2/18/2009

1

Is the treatment appropriate for the response you want

to study?

� Is studying the effects of eating red meat on cholesterol

values in a group of middle-aged men a realistic way to

study factors affecting heart disease problem in humans?

� What about studying the effects of hair spray

on rats to determine what will happen

to women with big hair?

One pitfall of Experimentation: Lack

of realism

1

The pitfalls in Experimentation • Example 1: Suppose researchers want to determine if the

drug Ecstasy causes memory loss. One possible design would be to take a group of volunteers and randomly assign some to take Ecstasy on a regular basis, while the others are given a placebo. Test them periodically to see if the Ecstasy group experiences more memory problems than the placebo group.

• The obvious flaw in this experiment is that it is unethical(and actually also illegal) to administer a dangerous drug like Ecstasy, even if the subjects are volunteers. The only feasible design to seek answers to this particular research question would be an observational study.

2

The pitfalls in Experimentation

• Example 2: Suppose researchers want to

determine whether females wash their hair

more frequently than males.

• It is impossible to assign some subjects to be

female and others male, and so an experiment

is not an option here. Again, an observational

study would be the way to proceed.

3

Summary: Observations

• explanatory variable's values allowed to occur

naturally

• because of the possibility of lurking variables,

it is difficult to establish causation

• if possible, control for suspected lurking

variables by studying groups of similar

individuals separately

• some lurking variables are difficult to control

for; others may not be identified.

4

Summary: Experiments• explanatory variable's values are controlled by researchers (treatment is

imposed)

• randomized assignment to treatments automatically controls for all lurking variables.

• making subjects blind avoids placebo effect.

• making researchers blind avoids conscious or sub-conscious influences on their subjective assessment of responses.

• randomized controlled double-blind experiment is generally optimal for establishing causation.

• lack of realism may prevent researchers from generalizing experimental results to real life situations.

• non-compliance may undermine an experiment. Volunteer sample might solve (at least partially) this problem.

• some treatments are impossible, impractical, or unethical to impose.

5

Explanatory variablesHirsch and Johnston from the Smell & Taste Treatment and Research

Foundation in Chicago believe that the presence of floral scent can

improve a person’s learning ability in some situations. To test this

hypothesis, they set up an experiment in which each of 22 subjects

completed 2 sets of three pencil and paper mazes, one set while wearing a

floral-scented mask. Each subject wore a floral-scented mask and an

unscented mask, and the order was randomized. The researchers measured

the length of time it took each subject to complete the sets of mazes.

What is the explanatory variable?

a) The amount of scent.

b) Presence or absence of the floral scent.

c) Time to complete the pencil and paper mazes.

d) Whether the subject was able to complete the mazes quicker while wearing

the floral-scented mask.

6

2/18/2009

2

Response variables

In the previous example what is the response

variable?

a) The amount of scent.

b)Presence or absence of the floral scent.

c) Time to complete the pencil and paper mazes.

d)Whether the subject was able to complete the

mazes quicker while wearing the floral-

scented mask.7

Individuals

In the previous example what are the

individuals?

a) The masks (floral-scented or unscented).

b)The 22 subjects.

c) The mazes.

8

Control

In the previous example the researchers

incorporated control/comparison by

a) Giving each subject a floral-scented and an

unscented mask.

b)Randomly assigning half of the subjects to

wear a floral-scented mask only and the other

half to wear the unscented mask only.

c) Giving each subject two sets of mazes.

9

RandomizationIn the previous example the researchers

incorporated randomization by

a) Randomly selecting the subjects to participate in the study.

b)Randomly assigning half of the subjects to wear the floral-scented mask and the other half to wear an unscented mask.

c) Randomly assigning the order that each subject receives the floral-scented and unscented masks.

10

Replication

In the previous example the researchers

incorporated replication by

a) Using two masks.

b)Using two sets of mazes.

c) Using three mazes within each set.

d)Using twenty-two subjects.

e) Repeating the entire experiment a second time.

11

Variables

If age affects whether the presence of a floral

scent improves learning ability and was not

included among the variables studied in the

experiment, then age is

a) An explanatory variable.

b)A response variable.

c) A lurking variable.

d)Confounded with floral scent.12

2/18/2009

3

Experimental design

The experimental design used in the floral scent

example is called a

a) Completely randomized design.

b)Randomized block design.

c) Matched pairs design.

13

Statistical significanceIf there is a statistically significant difference between

the average times to complete the mazes while

wearing the floral-scented mask and the unscented

mask, then the difference in average times to

complete the mazes between the floral-scented mask

and the unscented mask is

a) Too large to be due to chance alone.

b) Too small to be due to chance alone.

c) So large that we can reasonably attribute it to chance.

d) So small that it is likely due to chance.14

Experimental designAn Austrian study investigated whether maintaining a surgery

patient’s body temperature close to normal by heating the

patient during surgery decreases wound infection rates.

Patients included in the study were undergoing colon or rectal

surgery and were randomly assigned to one of two treatment

groups. In the normalthermic group, patients’ core

temperatures were maintained near normal 36.5 degrees

Celsius. In the hypothermic group, patients’ core temperatures

were allowed to decrease to about 34.5 degrees Celsius.

The design is called a

a) Completely randomized design

b) Randomized block design

c) Matched pairs design

15

Experimental designIn the previous experiment involving patients’

temperatures, both men and women were the

patients. If the men and women were

separately assigned to treatments, the design

would be a

a) Completely randomized design

b)Randomized block design

c) Matched pairs design

16

Randomized block designIn a randomized block design, a block contains

a) Individuals that are similar with respect to the characteristic that defines the block.

b) Individuals that are assigned to the same treatment.

c) Individuals that are similar with respect to the characteristic that defines the block and that are assigned to the same treatment.

17

Problems with experimentsA study claims that patients who receive surgery for intestinal

cancer live much longer after treatment than patients who are treated without surgery. However, doctors operated only on patients in relatively good condition so we cannot conclude from this study that surgery lengthens intestinal cancer patients’ lives.

This is an example of

a) Confounding.

b) A lurking variable.

c) A double-blind experiment.

d) The placebo effect.

18

2/18/2009

4

Double-blind experimentsMedical experiments are often double-blind. This means

that

a) All individual data are kept confidential.

b) Neither the subject nor the doctor/administrator knows which treatment the subject receives.

c) Doctors are not allowed to decide which treatment a patient will receive; subjects are randomly assigned to treatments.

d) The subjects in the control group receive a placebo treatment.

19

ExperimentsAn advantage of experiments over observational

studies is

a) An experiment can provide evidence of cause and effect.

b)An experiment can compare two or more groups.

c) An experiment can include explanatory and response variables.

20

Experiments

Which of the following principles of good

experimentation does an observational study

not incorporate?

a) Control or comparison

b)Random assignment to treatments

c) Replication

21 22

Chapter 1Chapter 1Section1-2: Types

of Data

Chapter 2Chapter 2Summarizing and

Graphing Data

The Big Picture of Statistics

23

Types of Data

• Any data set contains information about some

group of individuals, and the information is

organized in variables.

– Individuals: people, animals or things

– Variables: weight, speed, age, color, gender,

concentration of a certain chemical, distance,

test scores, etc.

24

2/18/2009

5

Variables

CategoricalPlaces an individual into

one of several categories

Examples:

Gender, color, favorite

movie, type of car,

religion, etc.

QuantitativeTakes numerical values (in a unit of

measurement) for which arithmetic operations make sense.

Examples: Height, weight, MPG, age, salary, etc.

Note: Quantitative variables

continuous discrete

can take on any can take on

numerical value only fixed values

in a range

25

Graphical DisplaysCategorical

• Pie charts

• Bar graphs

Quantitative

• Dotplot

• Stemplots

• Histograms

• Boxplots

(details later)

Show

individual

data points.

Okay for small

data sets.

Better for large

data sets.

26

GraphicalGraphical Displays Displays

for for

Categorical Variables Categorical Variables

27

1. Pie Charts

• Pie charts are useful for summarizing a single categorical variable if not too many categories

• A pie chart must include all the categories that make up a whole.

• Use a pie chart when

you want to emphasize

each category’s relation

to the whole.

28

2. Bar Graphs

• Bar graph uses a horizontal or vertical rectangular

bar that levels off at the appropriate level.

• Bar graphs are useful for summarizing one or two

categorical variables

and particularly useful

for making comparisons

when there are two

categorical variables.

29

GraphicalGraphical Displays Displays

for for

Quantitative Variables Quantitative Variables

30

2/18/2009

6

1. Dotplots

• Work best when

– You have a relatively small data sets

– Want to see (approximately) individual values

– Want to see shape

– Have one group or small number of groups to

compare

31

1. Dotplots

• One Axis: only a horizontal axis

• Scale

– Tick marks with numerical labels

– Equally spaced

• Simply record a dot for each data point above an

appropriate axis.

• If the data value repeats, the dots are piled up at that

location, one dot for each repetition.

32

1. Dotplots

• Example: As part of a study on the effects of

calcium on blood pressure, the following 21

blood pressure readings were recorded

• 107,123,102,110,112,98,136,112,119,109,111,

112, 129,117,110,102,130,112,123,114,107

33

1. Dotplots

34

1. Dotplots

35

2. Stemplots

• are used for relatively small data sets of

quantitative variables.

• show exact values

36

2/18/2009

7

Stemplot example

• Suppose we examine the following data: 55, 65,

66, 69, 71, 73, 79, 81, 83, 84, 84, 85, 86, 88, 89,

90, and 94

• The stems for these data are 5, 6, 7, 8, and 9

since the data start in the 50’s and end in the

90’s

37

Making the stemplot

5

6

7

8

9

Now, we record the leaves,

the ones digit for each value

38

Making the stemplot

Data set:

55, 65, 66, 69, 71, 73,

79, 81, 83, 84, 84,

85, 86, 88, 89, 90,

and 94

Stemplot:

5 5

6 569

7 139

8 13445689

9 04

39

Stemplots with split stem

• Split the stems so that the original stem

becomes two stems

– One for the digits 0, 1, 2, 3, 4 --placed on first

line of the stem

– One for digits 5, 6, 7, 8, 9 --placed on second

line of the stem

40

Making the split stemplot

Data set:

55, 65, 66, 69, 71, 73, 79, 81, 83, 84, 84, 85, 86, 88, 89, 90, and 94

Stemplot Split Stemplot:

5

5 5 5 5

6 569 6

7 139 6 569

8 13445689 7 13

9 04 7 98 1344

8 5689

9 04

9

41

Stemplots to compare distributions• Back-to to-back stemplots

• Example: speed of predators and nonpredators

Predator Nonpredator

1 12

2 05

900 3 025

2 4 00058

0 5

6

0 7

Nonpredator:

11, 12, 20, 25,

30, 32, 35, 40,

40, 40, 45, 48

Predator:

30, 30, 39, 42,

50, 70

42

2/18/2009

8

Stemplots: Example

• The age of Best Actress Oscar winners:

– 34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33

• To make a stemplot:1. Separate each observation into a stem and a leaf.

2. Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column.

3. Go through the data points, and write each leaf in the row to the right of its stem.

4. Rearrange the leaves in an increasing order.

43

Stemplots: Example• The age of Best Actress Oscar winners:

34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61

21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33

Split stemplot

2|1

2|56669

3|013333444

3|555789

4|11123

4|599

5|

5|

6|1

6|

7|4

7|

8|0

44

3. Histograms

• Shows groups of cases as rectangles or bars

• A dot plot with bars

• All bars must be same width

• Bars must be

touching

45

Histogram

Work best

– With a large number of values to plot

– Do not need to see individual values exactly

– Want to see general shape

– One distribution or small number of

distributions to examine

46

Histograms

Two Axes

• Horizontal Axis : The variable that you are analyzing

• Vertical Axis: Frequency (or Relative Frequency--

percent)

47

To make a histogram

To make a histogram we choose the

• classes or bins (usually use 5-15 bins, depending on the number of items, and make bin sizes equal)

• tally the data into the classes or bins (put each data point into one and only one bin)

• count the number of items in each bin

• and then draw each bar with its height proportional to the number of items in its bin

frequency histogram

48

2/18/2009

9

Note about the bins

How many bins should we use?

• No one right choice

– Too few bins will give a “skyscraper” graph

– Too many bins will give a “pancake” graph

• Neither choice will give a good picture of the

shape of the distribution

• Use your judgment

49 50

Example

• Step 1: Choose the bins (classes)

The data in Table 1.1 range from 17.0 to 44.2, so we decide to use 6 bins:

15.1-20.0

20.1-25.0

25.1-30.0

30.1-35.0

35.1-40.0

40.1-45.0

51

Example

• Step 2: count the individuals in each class

Class Count

15.1-20.0 5

20.1-25.0 21

25.1-30.0 14

30.1-35.0 9

35.1-40.0 1

40.1-45.0 1

52

Example

• Step 3: draw the histogram

– Mark the scale for the percent of the state’s adults

with college degree (horizontal axis). The scale

runs from 15-45.

– The vertical axis contains the scale of counts. The

scale runs from 0-21 (21 is the maximum count)

– Each bar represents a class—the bar height is the

class count

– No space between the bars unless a class is empty

53 54

2/18/2009

10

Summary: Histograms• Shows groups of cases as

rectangles or bars

• A dot plot with bars

• All bars must be same width

• Bars must be touching

• No one right choice for the bins

– Too few bins will give a “skyscraper” graph

– Too many bins will give a “pancake” graph

• Use your judgment

Graphical Displays for Quantitative

Variables

5 5

6 569

7 139

8 13445689

9 04

Interpreting Graphical Displays

• Once the distribution has been displayed

graphically, we can describe the overall pattern

of the distribution and mention any striking

deviations from that pattern.

Shape

Look for:

• Symmetry/skewness of the distribution

• Peakedness (modality) - the number of peaks

(modes) the distribution has.

Examples of Symmetric Distributions

• Symmetric,

unimodal

distribution (one

peak)

• Example: test scores

Examples of Symmetric Distributions

• Symmetric,

bimodal

distribution (two

peaks)

• Example: life

expectancy in

Europe and Asia

2/18/2009

11

Examples of Symmetric Distributions

• Symmetric,

uniform

distribution (no

peak)

• Example: random

numbers between 1

and 10 generated by

computer

Examples of Skewed Distributions

• Right –skewed

distribution

– Example: salary

• Left–skewed

distribution

– Example: age of death

from natural causes

Center

• The center of the distribution is its midpoint -

the value that divides the distribution so that

approximately half the observations take

smaller values, and approximately half the

observations take larger values.

– Note that from looking at the histogram we can get

only a rough estimate for the center of the

distribution.

Spread

• The spread (also called variability) of the

distribution can be described by the approximate

range covered by the data. From looking at the

histogram we can approximate the smallest

observation (min), and the largest observation

(Max), and thus approximate the range.

• Range=Max-Min.

– (More exact ways of finding measures of spread will

be discussed in the next section.)

Outliers

• Outliers are

observations that fall

outside the overall

pattern.

• One high outlier

Example

• Shape: Roughly symmetric

• Center is about 70

• Spread: range=approximate max-approximate min=95-45=50

• Outliers: no outliers

2/18/2009

12

Another example

• Shape: right-

skewed

• Center: about

6-7%

• Spread: the

range is 27.5-

2.5=25%

Another example

• Shape: neither symmetric nor skewed. There are three clusters.

• Center: about

$22,000

• Spread:

range=32,500-

5,500=$27,000• Note: the center and

spread nor very useful

here.

Individuals vs. variables

Airport administrators take a sample of airline baggage and record the number of bags that weigh more than 75 pounds. What is the individual?

a) Number of bags weighing more than 75 pounds.

b)Average weight of the bags.

c) Each piece of baggage.

d)The airport administrators.

Individuals vs. variablesAirport administrators take a sample of airline

baggage and record the number of bags that weigh more than 75 pounds. What is the variable of interest?

a) Number of bags weighing more than 75 pounds.

b)Average weight of the bags.

c) Each piece of baggage.

d)The airport administrators.

Individuals vs. variables

In a study of commuting patterns of people in a large metropolitan area, respondents were asked to report the time they took to travel to their work on a specific day of the week. What is the individual?

a) Travel time.

b) A person.

c) Day of the week.

d) City in which they lived.

Individuals vs. variablesIn a study of commuting patterns of people in a

large metropolitan area, respondents were asked to report the time they took to travel to their work on a specific day of the week. What is the variable of interest?

a) Travel time.

b) A person.

c) Day of the week.

d) City in which they lived.

2/18/2009

13

Categorical vs. quantitative variables

Would the variable “monthly rainfall in

Michigan” be considered a categorical or

quantitative variable?

a) categorical

b)quantitative

Categorical vs. quantitative variables

If we asked people to report their “weight,”

would that variable be considered a categorical

or quantitative variable?

a) categorical

b)quantitative

Categorical vs. quantitative variables

We then asked people to classify their weight as

underweight, normal, overweight, or obese.

Would this variable now be categorical or

quantitative variable?

a) categorical

b)quantitative

Categorical vs. quantitative variables

What type of data is produced by the answer choices for this question?

a) categorical

b) quantitative

How many times have you

accessed the Internet this

week?

1) None

2) Once or twice

3) Three or four times

4) More than four times

Graphing

For the Internet access data in the previous

question, what is the BEST method of

displaying the data?

a) bar graph

b)boxplot

c) histogram

d)scatterplot

Stemplots

In the dataset represented by the following stemplot, how many times does the number “28” occur? Leaf unit = 1.0.

a) 0

b) 1

c) 3

d) 4

0 9

1 246999

2 111134567888999

3 000112222345666699

4 001445

5 0014

6 7

7 3

2/18/2009

14

Histograms

Look at the following histogram. How many baseball players report a salary of less than $1,441,000?

a) 50

b)170

c) 220

d)350

HistogramsLook at the following histogram for salaries of

baseball players. What shape would you say the data take?

a) Bi-modal

b) Left-skewed

c) Right-skewed

d) Symmetric

e) Uniform