§ 13.4 - 14.1 Terminology, § 13.4 - 14.1 Terminology, Clinical Studies, Graphical Clinical Studies, Graphical

Representations of DataRepresentations of Data

TerminologyTerminology

A statistic is a piece of numerical information taken from a sample.

A parameter is a piece of numerical information about the population being studied.

In other words, a statistic is an estimate for a parameter.

TerminologyTerminology

Sampling error is the difference between a parameter and the statistic used to estimate it. The causes of this error are:1. Error due to chance or sampling variability.2. A poorly chosen sample--sample bias.

If we have a sample of size n from a population of size N then the sampling rate is the ratio n/N.

The Capture-Recapture The Capture-Recapture MethodMethod

Step 1: Capture (choose) a sample of size n1 and tag a certain number of the animals/objects/people.

Step 2: After some amount of time, capture a new sample of size n2 and take a count of the tagged individuals. Call this number k.

If the second sample is representative then the size of the population is N (n1)(n2)/k

Example: The Example: The N N - value of - value of the Monarch Butterflythe Monarch Butterfly

Suppose 150 monarchs are caught, tagged and released.

A few days later 200 more monarchs are caught, of which only 2 are found to be tagged.

Estimate the N - value of the local monarch population.

Clinical Studies

Clinical studies are concerned with determining whether a single variable is causes a certain effect.

The goal is to limit confounding variables--other possible causes.

In a controlled study the subjects are divided into two groups: the treatment group and the control group.

If the subjects are assigned to the two groups randomly then the study is a randomized controlled study.

Clinical Studies

If the control group is given a placebo then the study is a controlled placebo study.

If neither group of subjects knows whether they are receiving treatment or a placebo then the study is said to be blind.

If neither the subjects nor the scientists know who is receiving treatment and who is receiving a placebo then the study is referred to as double-blind.

Graphical Representations Graphical Representations of Dataof Data

• A data set is a collection of individual data points.

Below is a data set consisting of test scores:

40484840444563648406076364844

324040404040404476443648363648

407240444828727260324048447276

3236244040323676403672324472 96

444444364436722844647244324048

Frequency TableFrequency Table

14812191316106211Frequency

96767264605648444036322824

4Score

One way we might summarize the data is in the form of a Frequency Table. The number below each exam score is the number of students getting that score.

Bar GraphsBar Graphs Another convenient way to summarize the test scores is in the form of a bar graph:

Test Scores

0

2

4

6

8

10

12

14

16

18

4 24 28 32 36 40 44 48 56 60 64 72 76 96

Score

Frequency

§ 14.2 Variables§ 14.2 Variables

Variables:Variables:Quantitative Quantitative v.v. Qualitative Qualitative

A variable is any value or characteristic that varies with members of a population.

In the previous example, test scores would be considered a variable.

A variable is said to be quantitative if it represents a measurable quantity.

A variable that cannot be measured is called qualitative.

Variables:Variables:Continuous Continuous v.v. Discrete Discrete

If the possible values of a variable are ‘countable’--or if there is some smallest increment we can use--the variable is said to be discrete.

If the difference between values of a variable can be arbitrarily small, then the variable is called continuous.

Example: Blood TypesBlood TypesForty people recently donated blood and their types are listed below:

ABOOAOAAAOO

AOOAABAABAA

AAOOAOOBOB

OAAAOABAOO

Example: Blood TypesBlood TypesWhile this data is qualitative, it is still possible to make both a frequency table and a bar graph to represent it:

Blood Type Distribution

0

2

4

6

8

10

12

14

16

18

20

O A AB B

Antigen

Frequency

Example: Blood TypesBlood TypesAnother way to present the information is in the form of a pie chart. What differentiates this from the previous tables and graphs is that it shows the percentage, or relative frequency of each blood type in the sample.

Blood Type Distribution

O

A

AB

B

Let’s return for a moment to our test score example. . .

Suppose the instructor decided to allocate grades as follows:

A 80 - 100B 50 - 79C 30 - 49D 0 - 29

This is an example of using what are called class class intervals intervals

When there are too many different values or categories to display our data nicely, we will use these kinds of intervals to simplify the situation.

The test scores, when sorted into class intervals (in this case the letter grades), can be graphed like this:

Grade Distribution

0

10

20

30

40

50

60

F D C B A

Letter Grade

Frequency

HistogramsHistograms

You may have noticed that in all the cases where we have given a chart or graph that the variable used was discrete.

How can we graphically display continuous variables?

We can use a variation on the bar graph called a histogram.

Example: Age at first marriage.Based on a survey, the frequency table below was obtained for

the age of groom at first marriage in the state of Wisconsin

Using class intervals of length 10 (years) draw a histogram for the given data.

8345 - 50

40440 - 45

84035 - 40

3,30030 - 35

9,79625 - 30

11,76820 - 25

# of Grooms

Age Interval*

Example: Age at first marriage.Based on a survey, the frequency table below was obtained for

the age of groom at first marriage in the state of Wisconsin

Using class intervals of length 10 (years) draw a histogram for the given data.

8345 - 50

40440 - 45

84035 - 40

3,30030 - 35

9,79625 - 30

11,76820 - 25

# of Grooms

Age Interval* Age of First Marriage

0

5000

10000

15000

20000

25000

20-30 30-40 40-50

Age Range

Frequency

Series1

Example: Age at first marriage.

Now draw a histogram with intervals which are five years in length.

8345 - 50

40440 - 45

84035 - 40

3,30030 - 35

9,79625 - 30

11,76820 - 25

# of Grooms

Age Interval*

Example: Age at first marriage.

Now draw a histogram with intervals which are five years in length.

8345 - 50

40440 - 45

84035 - 40

3,30030 - 35

9,79625 - 30

11,76820 - 25

# of Grooms

Age Interval*

Age of First Marriage

0

2000

4000

6000

8000

10000

12000

14000

20-25 25-30 30-35 35-40 40-45 45-50

Age Group

Frquency