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)