Week 01 Introduction to Statistics Probability & Statistics 1

What is Statistic? 2

Statistics Statistics is the science of data which involves – collecting, – classifying, – summarizing, – organizing, – analyzing, – and interpreting numerical information 3 1-2

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Data collection methods Questionnaires. Interviews Observation 5 1-2

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. 6

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. 7

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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. 10

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 11

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. 12

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relationship between samples and populations. 14

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 population Eg: 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. 15

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 have climbed 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. 16

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 17

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 18

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. 19

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. 20

Types of Statistics Descriptive 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. 21

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 22

Types of Statistics Inferential 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 23

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. 24

Levels of Measurement There are four levels of measurement: nominal, ordinal, interval and ratio

Nominal level 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. 27

Ordinal level 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 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.

Interval level For example, When we measure temperature (in Fahrenheit), the distance from is same as distance from 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 30

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Ratio level 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.

33 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

34 SAMPLING BREAKDOWN

SAMPLING ……. 35 TARGET POPULATION STUDY POPULATION SAMPLE

Types of Samples 36 Probability (Random) Samples – Simple random sample – Systematic random sample – Stratified random sample – Cluster sample

Basic Methods of Sampling  Random Sampling  Selected 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 37

 The “pick a name out of the hat” technique  Random number table  Random number generator Random Sampling 38 Hawkes and Marsh (2004)

Simple Random Sample Every subset of a specified size n from the population has an equal chance of being selected Math Alliance Project 39

Simple random sampling 40

Systematic Sampling This is a form of random sampling, involving a system. Every n th item is selected throughout the list. – Not fully random and therefore there is a possibility of bias. 41

Basic Methods of Sampling  Systematic Sampling  Select a random starting point and then select every k th subject in the population  Simple to use so it is used often 42

 All data is sequentially numbered  Every nth piece of data is chosen Systematic Sampling 43 Hawkes and Marsh (2004)

Systematic Sample Every kth member ( for example: every 10th person) is selected from a list of all population members. Math Alliance Project 44

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. 45

Basic Methods of Sampling Stratified Sampling Divide 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 46

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. Math Alliance Project 47

Stratified Sampling  Data is divided into subgroups (strata)  Strata are based specific characteristic  Age  Education level  Etc.  Use random sampling within each strata 48 Hawkes and Marsh (2004)

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. 49

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 50

 Data is divided into clusters  Usually geographic  Random sampling used to choose clusters  All data used from selected clusters Cluster Sampling 51 Hawkes and Marsh (2004)

Cluster sampling Section 4 Section 5 Section 3 Section 2Section 1 52

Sampling Relationships 53 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. Math Alliance Project 54

In a class of 18 students, 6 are chosen for an assignment Example 1: Sampling Methods 55 Sampling Type Example RandomPull 6 names out of a hat SystematicSelecting every 3 rd student StratifiedDivide the class into 2 equal age groups. Randomly choose 3 from each group ClusterDivide the class into 6 groups of 3 students each. Randomly choose 2 groups ConvenienceTake the 6 students closest to the teacher

 Determine average student age  Sample of 10 students  Ages of 50 statistics students Example 2: Utilizing Sampling Methods

Example 2 – Random Sampling  Random number generator 57 Data Point Location Correspondin g Data Value Mean 25.1

Example 2 – Systematic Sampling  Take every data point 58 Data Point Location Correspondin g Data Value Mean 20.8