nature of statistics

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THE NATURE OF STATISTICS(Weiss, Chapter 1)

1.1 Two kinds of statistics

Statistics

Body of principles and methods concernedwith

Two basic areas:

1. Descriptive Statistics

Deals with methods of

Ex.

2. Inferential Statistics

Body of methods for drawing conclusionsabout a

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Key Statistical Concepts

- Population

- Parameter

- Sample

- Statistics

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Example

Population:

Parameters:

Sample:

Statistic:

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Classifying Statistical Studies

Descriptive

Describes & summarizes data.( ).

Ex. The voting results for

A bank manager determines the

Inferential

Ex. A consumer group wants informationabout

Biologist interested on body size of

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1.3 Simple Random Sampling (SRS)

Each item of a population is

Two types:

1. SRS with replacement

2. SRS without replacement

Here:Assume SRS without replacement.

How large should be a sample? Oftensurprisingly small.

Pre-election polls:

PopulationSampleError

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1.4 Other Sampling Designs

1.5 Experimental Design

Observational Studies

characteristics andtake measurements.

Design Experiment

then observe characteristicsand take measurements.

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DESCRIPTIVE STATISTICS(Weiss, Chapters 2 & 3)

2.1 Variables and Data

Variable

A characteristic that

Data

Information obtained by

Two types of data

Quantitative Qualitative

Response Response

Ex: Ex:

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Quantitative Data

DiscreteWhose possible values form a finite (orcountable infinite) set of numbers.

Ex.

ContinuousWhose possible values form some intervalof numbers.

Ex.

Important:

The statistical technique employed depends onthe type of data.

Qualitative -

Quantitative -

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Summary:

Variable

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2.2 Grouping Data

The Bureau of Economic Analysis gathersinformation on the length of stay in Europe byUS travelers. A sample of 36 US residentsyield the following data:

41 16 6 21 1 215 31 20 27 17 10

3 32 2 48 8 1221 44 1 56 5 123 13 15 10 18 31 11 14 12 64 10

Group data using classes of equal width

starting with the class 0 - (12).

0 - (12): 0 inclusive up to 12 exclusive - upto and not including.

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Lengthof Stay Frequency

0 - (12)

36 - (48) 248 - (60) 260 - (72) 1

Guidelines for grouping data:

1. # classes small enough to provide& large enough to

2. Each observation must

3. Whenever feasible, all classes

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Frequency & Relative-Frequency

Frequency

The number of

Relative Frequency

expressed as adecimal. Hence

Relative Frequency =

where

n :

fi :

Midpoint

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Length Relat.of Stay Freq. Freq. Midpoint

0 - (12) 15 0.41712 - (24) 13 0.36124 - (36) 3 0.08336 - (48) 2 0.056 4248 - (60) 2 0.056 5460 - (72) 1 0.028 66

Single-Value Grouping

Sometimes more appropriate to use classesthat each represents a single possible value:

- Discrete data with few observations,

- Qualitative data.

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Example 2.8, pg 50

Students in an introductory stats class wereasked for their political party affiliation:

- Democratic (D)- Republican (R)- Other (O)

Responses:

D R O R R R R RD O R D O O R DD R O D R R O RD O D D D R O D

O R D R R R R D

RelativeParty Frequency Frequency

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2.3 Graphs and Charts

Histograms

Ex. US travelers in Europe

Length Relat.of Stay Freq. Freq.

0 - (12) 15 0.41712 - (24) 13 0.36124 - (36) 3 0.08336 - (48) 2 0.05648 - (60) 2 0.05660 - (72) 1 0.028

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1. Frequency Histogram: Plot

ClassesFrequencies

2. Relative-frequency Histogram: Plot

ClassesRelat. Freq.

Note: Freq. and rel. freq. are bothrepresented byheight is the

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Frequency Histogram - Travel example

US travelers

1513

32 2

1

0

2

4

6

8

10

12

14

16

6 18 30 42 54 66

Length of Stay (days)

Frequency

Inspection of the histogram reveals the

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Relative-frequency Histogram - Travel ex.US travelers

0.417

0.361

0.0830.056 0.056

0.028

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.350.40

0.45

6 18 30 42 54 66


Relative

Frequency

Shape of both graphs are

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Cumulative Frequency Distributions

Usually cumulate for

Number of < class end pointobservations

Ex. US travelers:

< 36 days< 36 days

or relative cumulative frequency of

< 36 days

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Ogive (pg 65)Plot relative cumulative frequenciesagainst upper end points of classes.

Ex. US travelers

Class RelatLength end Cumul Cumulof stay point Freq Freq Freq

0 0 0.0000 -(12) 12 15 0.41712-(24) 24 13 0.77824-(36) 36 3 0.861

36-(48) 48 2 33 0.91748-(60) 60 2 35 0.97260-(72) 72 1 36 1.000

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OGIVE - US Travelers

0.00.10.20.30.40.50.60.70.80.9

1.01.1

0 12 24 36 48 60 72


Cum.

Relat.Freq.

Uses of ogives

- What is the length of stay that 50% of UStravelers fall above (below) of? (known asMedian.)

Read off the ogive:

- What is the minimum length of stay the top10% of travelers stay in Europe?

Approximately

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Percentiles

P%

Pth percentile

100%

The pth percentile is a number

Ex. US travelers

10th percentileFirst quartile th percentile

Median th percentile

Third quartile th percentile

90th percentile

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