This document provides an overview and lecture notes for chapters 3-5 of an introductory statistics course. It discusses preparing preliminary analyses of categorical and quantitative variables through graphs and descriptive statistics. For categorical variables, it describes frequency tables, contingency tables, bar charts, and pie charts. For quantitative variables, it introduces histograms, boxplots, dot plots, and stem-and-leaf displays. It emphasizes using graphs to understand the shape, center, and spread of variable distributions. The lecture also warns of Simpson's Paradox and provides examples of analyzing real data.

1. Introduction to Statistics
Lecture Notes
Chapters 3-5
Please sign in (SIGNATURES) as you come in to class. It will save
my voice instead of my taking attendance (this is only to settle the
class roster).

2. What’s up with the powerpoint?
 I don’t usually use slides, but am going to try to use
these to save my voice somewhat.
 Notes: Still working on getting the class roster
settled. Has been some movement on the waitlist,
will keep in touch as things develop. Be sure you’ve
signed in!
 First homework is posted (on our course website),
but isn’t due until next Friday (the 4th). The
additional problem is NOT optional, that just means it
is not a book problem.

3. Handouts for Today
 There is one handout on graphs/descriptive statistics
going around. Save this to use tomorrow in class.
 There is a second handout – the anonymous survey
largely designed by the class on Monday. Please go
ahead and take a few minutes to fill this out (no
names!) and get it back to me. We’ll take a look at
this data next week in lab.
 If you missed class Monday, I have extra course
syllabuses at the front as well.

4. The “W”’s of a Data Set
 Who – the observations (population – set of all objects
you are interested in obtaining the value of some
parameter for – since we usually can’t observe all
objects, we take a sample of objects – a subset of the
overall population of objects to observe)
 Note: There is NO such thing as a population sample or
sample population.
 What – the variables
 Why – why was the data collected
 How – how was the data collected (related to
design/sampling in chapters 12-13)
 When/Where – more information that could be relevant

5. Chapters 3-5 Overview
 Covers basic graphs and descriptive statistics for
both categorical and quantitative variables
 This is what you would do as a “preliminary analysis”
for a variable.
 Recall: a data set can have multiple variables in it.
 These chapters focus on mostly univariate (single
variable) analyses. There is one comparative graph
– a side-by-side boxplot in Chapter 5.

6. 3 Rules of Data Analysis
 Rule 1- Make a picture
 Rule 2 – Make a picture (really, before you do
anything else)
 Rule 3 – Make a picture (really, we mean a well-
chosen picture for your variables)

7. Categorical Variable Prelim Analysis
 Frequency tables (one variable) – summarize counts
by category
 Contingency tables (2 or more variables) –
summarize counts by category for multiple variables
 Bar charts
 Pie charts

8. Frequency
 What is frequency?
 Frequency is the number of objects/cases per category
 You can also look at relative frequency.
 Relative frequency is the number of objects/cases per
category divided by the total number of objects.
 Hence it gives proportions for each category out of the
total.
 It is often converted to %.

9. Bar Charts
 One bar per category – height is determined by
frequency or relative frequency
 Order of categories is arbitrary.
 Does NOT let you talk about the shape of a
distribution.
 “Area” principle – areas are supposed to be relative.
This is often violated when people try to make
graphs “cool” and go 3-D, etc. (see Example passed
around).

10. Pie Charts
 Take 100% of cases and divide up 360 degrees
based on relative frequencies.
 We will look at bar charts over pie charts.
 Note that for bar charts you do not need to create
bars for 100% of the cases. You could look at the top
three risk factors for a disease, etc. However, we
usually do have 100% of cases shown.

11. Contingency Tables - Example
 See first page of Handout
 Totals for rows/columns give marginal distributions
for each variable.
 You can also look at conditional distributions. Fix
a row or column and work solely within that row or
column.
 Concept of independence (will formalize later):
 If the distribution of one variable is the same for all
categories of another variable, then the two variables are
independent.

12. On Your Own
 Text has some discussion of segmented bar-charts
and side-by-side (feel free to read or skip)

13. Simpson’s Paradox
 Something that can happen when you aggregate
categorical data
 Looking at overall averages or % can be misleading
 Can get different results looking at breakdown
 Berkeley Discrimination Data Example (see bottom of
page one of the handout)
 Claims of Sexual Discrimination in1973 Graduate School
Admissions
 Overall, 44.28% of males who applied were admitted, while
only 34.58% of females were admitted.
 Look what happens when you breakdown by the 6 largest
departments though! (try this on your own or with a partner). Is
there evidence of discrimination against females at the dept.
level? What is going on?

14. Quantitative Variables Preliminary Analysis
 Graphs
 Dot plot – won’t use much – read about on your own
 Stem and leaf – won’t use much – read about on your own
 Histogram
 Boxplot (chapter 5)
 Qqplot (Friday or next week)
 Time plot (Friday or next week)
 Descriptive statistics
 Measures of center: mean, median
 Measures of spread: standard deviation, IQR, range

15. Describing the distribution of a quantitative
variable
 You should focus on three things when describing
the distribution of a quantitative variable:
 Shape – unimodal (one peak), bimodal (two peaks),
multimodal (many peaks), bell-shaped, skewed left (tail to
the left), skewed right (tail to the right), symmetric,
uniform (no peaks, basically flat)
 Center – estimate the center (or use a descriptive
statistic)
 If multiple peaks, report the peak locations
 Spread – estimate the spread (can use a descriptive
statistic)

16. Dot Plot – On Your Own
 Most basic quantitative graph
 Use for a low number of observations (<50)
 Basically use a number line and place a dot above it
for each value you have observed.
 Example from wikipedia:

17. Stem and Leaf – On Your Own
 Your book discusses lots of options for these,
including split leaves (which is something R/Rcmdr
will do).
 Basics: You take your values and set a stem –
maybe tens. Then the leaves are the ones place. For
each stem, you list the leaves that coincide in
numeric order.
 Usually works decently for fewer than 100
observations
 Try it. Suppose you have scores on a pre-test for an
at-risk youth group as follows:
 5, 11, 13, 21, 34, 36, 45, 47, 48, 48, 49

18. Histogram
 Take the quantitative variable and break it up into “piles”
or “bins” (usually the same width).
 Count the number of observations in each bin or pile.
 Plot the frequencies per bin.
 Usually no spaces between bins (if there is, it is a gap –
NOT like a bar chart).
 You DO need to know the boundaries. (5,10], (10,15] as
bins IS different from [5,10),[10,15). (If anyone needs me
to explain open/closed brackets, please ask).
 Technology lets us vary the width of bins (effectively the
number)
 You can also use unequal bin widths but then you need
something called density, not frequency.

19. Examples
 See page 2 of the handout
 Try to describe the shape of each histogram
 Then see page 3 of the handout
 We’re going to create a histogram by hand if there is time
 If no time, you can do this on your own.

20. Cookie Lab
 Time Permitting (otherwise, Friday)
 The last page (to turn in) is not due till the end of
class tomorrow. So don’t worry if we don’t get to it
today. You can look at it tonight or tomorrow in class
(I’ll give last five minutes of class for you to work on
it).