Introduction

30
Introduction Population – the entire group of concern Sample – only a part of the whole Based on sample, we’ll make a prediction about the population. sampling: convenience, bias, voluntary d sampling: simple random sample(SRS). Inferential Stats: making predictions or inferences about a population based on a sample

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Introduction. Population – the entire group of concern Sample – only a part of the whole Based on sample, we’ll make a prediction about the population. Bad sampling: convenience, bias, voluntary Good sampling: simple random sample(SRS). Inferential Stats: making predictions or - PowerPoint PPT Presentation

Transcript of Introduction

Page 1: Introduction

IntroductionPopulation – the entire group of concernSample – only a part of the wholeBased on sample, we’ll make a prediction about

the population.

Bad sampling: convenience, bias, voluntary

Good sampling: simple random sample(SRS).

Inferential Stats: making predictions orinferences about a population based on a sample

Page 2: Introduction

ExperimentsObservation – no attempt to influenceExperiment– deliberately imposes some treatment

Basic design principles: Control the effects of lurking variables Randomize which subject gets which treatment Use large sample size to reduce chance variation

Statistical Significance: An observed effect so big that it would rarely occur just by chance.

Page 3: Introduction

Picturing Distributions with Graphs

• Individuals – objects described by data– can be

• Variables– characteristic of individuals of particular interest– different values possible for different people

What makes up any set of data?

Page 4: Introduction

Two kinds of variables

Categorical (Qualitative)– describes an individual by category or quality.– examples like

Numerical (Quantitative)– describes an individual by number or quantity.– discrete for variables that are– continuous for variables that are– examples like

Page 5: Introduction

Describing Categorical Variables

Tables summarize the data set by– listing possible categories. – giving the number of objects in each category.– or show the count as a percentage.

Picture the distribution of a cat. var. with– Pie charts–Bar graphs

Page 6: Introduction

Pie Chartswhole is split into appropriate pieces.

Page 7: Introduction

Bar GraphHorizontal line keeps track of categorical values.

Vertical bars at each value keeps track of # or %.

A B C D E F

4

12

20#

5

15

25%

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Example 180 AASU students in an Elem. Stats class come from

one of four colleges (S & T, Edu, Health, Lib. Arts). The breakdown of these 80 students is given below.CollegeLiberal ArtsEducationHealth ProfessionsScience & TechnologyUndeclared

Count

17

432

23

480

Percent

Page 9: Introduction

Ex1 - Pie ChartCollegeLib ArtsEduHealth

S & T

Undeclared

Count17

4

32

23

4

80

Percent21.25%

5%

40%

28.75%

5%

100%

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Ex1 – Bar Graph

LA E H ST

10

20

30

%CollegeLib Arts

Edu

Health

S& T

Undeclared

Count17

4

32

23

4

80

Percent21.25%

5%

40%

28.75%

5%

100%

U

Page 11: Introduction

Describing Quantitative Variables

Tables summarize the data set by– listing possible intervals (ranges, classes).– giving the number of individuals in each class– or showing the number as a percentage.

Picture the distribution of a quant. var. with–Histogram (similar to bar graph but now vertical bars of neighboring classes touch)Where one class ends, the next begins.

Page 12: Introduction

Example 2Consider the ages of the full-time faculty in the math dept. The breakdown of these 19 individuals is given in the table.

Age Class

20-3030-4040-5050-60

60-70

Count

5354

219

Percent

26.3%

15.8%

26.3%21.1%

10.5%

100%

10 30 50 70

10

20

30%

Page 13: Introduction

Info from histograms

Helps to describe a distribution with– pattern (shape, center, spread)

– deviations (outliers) from the rest of the data• Could result from unusual observation or typo

– For shape, look at symmetric vs. skewed

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Examples 3 and 4

2 4 6 8

%

10

20 40 60 80

%

100

12

v

Page 15: Introduction

Example 4 without outliers%

v

20 40 60 80 100

%

v

20 40 60 80 100v

105

105

30

20

v

Page 16: Introduction

Describing Distributions with Numbers

Center: mean, median, mode

Spread: quartiles, standard deviation

There are better ways to describe a quantitative data set than by an estimation from a graph.

Page 17: Introduction

Center: Mean

The mean of a data set is the arithmetic average ofall the observations.

nxxx ,,, 21 Given a data set:

Page 18: Introduction

Mean – Example 1

Your test scores in a Stats Class are: 60, 75, 92, 80

Your mean score is:

Page 19: Introduction

Mean – Example 2Compare high temperatures in Savannah for July

2010 and July 2011.

July 2010 high temps: 83, 87, 84, …, 97, 100, 92

31

9287832010

x

July 2011 high temps: 94, 91, 93, …, 97, 99, 99

31

9991942011

x

Page 20: Introduction

Center: Median

The median of a data set is the middle value ofall the (ordered) observations.

nxxx ,,, 21 Given a data set:

Page 21: Introduction

Median – Examples 3/4

11 tests: 60, 77, 92, 80, 84, 93, 80, 95, 65, 66, 75Ordered data set: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95

10 dice rolls: 2, 4, 5, 5, 6, 7, 7, 8, 9, 10

Page 22: Introduction

Center: ModeThe mode of a data set is the value that appears the most.

Tests data set: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95

Dice rolls: 2, 4, 5, 5, 6, 7, 7, 8, 9, 10

2010 July High Temps mode:2011 July High Temps mode:

Page 23: Introduction

Spread: Quartiles

A measure of center is not useful by itself–Are other observations close or far from center?

Take an ordered data set and find:–M, – Q1, – Q3, – IQR =

Summary of data in the “Five-Number Summary”:

Page 24: Introduction

Quartiles – Example 511 tests: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95

5-num-sum:

Visualize 5-num-sum with a boxplot.• Draw rectangle with ends at Q1 and Q3.• Draw line in the box for the median.• Draw lines to the last observations within 1.5IQR of

the quartiles.• Observations outside 1.5IQR of the quartiles are

suspected outliers.

Page 25: Introduction

Boxplot – Example 65-Num-Sum: 60, ____, 80, ____, 95

• Draw rectangle with ends at Q1 and Q3• Draw line in the box for the median• Draw lines to last observations within 1.5IQR of the quartiles• Observations outside 1.5IQR of the quartiles are suspected outliers

50 60 70 80 90 100

Page 26: Introduction

Boxplot – Example 7July 2010 5-Num-Sum: 83, 92, 94, 97, 102

80 85 90 95 100 105

2010IQR = 97-92=5

July 2011 5-Num-Sum: 84, 91, 95, 98, 99

2010

20112011IQR = 98-91=7

Page 27: Introduction

Spread: Standard DeviationMore common measure of spread (in conjunction

with the mean) is the standard deviation.

A single deviation from the mean looks like

For every value in a data set, deviations are either positive, negative or zero.

Finding an average of those will be trouble, since when you add the deviations together, you’ll get 0.

Example 1 data: 60, 75, 92, 80 75.76x

Page 28: Introduction

To deal with this “adding to zero”, we get rid of any negative terms by squaring each deviation.

A single squared deviation from the mean looks like:

The average of the squared deviations is called the variance:

n-1 is called the degrees of freedom, since knowledge of the first (n-1) deviations will automatically set the last one.

Page 29: Introduction

The standard deviation is the square root of the variance.

2s

1

2

n

xxs i

Observations Deviations Squared Dev

60

75

92

80

mean=76.75

s

Page 30: Introduction

When to use what?For skewed data:

For (nearly) symmetric data:

Outliers have a big impact on mean and std. dev.

Consider two data sets:Set 1: 1, 1, 3, 5, 10

Set 2: 1, 1, 3, 5, 70


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