity is the only force acting on the object, ignoring drag force, electrical attractions, etc.
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. ]