1. Introductory Statistics and Data Analysis
MAT 135
Introductory Statistics and Data Analysis
Adjunct Instructor
Kenneth R. Martin
Lecture 2
September 7, 2016

2. Confidential - Kenneth R. Martin
Agenda
Housekeeping
Readings
Collect HW #1
Review HW #1
Chapter 1, 14 & 10
Quiz #1
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Housekeeping
Read, Chapter 1.1 – 1.4
Read, Chapter 14.1 – 14.2
Read, Chapter 10.1
HW #2 to be issued
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Housekeeping
Connect Math
Course code: HHGCW-JNEMX
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5. Confidential - Kenneth R. Martin
Housekeeping
Collect HW #1
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6. Confidential - Kenneth R. Martin
Housekeeping
Review HW #1
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Housekeeping
Quiz #1
Beginning of next class, 9/14/2016
Material and associated readings covered through today.
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8. Statistics - Definition
Statistics: the science of the collection, analysis, interpretation and use of data relating to any group or groups of individuals, experiments, or data points (scores or raw scores).
Also includes the planning of the collection of data, in terms of the design of surveys, experiments, etc.
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9. Statistics – Application to Research
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10. Statistics - Definition
Descriptive Statistics: summarizing the data by describing what was observed in the sample taken, either numerically or graphically (tables, graphs, or single values).
Inferential (Inductive) Statistics: drawing generalizations about the population represented from a smaller data sample, taking into account randomness.
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11. Statistics - Definition
Variable: a characteristic or attribute that can take on different values.
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12. Statistics - Definition
Data (plural): measurements or observations that are usually numeric.
Datum (singular): a single measurement or observation, usually referred to as a score or raw score.
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Statistics
Population and Sample
Population: The collection of all possible individuals, subjects, times, places, units, etc., which we wish to study
The data collected in the population is referred to as “Parameters”
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Statistics
Population and Sample
Sample: The particular subset of individuals, subjects, times, places, units etc. from which measures are obtained
It is a representative subset of the population
The data collected in the sample is referred to as “Statistics”.
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Statistics
Population and Sample
Sample data shall be randomly drawn and representative of the entire population.
Random means the sample drawn has an equal chance of being selected
Samples are a limited number (n) of a larger source N (population)
Samples shall be ~  30 to be statistically valid.
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Statistics
Population and Sample
SAMPLE
The particular individuals, subjects, times, places, units etc. from which measures are obtained
POPULATION
The collection of all possible individuals, subjects, times, places, units, etc. which we wish to study
SAMPLING SCHEME
The rules by which we will choose which individuals, etc. to include in the sample
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Statistics
Why collect samples ?
Often impractical to collect all the data from the entire population (i.e. U.S. census).
Some test methods are destructive – we wouldn’t have any products or services left to ship to a customer!
Too expensive to sample the entire population.
Don’t have to collect 100% of the population ! We can use inferential statistics to make sound conclusions about the population.
Population and Sample
Sampling
Scheme
POPULATION
SAMPLE
Measure
Data!
Use data from the SAMPLE to make conclusions about the POPULATION
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Statistics
Sample vs. Population (differentiation)
Bradford County
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Statistics
Sample vs. Population (differentiation)
United States
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Statistics
Sample vs. Population (differentiation)
When the sample size becomes large, we start treating the data as if it is the population.
When a particular “product / process” has a finite life, then you have all the data, and thus it is the population.
When you’re confident you’ve seen all possible variations, then you can treat the sample data as the population.
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Statistics
Sample vs. Population (differentiation)
250 dot sample
1000 dot sample
2000 dot sample
This example is simply designed to illustrate that as the sample size goes up the sample will more clearly illustrate with is in the population.
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22. Statistics -Characteristics of a Good Sampling Scheme
Generates samples that are representative of the population
Uses a random selection mechanism
Each element of the population has an equal chance of being in the sample
Unbiased selection of samples
Some sort of objective random process used to select the elements in the sample
The 3 items in red are the primary characteristics of good samples that students need to know.
Sampling
Scheme
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23. Statistics – Random Sampling
Population
Sample
XXXXXXXXXX
XXXXX
Basic Idea of Random Sampling
Each unit in the population has an equal chance of being selected
Elements are not selected based on a preliminary look at their measurement values
Some additional rules may be used, however, to assure all elements of the population are considered
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24. Statistics – Sampling Variations
Purely Random
In a figurative sense: Put all the population in a hat; reach in and randomly select the individuals to include in the sample. Or use a random number generator.
Example: Randomly select 100 inquiries from the 5000 received last month.
Systematic Sampling
Select every kth individual in the population (assumes that the population individuals are ordered in some way that is not related to what you are measuring)
Example: Randomly select every 50th inquiry from the 5000 received last month.
Stratified Sampling
If population has subgroups that may be related to what you are measuring, randomly select a certain number from each subgroup, for the sample.
Select more from larger subgroups, fewer from smaller ones, assuring proportionality.
Example: “Since 35% of the inquiries were about our software products, 40% about hardware, and 25% were about other things, randomly select 35 software inquiries, 40 hardware inquiries, and 25 other inquiries from the total of 100 received.”
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25. Statistics – Research Methods
Definitions
Science: The study of some phenomena, through strict observation, evaluation, interpretation, and theoretical explanation.
Experiment: Any study that demonstrates cause, by following strict procedures to ensure all other possible causes are eliminated or highly unlikely.
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26. Statistics – Research Methods
Definitions
Observational study: the researcher simply observes what is or what did happen, and tries to draw conclusions based upon these observations.
Experimental study: the researcher manipulates one (or more) of the variables and tries to determine how the manipulation influences the other variables.
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27. Statistics – Research Methods
Definitions
Independent Variable (IV): The variable that is manipulated in an experiment. It is the “presumed cause”.
Dependent Variable (DV): The variable that is measured in each group of the study. It is the “presumed effect”, and is believed to change in the presence of the IV.
Confounding Variable (CV): a variable that influences the DV, but was not completely separated from the IV.
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28. Statistics – Research Methods
Definitions
Operational Definition: A definition of how we will measure the DV.
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29. Statistics – Research Methods
Experimental Study
Requirements that must be satisfied for the Experimental Study, to allow the researcher to be able to draw cause-effect conclusions.
Manipulation - of the variables that operate in an experiment.
Randomization - of assigning participants to conditions
Comparison / Control - something to compare the results
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30. Statistics – Research Methods
Experimental Study
Q: Do distractions during an exam effect student scores ?
Requirement 1
Requirement 2
Requirement 3
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31. Statistics – Research Methods
Correlational Method (Scatter Diagram)
Mathematical representation of two (or 3) variables, using Cartesian coordinates, for a common data set
Identifies potential relationships and may* suggest causation (if applicable)
Independent Variable on x-axis, Dependent Variable on y-axis
X is input to some “process”, and Y is the output
X variable can be incremented or decremented.

32. Statistics – Research Methods
Correlational Method (Scatter Diagram)
Will confirm a hypothesis if variables are related
Observe direction and “tightness” of data
Closer to a straight line, the greater the correlation between the variables.

33. Statistics – Research Methods
Correlational Method (Scatter Diagram)
Can test the strength of the relationship mathematically (Correlation coefficient)
Can only say X & Y are related, not one caused other

34. Statistics – Research Methods
Scatter Diagram (positive correlation)

35. Statistics – Research Methods
Scatter Diagram (negative correlation)

36. Statistics – Research Methods
Scatter Diagram – 3D Scatter & Contour Plot

37. Statistics - Definition
Data:
Variables: those items that are measurable
Continuous: measurable characteristics that can take on any value, within measurement capability. (i.e. temperature, distance, etc.) [Quantitative]
Discrete: countable characteristic that can only take on specific values (integer). (i.e. # defects, # failure, # people, etc.) [Quantitative or Qualitative]
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38. Statistics - Definition
Data:
Attributes: those items that are either conforming or non-conforming to a specification. (i.e. go-no/go; pass / fail; black / white; I / O, etc.) [Qualitative]
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39. Statistics - Definition
Data:
Quantitative: varies by amount
Qualitative: varies by class / category
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40. Continuous vs. Discrete vs. Attribute Data
infinite # of possible measurements in a continuum 1 2 3 4 5 6 7 8 9 10
Discrete:
Count
Discrete:
Ordinal 1 2 3 4 5 6 7 8 9 10
“low”/“small”/“short”
“high”/”large”/”tall”
“medium” / “mid”
Discrete:
Nominal or
Categorical
defines several groups - no order or ranking
Group A
Group B
Group C
Group D
Group E
Group F
Attribute:
Binary
“bad”/“no-go”/”group #1”
“good”/“go”/”group #2
defines TWO groups - no order
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41. Continuous vs. Discrete vs. Attribute Data
Continuous: can take on any measurement value, within measurement capability. i.e. weight of people
Count: can only take on a specific integer value. i.e. the number of people in a room
Ordinal: shows order and simply one rank is greater than or less than another. i.e. shirt sizes.
Nominal / Categorical: groupings of common items / values. Groups do not imply any importance, order, or additional information. i.e. seasons of birth
Binary: defines two possible outcome, and no order. i.e. standard light switch (on / off)
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42. Statistics - Definition
Accuracy: how close are measured values to the true (actual) value.
Precision: how close the measured values are to each other.
Low accuracy Low precision High accuracy
High precision High accuracy High precision
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Statistics
Resolution:
The measurement device shall be able to read to at least one decimal place beyond the discerned level needed.
Example, if you are interested in weight of a person, and only interested to the nearest pound, the scale shall be able to measure to a tenth of a pound.
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Statistics
Numbers:
When rounding, determine the resolution of data needed. Then look to the digit immediately to the right.
If that digit ends with 0-4, keep the previous number (round down)
If that digit ends with 5-9, round the previous digit up
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Statistics
Numbers:
Significant figures: the number of digits needed to locate the decimal point, exclusive of leading zeros
significant digits
significant digits
significant digits
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Statistics
Numbers:
Multiplication / division / exponentiation: the number with the fewest sig. figures dictates
6.59 * 2.3 = = 15 (2 sig. digits)
Addition / subtraction: the answer has no more sig. figures after decimal than # with fewest after decimal.
= 4.29 = 4.3
8.1 x = 6.9 x 103
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Statistics
Math rule:
Please (Parenthesis)
Routinely (Roots)
Excuse (Exponents)
My (Multiplication)
Dear (Division)
Aunt (Addition)
Sally (Subtraction)
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