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Introduction
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STATISTICS
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by Jim Mirabella
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Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
Copyright 2011, Savant Learning SystemsTM
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Table of Contents
Chapter One: Graphing & Descriptive Statistics
What is Statistics?………………………………………………………………………………..1-1
Why Do We Learn Statistics?…………………………………………………………………1-1
Descriptive vs. Inferential Statistics………………………………………………………..1-1
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Basic Terminology………………………………………………………………………………..1-1
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Types of Variables ……………………………………………………………………………….1-2
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Levels of Measurement ………………………………………………………………………..1-2
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Organizing and Graphing Categorical Data …………………………………………….1-3
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Organizing and Graphing Numerical Data ………………………………………………1-5
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Descriptive Statistics ……………………………………………………………………………1-6
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Central Tendency …………………………………………………………………………………1-7
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Dispersion …………………………………………………………………………………………..1-8
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Percentiles and Quartiles …………………………………………………………………….1-10
Discussion Questions …………………………………………………………………………
1-11
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Practice Problems ……………………………………………………………………………… 1-11
Assigned Problems …………………………………………………………………………….1-12
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A Distribution
Chapter Two: Probability & the Normal
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Basics of Probability ……………………………………………………………………2-1
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Solving Probability Problems ……………………………………………………….2-2
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The Normal Distribution ………………………………………………………………2-4
The Central Limit Theorem ………………………………………………………….2-7
Discussion Questions …………………………………………………………………..2-9
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Practice Problems ……………………………………………………………………….2-9
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Assigned Problems ……………………………………………………………………2-10
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Chapter Three: Confidence Intervals 7& Sample Sizes
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Sampling …………………………………………………………………………………….3-1
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Confidence Intervals ……………………………………………………………………..3-6
Sample Size ……………………………………………………………………………….3-12
Discussion Questions ………………………………………………………………….3-15
Practice Problems ……………………………………………………………………….3-15
Assigned Problems ……………………………………………………………………..3-16
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
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Table of Contents
(continued)
Chapter Four: Hypothesis Testing for One Sample
The Logic of Hypothesis Testing …………………………………………………..4-1
Z-tests vs. t-tests …………………………………………………………………………4-6
Closing Notes ……………………………………………………………………………..4-6
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Discussion Questions …………………………………………………………………..4-7
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Practice Problems ……………………………………………………………………….4-7
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Assigned Problems ……………………………………………………………………..4-8
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Chapter Five: Hypothesis Testing for Two
S or More Samples
Choosing the Correct Test …………………………………………………………………….5-1
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T-test for Independent Samples ……………………………………………………………..5-2
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T-test for Paired Samples ………………………………………………………………………5-5
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Analysis of Variance …………………………………………………………………………….5-6
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Discussion Questions …………………………………………………………………………..5-8
Practice Problems ………………………………………………………………………………..5-8
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Assigned Problems ………………………………………………………………………………5-9
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Categorical Analysis …………………………………………………………………………….6-1
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Goodness of Fit …………………………………………………………………………………..6-1
Crosstabulations ………………………………………………………………………………….6-5
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Discussion Questions …………………………………………………………………………..6-9
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Chapter Six: Chi Square Testing
Practice Problems ………………………………………………………………………………..6-9
Assigned Problems …………………………………………………………………………….6-10
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Chapter Seven: Correlation & Regression
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Correlation ………………………………………………………………………………………….7-1
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Regression ………………………………………………………………………………………….7-5
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Multiple Regression …………………………………………………………………………….7-6
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Discussion Questions …………………………………………………………………………7-10
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Practice Problems ………………………………………………………………………………7-10
Assigned Problems ……………………………………………………………………………. 7-11
Chapter Eight: Data Analysis Project
Appendix: Pivot Table Tool Tutorial
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
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CHAPTER ONE
GRAPHING & DESCRIPTIVE STATISTICS
What is Statistics?
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We use numbers every day to analyze theHworld or our piece of it, whether it be related to
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sports, finance, weather, politics, etc. Statistics
is essentially a compilation of tools and
techniques used for describing, organizingI and interpreting these numbers; in other words,
it is a means for converting data into information.
In the business world, there are four
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main uses of statistics à to summarize business
data, to draw conclusions from that data,
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to make reliable forecasts, and to make decisions important to the business.
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Why Do We Learn Statistics?
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While it would be ideal if everyone could ,strive to be a doer of statistics, the aim here is for
you to become an adept user of statistics. Rather than trust headlines without substance,
you might be able to see through the numbers and question results intelligently. You won’t
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just trust the media or politicians blindly. You can play fantasy sports with greater success.
A and figures that you can interpret and not just
You can make business decisions with facts
with gut instinct. Above all else, you willM
think differently and see the world in a way few
people do.
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Descriptive vs. Inferential Statistics
Statistics can be broken down into two 5distinct branches. The first part of the course
focuses on descriptive statistics, which5is the collecting, summarizing, presenting and
6 charts and graphs to display and picture the
analyzing of data sets. Here we use tables,
data, and we compute statistics to describe
7 the data with single numbers. The second part
of the course focuses on inferential statistics,
which involves taking the results from a
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data set and drawing inferences to the larger data sets that it came from. This is where the
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power of statistics truly comes alive. It is here that conclusions are drawn, decisions are
made, and forecasting models are created.
Basic Terminology
A variable is a characteristic of an item or individual that is measured. If you break up the
word, you will note that a variable must be able to vary. Ages, heights, weights, genders are
variables because they each have varying values, but if were studying only those who wear
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
1-1
Chapter One: Graphing & Descriptive Statistics
contact lenses, that would be a constant and not a variable. For a given variable, an observation is
the simplest form of data; it is a single value in a data set. When we have all possible observations
of a variable, we have a population. It is the population that we typically wish to evaluate with our
analysis, and we do so by utilizing a subset of that population, known as a sample. The numerical
measures we compute from a population are known as parameters, while the numerical measures
we computer from a sample are known as samples. To help remember this, note that parameter
and population begin with “p”, while sample and statistic begin with “s.”
Types of Variables
Variables can be classified as either categorical or numerical. A categorical variable is essentially
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qualitative and has non-numerical values, like yes / no, or red / white / blue, or republican /
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democrat, or freshman / sophomore / junior / senior.
Most survey questions have categorical
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responses, and the analysis of such data is limited to counting the frequency of each value and
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computing their relative percentages. A numerical
variable is quantitative in nature and its
values are a measurement; these variables can beS
further subdivided into discrete and continuous
variables. Discrete numerical variables are finite
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household, or the number of books on a shelf or the number of cars in a parking lot. Continuous
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numerical variables can have any numerical value along a continuum, and can be fractions or
have many decimal places; with continuous data,Ait is not unusual to see no repetition in the data
set. For example, if you measured the times ofNrunners in the NYC marathon, you will see as
many different times as there are runners, with a,rare tie, and the times are computed in fractions
of a second. Likewise if we measured the heights or weights of people, while we tend to round off
weight to the nearest pound and height to the nearest
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rarely if ever see a tie. And because of this lack of repetition, analysis of continuous data must be
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handled differently from discrete data, as shown in the later section on organizing and graphing
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numerical data.
Levels of Measurement
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Before beginning to analyze data, you must determine the appropriate analytical techniques to use,
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and that question is answered by knowing the level of measurement of the variables in question.
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There are four levels of measurement à nominal, ordinal, interval and ratio. A nominal
variable, also known as categorical, is essentially6a naming variable. It has no measurement tied
7 no numbers (e.g., yes / no, male / female,
to it whatsoever. Most of the time its values have
Caucasian / African-American / Asian / etc.) butBsometimes it can have numerical values which
are truly categorical in nature and have no quantitative
meaning (e.g., a zip code, an area code,
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or even on surveys where codes are used and maybe 1 = female and 0 = male). You would never
average the zip codes in a class to determine where the typical student lives, but you might count
how many live in each zip code to understand the commutes for the entire class or how the school
attracts potential students by location. Whether the values have numbers or just text, nominal data
can only be analyzed at the simplest level (i.e, you can count the occurrences of each value and
compute percentages too).
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
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Chapter One: Graphing & Descriptive Statistics
The next level up from nominal data is ordinal, which includes a ranking or order in that we
know one value is greater than another which is greater than another, but it is not apparent as to
how much they differ. Surveys often use Likert scales (i.e., Strongly Agree – Agree – Neutral
– Disagree – Strongly Disagree) and you are asked to mark 1 for Strongly Disagree up to 5 for
Strongly Agree. We know that Strongly Agree is higher than Agree and Disagree is higher than
Strongly Disagree, but the gaps are not consistent, and they vary by respondent. Even letter grades
at school are ordinal à while an A is always higher than a B, and a C is always higher than a D,
grades of 90, 89, 79 and 60 should illustrate that the A (90) is only slightly higher than the B (89),
while the C (79) is significantly higher than the D (60), and so the ranking is clear but the gaps are
inconsistent.
C and ratio, both of which maintain the ranking
The two highest levels of measurement are interval
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/ order of magnitude but now they also have consistent
gaps in the data. The only difference
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between the two is that ratio data also has an absolute
zero, meaning that zero on its scale truly
means “the absence of”, and as a result, we can do
I ratio comparisons on the data. 200 pounds is
twice as heavy as 100 pounds and four times as heavy
S as 50 pounds (note the ratio comparisons),
and this is possible because 0 pounds means the absence of weight, so weight in pounds is a ratio
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variable. A golf score of +2 (i.e., 2 over par) is 1 stroke worse than +1 but it is not twice as bad,
I strokes, but is instead just another value on the
and that is because 0 (i.e., par) is not the absence of
A ratio variables include height, weight, age and
scale; thus golf scores are interval data. Common
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income, while common interval variables include
Celsius. Truthfully, when it comes to statistical analysis,
it doesn’t matter if the variable is interval
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or ratio, as you can perform the same techniques on both (with the exception of a geometric mean,
which is rarely used and which will not be part of this course). Some software packages even
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simplify matters further by using the term scale to represent interval or ratio data, so we would
really only have nominal, ordinal and scale data.ASince the intent here is to determine the correct
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statistical technique, it seems a sensible idea.
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Organizing and Graphing Categorical Data
When we wish to analyze categorical / nominal data, we are limited in what we can do. Essentially
we can count how many of each value we have,5and compute the percentage of each value, and
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then we can graph the results in a bar chart or pie chart.
Just because it is a limitation doesn’t mean
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it isn’t valuable; in fact, one advantage is that you know exactly what you need to do with the data.
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Suppose we had a sample of 25 students and we had a variable that represented the student’s class
B file provided (shown in Figure 1), we would
year. Using the Categorical_Data_Analysis.xls
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copy the 25 values into the Data column (after deleting
in the Values column we would list all of the unique values in the data set (and you can see there
are only four class years to list à Freshman, Sophomore, Junior and Senior, spelled exactly as it is
spelled in the data set). The Frequency column counts the total occurrences of each value, and it
shows 4 Freshmen, 5 Sophomores, 7 Juniors and 9 Seniors, for a total of 25 students as the sample
size. The Relative Frequency column computes the percentages of the sample size represented
by each value, and the 4 Freshmen in the sample of 25 accounts for 4 / 25 = 16% of the sample.
Similarly, 20% of the sample are Sophomores, 28% are Juniors and 36% are Seniors, for a total
of 100%.
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
1-3
Chapter One: Graphing & Descriptive Statistics
This is a complete Frequency table of the class years in our data set and we can now begin to make
analytical statements about the data that we couldn’t do before (i.e., if you look at the list under the
Data column, what can you really say?). Things to note might be that there are increasing numbers
of students at the higher class levels, or that there are as many Seniors as there are Freshmen and
Sophomores combined. Note that we cannot give reasons for this, as it is only a small sample and
we don’t know much about the population. It might be tempting to make a comment about how
new enrollments are down or how there are greater numbers of late transfers into the school or
how dropout rates are highest at the lowest levels, but we know nothing except what we see. So
all we can do is to describe what
is there and leave it to experts
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(i.e, those with a working
knowledge of this particular
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data set) to draw conclusions or
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make guesses about what might
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be occurring.
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Figure 1: Screen display from
Categorical_Data_Analysis.xls
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There was a great song about how a picture paints a thousand words. The “thousand” might be
debatable but the point is well-taken. If you click 5on the Charts tab at the bottom of the screen, you
will be taken to another sheet in the Excel file where
5 the tabulated data is charted (see Figure 2).
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Figure 2: Screen display from Categorical_Data_Analysis.xls – Charts
Copyright 2011, Savant Learning SystemsTM
Introduction to Statistics by Jim Mirabella
1-4
Chapter One: Graphing & Descriptive Statistics
On the left is a Bar Chart in which each bar is one row from the Frequency Table and illustrates the relative
frequencies. Note that the Freshman bar lines up with 16% and the Sophomore bar lines up with 20%,
and so on. Visually you can see increases in class sizes over the years. On the right is a Pie Chart which
treats the whole pie as 100% and each value has a slice of the pie, sized in accordance with the relative
frequencies. While it has the disadvantage of getting cluttered at times and not being as visually appealing
as bar charts, it has the advantage of being out of 100%, so each slice tells you how much of the whole
it truly represents (which the bar chart doesn’t do as well). Together the two charts should help to paint
a more vivid description of the data, but still be warned not to overstep your bounds and do more than
describe what you see (which is the essence of descriptive statistics anyway). If a pattern looks interesting
and warrants further investigation, then get more data and keep digging, but don’t conclude beyond the
scope of the data that you have, especially if you are C
not extremely knowledgeable about that sample.
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Organizing and Graphing Numerical Data R
When analyzing numerical variables, first determineI if the data is discrete. If it is, you could use the
Categorical_Data_Analysis.xls file as an option providing
there aren’t too many different values in the
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data set. For example, if a subdivision had some homes
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work just fine to count the totals, compute relative percents, and create bar and pie charts. But if there
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are a lot …
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