Upload
joel-carr
View
216
Download
2
Tags:
Embed Size (px)
Citation preview
Welcome to MATH171!
Overview of Syllabus Technology Overview Basic Skills Quiz Start Chapter 1!
Displaying data with graphs
BPS chapter 1
© 2006 W. H. Freeman and CompanyWith Modifications by Dr. M. Leigh Lunsford
The Collection and Analysis of Data
What is Statistics?
Sampling and Experimental DesignChapters 8 & 9
Descriptive Statistics(Data Exploration)
Chapters 1 - 5
Inferential StatisticsChapters 14 - 21
Probability & Sampling DistributionsChapters 10 & 11
Statistics is the Science of Learning from Data
Objectives for Chapter 1
Picturing Distributions with Graphs
Individuals and variables
Two types of data: categorical and quantitative
Ways to chart categorical data: bar graphs and pie charts
Ways to chart quantitative data: histograms and stemplots
Interpreting histograms
Time plots
Individuals and variables (page 3)Individuals are the objects described by a set of data. Individuals may be people, but they may also be animals or things.
Example: Freshmen, 6-week-old babies, golden retrievers, fields of corn, cells
A variable is any characteristic of an individual. A variable can take different values for different individuals.
Example: Age, height, blood pressure, ethnicity, leaf length, first language
Two types of variables (page 4)A variable can be either
quantitative Something that can be counted or measured for each individual and
then added, subtracted, averaged, etc., across individuals in the population.
Example: How tall you are, your age, your blood cholesterol level, the number of credit cards you own.
OR
categorical Something that falls into one of several categories. What can be
counted is the count or proportion of individuals in each category. Example: Your blood type (A, B, AB, O), your hair color, your ethnicity,
whether you paid income tax last tax year or not.
Example 1.1 (page 4-5) How do you determine if a variable is categorical or quantitative? Identify individuals, variables and types of variables.
Ways to graph categorical dataBecause the variable is categorical, the data in the graph can be ordered any way we want (alphabetical, by increasing value, by year, by personal preference, etc.).
Bar graphsEach category isrepresented by a bar.
Pie chartsUse when you want to emphasize
each category’s relation to the whole.
Variable
Variable Values
Example: Top 10 causes of death in the United States, 2001
Rank Causes of death CountsPercent of
top 10s
Percent of total
deaths
1 Heart disease 700,142 37% 29%
2 Cancer 553,768 29% 23%
3 Cerebrovascular 163,538 9% 7%
4 Chronic respiratory 123,013 6% 5%
5 Accidents 101,537 5% 4%
6 Diabetes mellitus 71,372 4% 3%
7 Flu and pneumonia 62,034 3% 3%
8 Alzheimer’s disease 53,852 3% 2%
9 Kidney disorders 39,480 2% 2%
10 Septicemia 32,238 2% 1%
All other causes 629,967 26%
For each individual who died in the United States in 2001, we record what was
the cause of death. The table above is a summary of that information.
How did they get these numbers?
Top 10 causes of death in the U.S., 2001
Bar graphsEach “value” of the categorical variable is represented by one bar. The bar’s
height shows the count (or sometimes the percentage) for that particular
category.
The number of individuals who died of an accident in 2001 is
approximately 100,000.
0100200300400500600700800
Co
un
ts (
x10
00
)
Bar graph sorted by rank Easy to analyze
Top 10 causes of death in the U.S., 2001
0100200300400500600700800
Co
un
ts (
x10
00
)
Sorted alphabetically Much less useful
Percent of people dying fromtop 10 causes of death in the U.S., 2000
Pie chartsEach slice represents a piece of one whole.
The size of a slice depends on what percent of the whole this category represents.
Percent of deaths from top 10 causes
Percent of deaths from
all causes
Make sure your labels match
the data!
Make sure all percents
add up to 100!!
Apply Your Knowledge Problem 1.4 Let’s work Problem 1.4 (page 10)
together! Bar graph (in count & percent) Pie chart?
Day of Week Births
Sun 7563
Mon 11733
Tues 13001
Wed 12598
Thurs 12514
Fri 12396
Sat 8605
Number of Babies Born on Each Day of
the Week in 2003
Births in 2004 by Day of Week
Sun10%
Mon15%
Tues16%
Wed16%
Thurs16%
Fri16%
Sat11%
Ways to chart quantitative data
Histograms and stemplots
These are summary graphs for a single variable. They are very useful to
understand the pattern of variability in the data.
Line graphs: time plots
Use when there is a meaningful sequence, like time. The line connecting the
points helps emphasize any change over time.
Other graphs to reflect numerical summaries (see Chapter 2)
An Example Suppose we want to determine the following:
What percent of all fifth grade students in our district have an IQ score of at least 120?
What is the average IQ score of all fifth grade students in our district?
It is too expensive to give an IQ test to all fifth grade students in our district.
Below are the IQ test scores from 60 randomly chosen fifth graders in our district. (Individuals (subjects)?, Variable(s)?)
Previews of Coming Attractions! We are interested in questions about a population (all fifth grade
students in our district). We want to know the percent (or proportion) of the population in a
particular category (IQ score of at least 120) and the average value of a variable for the population (average IQ score).
We have taken a random sample from the population. Eventually we will use the data from the sample to infer about the
population. (Inferential Statistics) For now we will describe the data in the sample. (Descriptive
Statistics) We will graphically represent the IQ scores for our sample (histogram &
stem and leaf) We will find the percent of students in our sample with an average IQ
score of at least 120 and understand how that percent relates to the graph.
Later (Chapter 2) we will also be able to describe the data with numerical summaries and other types of plots (boxplots)
Stemplots (page 19)
How to make a stemplot:
1) Separate each observation into a stem, consisting of
all but the final (rightmost) digit, and a leaf, which is
that remaining final digit. Stems may have as many
digits as needed, but each leaf contains only a single
digit.
2) Write the stems in a vertical column with the smallest
value at the top, and draw a vertical line at the right of
this column.
3) Write each leaf in the row to the right of its stem, in
increasing order out from the stem.
Let’s try it with this data: 9, 9, 22, 32, 33, 39, 39, 42,
49, 52, 58, 70
STEM LEAVES
Now Let’s Make a Stemplot for Our IQ Data
Stem & Leaf Plot for IQ Data IQ Test Scores for 60 Randomly
Chosen 5th Grade Students
Stem and Leaf plot for IQ Scores
stem unit = 10
leaf unit = 1
Frequency Stem Leaf
3 8 1 2 9
4 9 0 4 6 7
14 10 0 1 1 1 2 2 2 3 5 6 8 9 9 9
17 11 0 0 0 2 2 3 3 4 4 4 5 6 7 7 7 8 8
11 12 2 2 3 4 4 4 5 6 7 7 8
9 13 0 1 3 4 4 6 7 9 9
2 14 2 5
60
Now Let’s Make a Histogram (pages 10-12) Use the Same IQ Data We will start by hand….using class (bin) widths of 10
starting at 80… Make a Frequency Table for the data:
Variable: X = IQ score
Frequency Table:
Bins Frequency Percent80X90X100X110X120X130X140Xtotals: 60 99.9%
Now Let’s Make a Histogram (pages 10-12) Use the Same IQ Data We will start by hand….using class (bin) widths of 10
starting at 80… Make a Frequency Table for the data:
IQ Scores of Randomly Chosen Fifth Grade Students
0
5
10
15
20
25
30
80
90
100
110
120
130
140
150
IQ Score
Per
cent
What is the meaning of this
bar?Percent of
What?
5.06.7
23.3
28.3
18.315.0
3.3
What percent of the 60 randomly chosen fifth grade students have an IQ score of at least 120? Numerically?
How to Represent Graphically?
Back to Our Question:
18.3%+15%+3.3%=36.6%
(11+9+2)/60=.367 or 36.7%
Grey Shaded Region corresponds to the 36.6% of students
What is Different Fromthe Histogram we Generated
In Class?
Another Histogram of the IQ Data!
How to create a histogram
It is an iterative process—try and try again.
What bin (class) size should you use?
Not too many bins with either 0 or 1 counts
Not overly summarized that you lose all the information
Not so detailed that it is no longer summary
Rule of thumb: Start with 5 to10 bins.
Look at the distribution and refine your bins.
(There isn’t a unique or “perfect” solution.)
Not summarized enough
Too summarized
Same data set
GOAL: Capture Overall Pattern
Apply Your Knowledge Let’s try problem 1.7 (page 14)
What is the difference between a histogram and a bar chart? See pages 12-13
Interpreting histograms
When describing a quantitative variable, we look for the overall pattern and for
striking deviations from that pattern. We can describe the overall pattern of a
histogram by its shape, center, and spread.
Histogram with a line connecting
each column too detailed
Histogram with a smoothed curve
highlighting the overall pattern of
the distribution
Most common distribution shapes
A distribution is symmetric if the right and left sides
of the histogram are approximately mirror images of
each other.
Symmetric distribution
Complex, multimodal distribution
Not all distributions have a simple overall shape,
especially when there are few observations.
Skewed distribution
A distribution is skewed to the right if the right
side of the histogram (side with larger values)
extends much farther out than the left side. It is
skewed to the left if the left side of the histogram
extends much farther out than the right side.
Alaska Florida
Outliers
An important kind of deviation is an outlier. Outliers are observations
that lie outside the overall pattern of a distribution. Always look for
outliers and try to explain them.
The overall pattern is fairly
symmetric except for two
states clearly not belonging
to the main trend. Alaska
and Florida have unusual
representation of the
elderly in their population.
A large gap in the
distribution is typically a
sign of an outlier.
IMPORTANT NOTE:
Your data are the way they are.
Do not try to force them into a
particular shape.
It is a common misconception
that if you have a large enough
data set, the data will eventually
turn out nice and symmetrical.
Line graphs: time plots
This time plot shows a regular pattern of yearly variations. These are seasonal
variations in fresh orange pricing most likely due to similar seasonal variations in
the production of fresh oranges.
There is also an overall upward trend in pricing over time. It could simply be
reflecting inflation trends or a more fundamental change in this industry.
Let’s Start Problem 1.41 on Page 35….
Time always goes on the
horizontal, or x, axis.
The variable of interest—
here “retail price of fresh
oranges”—goes on the
vertical, or y, axis.
Death rates from cancer (US, 1945-95)
0
50
100
150
200
250
1940 1950 1960 1970 1980 1990 2000
Years
Death
rate
(per
thousand)
Death rates from cancer (US, 1945-95)
0
50
100
150
200
250
1940 1960 1980 2000
Years
Dea
th r
ate
(per
thou
sand
)
Death rates from cancer (US, 1945-95)
0
50
100
150
200
250
1940 1960 1980 2000
Years
Death
rate
(per
thousand)
A picture is worth a thousand words,
BUT
there is nothing like hard numbers.
Look at the scales.
Scales matterHow you stretch the axes and choose your scales can give a different impression.
Death rates from cancer (US, 1945-95)
120
140
160
180
200
220
1940 1960 1980 2000
Years
Death
rate
(pe
r th
ousan
d)