Copyright (c) Bani Mallick1 Lecture 2 Stat 651. Copyright (c) Bani Mallick2 Topics in Lecture #2 Population and sample parameters More on populations.

Copyright (c) Bani Mallick 1

Lecture 2

Stat 651


Topics in Lecture #2 Population and sample parameters

More on populations and samples

The Median and Percentiles

Robustness of the median and IQR, lack of robustness for the mean and standard deviation

Variability: standard deviation and interquartile range (IQR)

Boxplots


Book Sections Covered in Lecture #2

Chapter 3.4

Chapter 3.5, up to page 88

Chapter 3.6


Review of Lecture #1

We described samples and populations

We want to make inference about populations

We draw samples from the population to do so

Different samples give different results



I will make quite a big deal about the difference between samples and populationsOne major thing we do in statistics is

to quantify: how far is what we see in the sample from what we cannot see in the population?



We defined the relative frequency histogram

This counts the percentage of the sample that falls into computer-generated categories

We studies the NHANES case-control study

This had samples from 2 (sub)populations

Those who developed breast cancer

Those who did not



In the NHANES data, the sample who developed breast cancer seemed to have smaller values of saturated fat than the sample that did not develop breast cancer

What we will try to do today is to quantify those differences


Review of Lecture #1: The Population Mean

• In many problems, the goal is to make inference about the population mean of a numerical variable, e.g., saturated fat intake

• The only thing we have available is data from a sample, e.g., the sample mean.

• Define in words what you mean by the population mean and the sample mean!



• In many problems, the goal is to make inference about the population mean of a numerical variable, e.g., saturated fat intake

• You’re right! The population mean is the average of all the outcomes in the population

• It can’t be measured, hence we take samples.

• BTW, what’s an average?



Formal definition. If the sample is of size n and the data are X1,…, Xn , then the sample mean is

Total sum of the values in the sample

n

i1 2 n i=1

Σ ΧΧ +Χ +...ΧΧ= =

n n

Number of observations in the sample


Parameters and Statistics

Parameter: numerical characteristic of a population

Statistic: numerical characteristic of a sample

PopulationSample



Parameter: numerical characteristic of

a population, called Statistic: numerical characteristic of a

sample, called

PopulationSample



Parameter: numerical characteristic of a

population, called never known!!!)

Statistic: numerical characteristic of a sample, called

PopulationSample

We want to make inference about populationfrom sample


Case-control data: NHANES log(Saturated

Fat)

Which sample has the larger sample mean?

0%

5%

10%

15%

Per

cen

t

Cancer

Healthy

2.00 3.00 4.0

Log(Saturated Fat)

0%

5%

10%

15%

Per

cen

t


Case-control data: NHANES log(Saturated

Fat)

Cancer: = 2.70

Healthy: = 2.99

0%

5%

10%

15%

Per

cen

t

Cancer

Healthy

2.00 3.00 4.0

Log(Saturated Fat)

0%

5%

10%

15%

Per

cen

t


The Concept of a Median

When we talk about the population median age of graduate students at Texas A&M, what do we mean?

When we talk about the sample median of graduate students at Texas A&M, what do we mean?



The population median is the central point

1/2 the population falls below the population median

1/2 the population falls above the population median

Look in newspapers for the use of the median and the mean



The sample median is the central point of the sample

1/2 the sample falls below the sample median

1/2 the sample falls above the sample median

We can use the sample median to try to estimate the population median

Remember though, different samples give different numbers



The sample median is computed by SPSS, or by hand as follows

Order the data

If n is an odd number, the sample median is the (n+1)/2 point in order

If n is even, it is the average of the n/2 point in order and the (n/2+1) point in order



Data 97 99 93 96 91 90 95: n = 7

Ordered: 90 91 93 95 96 97 99

If n is an odd number, the sample median is the (n+1)/2 point in order

(n+1)/2 = 4

Sample median = 95



Data 97 92 96 91 90 95: n = 6

Ordered: 90 91 93 95 96 97

If n is even, it is the average of the n/2 point in order and the (n/2+1) point in order

n/2 = 3, (n/2+1) = 4

Sample median = average of 93 and 95 = 94


Summary Statistics in SPSS

Select “analyze”, “descriptive statistics”, “explore”

Select your variables (“Dependent”) and you populations (“Factor List”)

Ask for “Statistics”

Cut and paste as needed


Descriptives

2.9905 7.969E-02

2.8310

3.1500

3.0015

2.9957

.381

.6173

1.39

4.26

2.88

.9130

-.332 .309

-.138 .608

2.6969 8.362E-02

2.5295

2.8642

2.6886

2.8332

.413

.6423

1.39

4.77

3.38

.8755

.156 .311

.748 .613

Mean

Lower Bound

Upper Bound

95% ConfidenceInterval for Mean

5% Trimmed Mean

Median

Variance

Std. Deviation

Minimum

Maximum

Range

Interquartile Range

Skewness

Kurtosis

Mean

Lower Bound

Upper Bound


5% Trimmed Mean

Median

Variance

Std. Deviation

Minimum

Maximum

Range

Interquartile Range

Skewness

Kurtosis

Healthy Status Numerical:0 = Healthy, 1 = Cancer

Healthy

Breast Cancer

Log(Saturated Fat)

Statistic Std. Error

SPSS Output for NHANES: you see variable, populations, sample means, medians, minimum and maximum


Summary Statistics in SPSS

For both measures of central tendency, healthy women had reported more saturated fat on the day they were interviewed


VARIABILITY

Variability is one of the hardest concepts to understand, and to measure.

Variability means how widely spread out the population is

Populations with large variability should have samples that are more spread out than are samples from populations with low variability

Variability is measured by the average distance of points to center


Shelf life of two drugs

(past shelf life the drug is harmful).

Which drug would you

take, assuming they are equally

effective on average?

Sample from Drug A

Sample fromDrug B

0%

5%

10%

15%

20%

Per

cen

t

.00

1.00

-4.0000 -2.0000 0.0000 2.0000 4.0000

v2

0%

5%

10%

15%

20%

Per

cen

t


Absolute distance from sample mean

Measuring Variability by Distances

Absolute distance from sample median

iX

i

Squared distance to sample mean

2

i

All numerical measures of variability are measures of “the center of the distance”


Variability as average squared distance

The sample variance, s2 is defined as follows

Compute (X - sample mean) for every observation: squared distances

Sum these numbers up

Divide by n-1

Except for the n-1, this is the sample mean of the squared differences


COMPUTING FORMULA for the Sample Variance s2

2n

ii 12

X

n 1s

Note how s2 measures how far the data are from the sample mean


The Standard Devation, s

The sample standard deviation is called s

It is the square root of the sample variance

Its units are the same as the units of the data

2s s


Aortic Stenosis Data

• Two populations: healthy kids and kids with aortic stenosis

• Two outcomes: body surface area and aortic valve area

• Size adjusted aortic valve areas is the ratio of aortic valve area to body surface area


Aortic Valve Area for

Healthy and Stenotic Kids

Which has larger mean,

larger variance?

5%

10%

15%

20%

25%

Per

cen

t

Healthy

Stenotic

0.000 1.000 2.000 3.000

Aortic Valve Area

5%

10%

15%

20%

25%

Per

cen

t





larger variance?

5%

10%

15%

20%

25%

Per

cen

t

Healthy

Stenotic

0.000 1.000 2.000 3.000

Aortic Valve Area

5%

10%

15%

20%

25%

Per

cen

t

Mean = 1.06Median = 0.80std dev = 0.78

Mean = 0.83Median = 0.69std dev = 0.64


SAMPLE MEANS ARE NOT ROBUST

HEIGHTS

-4 -2 0 2 4 6

MED=

-40 -2 0 2 4 6

MED=

-2 0 2 4 6 40

MED=


SAMPLE MEANS ARE NOT ROBUST

Outliers affect sample means much more than they do sample medians.

You should look out for wild points

They may be errors, or naturally occurring variability, but they have the potential for mischief

We will develop statistical methods that help us understand whether our conclusions are being driven by a few wild points.


PERCENTILESSAT-

SCORES•If your are in the 90th percentile of the population, what % scored higher than you?•If you are in the 25th percentile, what percent scored less than or equal to you?

•What percent lie between the 25th and 75th percentiles?

•What percentile is the median?


INTERQUARTILE RANGE (IQR) Defined as the difference between the 75th and 25th percentiles

The length of data needed to cover 50% of the data.

This is a natural, robust measure of variability

Why do I say it is robust?

Why do I say it is natural?





larger variance?

5%

10%

15%

20%

25%

Per

cen

t

Healthy

Stenotic

0.000 1.000 2.000 3.000

Aortic Valve Area

5%

10%

15%

20%

25%

Per

cen

t

Mean = 1.06Median = 0.80std dev = 0.78IQR = 0.98

Mean = 0.83Median = 0.69std dev = 0.64IQR = 0.59


INTERQUARTILE RANGE Defined as the difference between the

75th and 25th percentiles

The length of data needed to cover 50% of the data.

This is a natural, robust measure of variability

If comparisons of variability are different for the standard deviation and the IQR, good chance of an outlier


Descriptives

2.9905 7.969E-02

2.8310

3.1500

3.0015

2.9957

.381

.6173

1.39

4.26

2.88

.9130

-.332 .309

-.138 .608

2.6969 8.362E-02

2.5295

2.8642

2.6886

2.8332

.413

.6423

1.39

4.77

3.38

.8755

.156 .311

.748 .613

Mean

Lower Bound

Upper Bound


5% Trimmed Mean

Median

Variance

Std. Deviation

Minimum

Maximum

Range

Interquartile Range

Skewness

Kurtosis

Mean

Lower Bound

Upper Bound


5% Trimmed Mean

Median

Variance

Std. Deviation

Minimum

Maximum

Range

Interquartile Range

Skewness

Kurtosis

Healthy Status Numerical:0 = Healthy, 1 = CancerHealthy

Breast Cancer

Look for variance, standard deviation and IQR

Log(Saturated Fat)Statistic Std. Error


Box Plots

Box plots are a means of visualizing data from different populations, especially comparing them

Far less clunky than histograms

Clear definitions, don’t have to worry about # of bars, class intervals, etc.

Easily available


BASIC FORM OF THE BOXPLOT

75th percentile

Median

25th percentile

IQR canbe read off


Box Plot Additions: Technical

To the basic box, whiskers are added:

Go out to furthest point 1.5 IQR from 75th and 25th percentiles

• Any other points are called “Suspicious” “Extreme” “Outliers”


BASIC FORM OF THE BOXPLOT

75th percentile

Median

25th percentile

IQR canbe read off

Point here

Point here

Outlier


IQR AS A MEASURE OF VARIABILITY

The box is from 25th to 75th sample percentileThis means 50% of the data are in the boxHence, IQR= range needed to cover 50% of the data IQR is a very robust measure of variability which can be judged graphically


NHANES Saturated Fat Data:

moderately outlying unusually outlying

Cancer Healthy

0.00

25.00

50.00

75.00

100.00

Sat

ura

ted

Fat


Box Plots

The SPSS plot actually displays the median line, but it did not translate into powerpoint

You go to graphs, interactive and boxplot to get these things

Here’s about what the thing looks like in SPSS (then I’ll show you SPSS)


NHANES Data not done interactively (just graphs and boxplot): cannot edit

6059N =

Health Status

HealthyCancer

Sa

tura

ted

Fa

t140

120

100

80

60

40

20

0

-20

60

118

119


Aortic Valve Area for Healthy and Stenotic Kids: done interactively, hence the median is not

labeled when imported to powerpoint

Healthy Stenotic

0.000

1.000

2.000

3.000

Ao

rtic

Va

lve

Are

a

Copyright (c) Bani Mallick1 Lecture 2 Stat 651. Copyright (c) Bani Mallick2 Topics in Lecture #2 Population and sample parameters More on populations.

Documents