1 1-1 overview 1- 2 types of data 1- 3 random sampling 1- 4 design of experiments 1- 5 abuses of...

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1-1 Overview

1- 2 Types of Data

1- 3 Random Sampling

1- 4 Design of Experiments

1- 5 Abuses of Statistics

Chapter 1 Introduction to Statistics

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Statistics (Definition)A collection of methods for planning experiments, obtaining data, and then organizing, summarizing,

presenting, analyzing, interpreting, and drawing conclusions based on the data

1-1 Overview

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DefinitionsPopulation The complete collection of all

data to be studied.

Census The collection of data from every member of the population.

SampleThe collection of data from a subset

of the population.

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ExampleIdentify the population and

sample in the studyA quality-control manager randomly selects 50 bottles of Coca-Cola to assess the calibration of the filing machine.

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Statistics

Broken into 2 areas

Descriptive Statistics Inferencial Statistics

Definitions

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DefinitionsDescriptive Statistics

Describes data usually through the use of graphs, charts and pictures. Simple calculations like mean, range, mode, etc., may also be used.

Inferencial Statistics Uses sample data to make inferences (draw

conclusions) about an entire population

Test QuestionTest Question

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Parameter vs. Statistic

Variables

Quantitative Data vs. Qualitative Data

Nominal Data vs. Ordinal Data

Discrete Data vs. Continuous Data

Univariate Data vs. Bivariate Data

1-2 Types of Data

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Parameter

a numerical measurement describing some characteristic of a population

population

parameter

Definitions

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Statistic

a numerical measurement describing some characteristic of a sample

sample

statistic

Definitions

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Examples

Parameter

51% of the entire population of the US is Female

Statistic

Based on a sample from the US population is was determined that 35% consider themselves overweight.

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Variable

» characteristics of the individuals (data) being measured or observed

» represented as a symbol (x, Y, s, σ, µ, etc.).

Definitions

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We further describe variables by

distinguishing between Qualitative and

Quantitative data (variables)

Definitions

VariablesVariables

QualitativeQualitative

QuantitativeQuantitative

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Definitions

Quantitative data

Numbers representing counts or measurements

Qualitative (or categorical or

attribute) dataCan be separated into different categories that are distinguished by some nonnumeric characteristics

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Examples

Quantitative data

x = The number of FLC students with blue eyes

Qualitative (or categorical or

attribute) datay = The eye color of FLC students

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ExampleOf the adult U.S. population, 36% has an allergy. A sample of 1200 randomly selected adults resulted in 33.2% reporting an allergy.

1. Describe the variable and give its type

2. Describe the population

3. Describe the sample

4. What is the value of the parameter?

5. What is the value of the statistic?

6. Which statement is descriptive in nature?

7. Which statement is inferential in nature?

Quiz Question

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We further describe qualitative data by

distinguishing between Nominal and

Ordinal data

Definitions

Qualitative Qualitative DataData

NominalNominal

OrdinalOrdinal

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Nominal Nominal data are categorical data where the order

of the categories is arbitrary Example: race/ethnicity has values 1=White, 2=Hispanic,

3=American Indian, 4=Black, 5=Other. Note that the order of the categories is arbitrary.

Ordinal Ordinal data are categorical data where there is a

logical ordering to the categoriesExample: scale that you see on many surveys: 1=Strongly

disagree; 2=Disagree; 3=Neutral; 4=Agree; 5=Strongly agree.

Definitions

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We further describe quantitative data by

distinguishing between discrete and

continuous data

Definitions

Quantitative Quantitative DataData

DiscreteDiscrete

ContinuousContinuous

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Discrete data result when the number of possible values is

either a finite number or a ‘countable’ number of possible values

0, 1, 2, 3, . . .

Continuous (numerical) data result from infinitely many

possible values that correspond to some continuous scale or interval that covers a range of values without gaps, interruptions, or jumps

Definitions

2 3

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Discrete

The number of eggs that hens lay; for

example, 3 eggs a day.

Continuous

The amounts of milk that cows produce;

for example, 2.343115 gallons a day.

Examples

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Univariate Data » Involves the use of one variable (X)

» Does not deal with causes and relationship

Bivariate Data» Involves the use of two variables (X and Y)

» Deals with causes and relationships

Definitions

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Univariate Data How many first year students attend FLC?

Bivariate DataIs there a relationship (association) between then

number of females in Computer Programming and their scores in Mathematics?

Example

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1. Center: A representative or average value that indicates where the middle of the data set is located

2. Variation: A measure of the amount that the values vary among themselves or how data is dispersed

3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed)

4. Outliers: Sample values that lie very far away from the vast majority of other sample values

5. Time: Changing characteristics of the data over time

Important Characteristics of Data

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Uses of Statistics

Almost all fields of study benefit from the application of statistical methodsSociology, Genetics, Insurance, Biology, Polling,

Retirement Planning, automobile fatality rates, and many more too numerous to mention.

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1- 3

Design of Experiments

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Designing an Experiment

Identify your objective

Collect sample data

Use a random procedure that avoids bias

Analyze the data and form conclusions

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Observational Study measures the characteristics of a population by studying individuals in a sample, but does not attempt to manipulate or influence variables of interest

Experimentapplies treatments to experimental units or subjects and attempts to isolate the effects of the treatments on a response variable

Definition

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Observational Study A poll is conducted in which 500 people are

asked whom they plan to vote for in the upcoming election

ExperimentTo determine the effect of type of fertilizers a

farmer might divide 20 tomato plants into two groups. Group 1 received fertilizer 1 and Group 2 receives fertilizer 2. All other factors for the two groups are kept the same (sunlight, water, etc).

Examples

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Define the treatment, experimental unit and response variable in the following experiment.To determine the effect of type of fertilizers a

farmer might divide 20 tomato plants into two groups. Group 1 received fertilizer 1 and Group 2 receives fertilizer 2.

All other factors for the two groups are kept the same (sunlight, water, etc). i.e., Confounding does not occur

Experimental Design

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Lurking variables: A variable that was not considered in a study but

may affect study.

Confounding: Occurs in a study when lurking variables affect the outcome.

Confounding

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Example: Flu shots are associated with a lower risk of being

hospitalized or dying from influenza.

Possible Lurking Variables: • age

• health status

• mobility of the senior

Confounding

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Experiment apply some treatment (Action)

Event (Response)observe its effects on the subject(s) (Observe)

Example: Experiment: Toss a coin

Event: Observe a tail

Probability Experiment

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1- 4

Sampling

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Random (type discussed in this class)

Systematic

Convenience

Stratified

Cluster

Methods of Sampling

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TI-83 CalculatorUsing a random number generator

1. Press Math

2. Cursor over to PRB

3. Press “5” RandInt

4. Enter (low value, high value, sample size)

Example:

RandInt(1,30,5) will select 5 random numbers between 1 and 30

Note: if you get duplicate numbers you should draw more numbers than you need and ignore the duplicates.

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Simple Random Sample members of the population are selected in such a way that each has an equal chance of being selected (if not then sample is biased)

Definitions

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Random Sampling - selection so that

each has an equal chance of being selected

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Systematic SamplingSelect some starting point and then

select every K th element in the population

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Convenience Samplinguse results that are easy to get

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Stratified Samplingsubdivide the population into at

least two different subgroups that share the same characteristics, then draw a sample from each

subgroup (or stratum)

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Cluster Sampling - divide the population into sections (or clusters); randomly select some of those clusters; choose all members from selected clusters

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Sampling Error the difference between a sample result and the true

population result; such an error results from chance sample fluctuations.

Nonsampling Error sample data that are incorrectly collected, recorded, or analyzed (such as by selecting a biased sample, using a defective instrument, or copying the data incorrectly).

Errors in Sampling

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Sampling Error (Example) A recent poll showed potential voters favored the

proposition 52% to 48%. The margin of error for the poll was 3%.

Nonsampling Error (Example)During presidential election or 2000, early results from an Florida exit poll were skewed by a programming error.

Errors in Sampling

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Bad Samples Small Samples Loaded Questions Misleading Graphs Pictographs Precise Numbers Distorted Percentages Partial Pictures Deliberate Distortions

1-5 Abuses of Statistics

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Abuses of StatisticsBad Samples

Inappropriate methods to collect data. BIAS (on test) Example: using phone books to sample data.

Small Samples (will have example on exam)

We will talk about same size later in the course. Even large samples can be bad samples.

Loaded Questions

Survey questions can be worked to elicit a desired response

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Abuses of Statistics

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Bachelor High SchoolDegree Diploma

Salaries of People with Bachelor’s Degrees and with High School Diplomas

$40,000

30,000

25,000

20,000

$40,500

$24,400

35,000

$40,000

20,000

10,000

0

$40,500

$24,40030,000

Bachelor High SchoolDegree Diploma

(a) (b)(test question)

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We should analyze the numerical information given in the graph instead of being mislead by its general shape.

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Double the length, width, and height of a cube, and the volume increases by a factor of eight

What is actually What is actually intended here? 2 intended here? 2 times or 8 times?times or 8 times?

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Precise NumbersThere are 103,215,027 households in the US. This is actually an estimate and it would be best to say there are about 103 million households.

Distorted Percentages

100% improvement doesn’t mean perfect.

Deliberate Distortions

Lies, Lies, all Lies


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Abuses of StatisticsPartial Pictures“Ninety percent of all our cars sold in this country in the last 10 years are still on the road.”

Problem: What if the 90% were sold in the last 3 years?

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Factorial Notation 8! = 8x7x6x5x4x3x2x1

Order of Operations 1. ( )2. POWERS3. MULT. & DIV.4. ADD & SUBT.5. READ LIKE A BOOK

Keep number in calculator as long a possible

Using Formulas

1 1-1 overview 1- 2 types of data 1- 3 random sampling 1- 4 design of experiments 1- 5 abuses of...

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