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Week 01Introduction to Statistics

Probability & Statistics

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What is Statistic?

StatisticsStatistics is the science of data which involves

– collecting, – classifying, – summarizing, – organizing, – analyzing, – and interpreting numerical information

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Data collection methods

•Questionnaires.•Interviews•Observation

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Why study Statistic?

You’ll be able to • make objective decisions, • make accurate predictions that seem inspired• convey the message you want in the most

effective way possible.• Statistics can be a convenient way of

summarizing key truths about data• need a way of visualizing data for everyone else.

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Why not just go on the data? Why chart it?

• Sometimes it’s difficult to see what’s really going on just by looking at the raw data.

• There can be patterns and trends in the data, but these can be very hard to spot if you’re just looking at a heap of numbers.

• Charts give you a way of literally seeing patterns in your data.

• They allow you to visualize your data and see what’s really going on in a quick glance.

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What’s the difference between information and data?

• Data refers to raw facts and figures that have been collected.

• Information is data that has some sort of added meaning.

Definitions

Populations and Parameters• A population is the entire collection of all

observations of interest.• E.g. All 2.5 million registered voters in Sri Lanka

• A parameter is a descriptive measure of the entire population of all observations of interest

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Definitions

Samples and Statistics• A sample is a representative portion of the population

which is selected for study.• Potentially very large, but less than the population.• E.g. a sample of 765 voters exit polled on election day.

• A statistic describes a sample and serves as an estimate of the corresponding population parameter.

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relationship between samples and populations.

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Parameters are numbers that summarize data for an entire population. Statistics are numbers that summarize data from a sample, i.e. some subset of the entire populationEg:A nutritionist wants to estimate the mean amount of sodium consumed by children under the age of 10. From a random sample of 75 children under the age of 10, the nutritionist obtains a sample mean of 2993 milligrams of sodium consumed.

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Individuals and Variables• Individuals are the people or objects included in the study. • A variable is the characteristic of the individual to be

measured or observed.

For example, if we want to do a study about the people who haveclimbed Mt. Everest, then the individuals in the study are the actual people who made it to the top. The variables to measure or observe might be the height, weight, race, gender, income, etc of the individuals that made it to the top of Mt. Everest.

Definitions

Variables• A variable is a the characteristic of the

population that is being examined in the statistical study.

• There are two basic types of data: Qualitative & Quantitative

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Types of Variables

• Qualitative or Attribute variable(Categorical): the characteristic or variable being studied is nonnumeric.

• EXAMPLES: Gender, religious affiliation, type of automobile owned, state of birth, eye color, type of dessert

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Types of Variables

• Quantitative variable: the variable can be reported numerically.

• EXAMPLE: balance in your savings account, minutes remaining in class, number of children in a family.

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Types of Variables• Quantitative variables can be classified as either

discrete or continuous.– Discrete variables: can only assume certain values

and there are usually “gaps” between values. EXAMPLE: the number of bedrooms in a house.

(1,2,3,..., etc...).

– Continuous variables: can assume any value within a specific range.

EXAMPLE: The time it takes to fly from Sri Lanka to New York.

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Types of StatisticsDescriptive Statistics:

• Methods of organizing, summarizing, and presenting data in an informative way.

• Descriptive statistics do not allow us to make conclusions beyond the data we have analyzed or reach conclusions regarding any hypotheses we might have made.

• Frequency distributions, measures of central tendency (mean, median, and mode), and graphs like pie charts and bar charts that describe the data are all examples of descriptive statistics.

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EXAMPLE for Descriptive Statistics:

• if we look at a basketball team's game scores over a year, we can calculate the average score, variance etc. and get a description (a statistical profile) for that team

• According to Consumer Report of Ceylon Pencil Company, 9 defective pens per 100. The statistic 9 describes the number of problems out of every 100 pens

Types of StatisticsInferential Statistics:

• Inferential statistics is concerned with making predictions or inferences about a population from observations and analyses of a sample

• The methods of inferential statistics are (1) the estimation of parameter(s) and (2) testing of statistical hypotheses. A Chi-square or T-test

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Inferential Statistics:

EXAMPLE: • TV networks constantly monitor the popularity of their

programs by hiring people to sample the preferences of TV viewers.

• To infer the success rate of a drug in treating high temperature, by taking a sample of patients, giving them the drug, and estimating the rate of effectiveness in the population using the rate of effectiveness in the sample.

Levels of Measurement

• There are four levels of measurement: nominal, ordinal, interval and ratio.

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Nominal level

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Nominal level (scaled): Data that can only be classified into categories and cannot be arranged in an ordering scheme.EXAMPLES: eye color, gender, religious affiliation• Religion (Catholic, Buddhist, etc) • Race ( African-American, Asian, etc) • Marital Status (Married, Single, Divorced)

These categories are mutually exclusive and/or exhaustive.

Nominal level• Mutually exclusive: An individual or item that,

by virtue of being included in one category, must be excluded from any other category.

• Two events are mutually exclusive if they cannot occur at the same time.

• An example is tossing a coin once, which can result in either heads or tails, but not both.

EXAMPLE: eye color.

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Ordinal level

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Ordinal level: involves data that may be arranged in some order, but differences between data values cannot be determined or are meaningless.

EXAMPLE: During a taste test of 4 colas, cola C was ranked number 1, cola B was ranked number 2, cola A was ranked number 3,cola D was ranked number 4.

Rankings (1st, 2nd, 3rd, etc) Grades (A, B, C, D. F) Evaluations Hi, Medium, Low

Interval level

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Interval data have meaningful intervals between measurements, but there is no true starting point (zero).

Variables or measurements where the difference between values is measured by a fixed scale.

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Interval levelFor example, • When we measure temperature (in Fahrenheit), the

distance from 30-40 is same as distance from 70-80. The interval between values is interpretable. Because of this, it makes sense to compute an average of an interval variable, where it doesn't make sense to do so for ordinal scales. But note that in interval measurement ratios don't make any sense - 80 degrees is not twice as hot as 40 degrees

• However 0 degrees (in both scales) cold as it may be does not represent the total absence of temperature

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Ratio level

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Ratio level: the interval level with an inherent zero starting point. Differences and ratios are meaningful for this level of measurement.

EXAMPLES: money, heights of students. A measurement such as 0 feet does make sense, as

it represents no length. Furthermore 2 feet is twice as long as 1 foot. So

ratios can be formed between the data.

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Level of data

Nominal

Data may only

be classified

Classification of

students by district

Ordinal

Data are ranked

Your rank for

this course module

Interval

Meaningful difference

between values

Temperature

Ratio

Meaningful 0 point &

ratio between values

Number of study hours

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SAMPLING…….

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TARGET POPULATION

STUDY POPULATION

SAMPLE

Types of Samples

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• Probability (Random) Samples– Simple random sample– Systematic random sample– Stratified random sample– Cluster sample

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Basic Methods of Sampling

Random SamplingSelected by using

chance or random numbers

Each individual subject (human or otherwise) has an equal chance of being selected

Examples: Drawing names from a

hat

Random Numbers

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TOC

The “pick a name out of the hat” technique Random number table Random number generator

Random Sampling

Hawkes and Marsh (2004)

Simple Random Sample•Every subset of a specified size n from the population has an equal chance of being selected

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Simple random sampling

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Systematic Sampling

• This is a form of random sampling, involving a system. Every nth item is selected throughout the list.– Not fully random and therefore there is a

possibility of bias.

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Basic Methods of SamplingSystematic Sampling

Select a random starting point and then select every kth subject in the population

Simple to use so it is used often

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All data is sequentially numbered Every nth piece of data is chosen

Systematic Sampling

Hawkes and Marsh (2004)

Systematic Sample

• Every kth member ( for example: every 10th person) is selected from a list of all population members.

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Stratified random sample• In this method all the people or items in the

sampling frame are divided into ‘categories’ which are mutually exclusive. Within each level a simple random sample is selected.– Within the categories the samples are random.– But the categories are not clear.

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Basic Methods of Sampling

Stratified SamplingDivide the population into at least two

different groups with common characteristic(s), then draw SOME subjects from each group (group is called strata or stratum)

Results in a more representative sample

Stratified Random Sample

• The population is divided into two or more groups called strata, according to some criterion, such as geographic location, grade level, age, or income, and subsamples are randomly selected from each strata.

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Stratified SamplingData is divided into

subgroups (strata)Strata are based

specific characteristic Age Education level Etc.

Use random sampling within each strata

Hawkes and Marsh (2004)

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Cluster sampling• Clusters are formed by breaking down the

area to be surveyed into smaller areas a number of which are selected by random methods for survey. Within the selected clusters are chosen by random methods for the survey.

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Basic Methods of Sampling

Cluster Sampling

Divide the population into groups (called clusters), randomly select some of the groups, and then collect data from ALL members of the selected groups

Used extensively by government and private research organizations

Examples:

Exit Polls

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Data is divided into clusters Usually geographic

Random sampling used to choose clusters All data used from selected clusters

Cluster Sampling

Hawkes and Marsh (2004)

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Cluster sampling

Section 4

Section 5

Section 3

Section 2Section 1

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TOC

Sampling Relationships

Random Sampling

Cluster Sampling

Stratified Sampling

Cluster Sample

• The population is divided into subgroups (clusters) like families. A simple random sample is taken of the subgroups and then all members of the cluster selected are surveyed.

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In a class of 18 students, 6 are chosen for an assignment

Example 1: Sampling Methods

Sampling Type

Example

Random Pull 6 names out of a hat

Systematic Selecting every 3rd student

Stratified Divide the class into 2 equal age groups. Randomly choose 3 from each group

Cluster Divide the class into 6 groups of 3 students each. Randomly choose 2 groups

Convenience Take the 6 students closest to the teacher

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Determine average student age Sample of 10 students Ages of 50 statistics students

Example 2: Utilizing Sampling Methods

18 21 42 32 17 18 18 18 19 22

25 24 23 25 18 18 19 19 20 21

19 29 22 17 21 20 20 24 36 18

17 19 19 23 25 21 19 21 24 27

21 22 19 18 25 23 24 17 19 20

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Example 2 – Random Sampling

Random number generator

Data Point Location

Corresponding Data Value

35 25

48 17

37 19

14 25

47 24

4 32

33 19

35 25

34 23

3 42

Mean 25.1

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Example 2 – Systematic Sampling

Take every

data point

Data Point Location

Corresponding Data Value

5 17

10 22

15 18

20 21

25 21

30 18

35 21

40 27

45 23

50 20

Mean 20.8

5th