1 RES 341 RESEARCH AND EVALUATION WORKSHOP 2 By Dr. Serhat Eren University OF PHOENIX.

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RES 341RES 341RESEARCH AND RESEARCH AND

EVALUATIONEVALUATION

WORKSHOP 2

By Dr. Serhat Eren

University OF PHOENIX

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CHAPTER IICHAPTER II

THE LANGUAGE OF RESEARCH AND STATISTICS

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2.1 CHAPTER OBJECTIVES2.1 CHAPTER OBJECTIVES

The Difference between the Population and a Sample of the Population

The Difference between a Parameter and a Statistic

Factors That Influence Sample Size: Some Sampling and Sample Size Considerations

Selecting the Sample Types of Data The Difference between Descriptive

Statistics and Inferential StatisticsBasic Summation Notation

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2.2 THE DIFFERENCE BETWEEN THE POPULATION 2.2 THE DIFFERENCE BETWEEN THE POPULATION AND A SAMPLE OF THE POPULATIONAND A SAMPLE OF THE POPULATION

2.2.1 The Population of Interest There is always a group of people or things

that needs to be studied and understood to make the necessary decision.

For the tissue manufacturer it was all the boxes of tissues that the company makes, for the company considering dress down day it was all the employees of the company.

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These are examples of what is called the population in the statistical thinking paradigm.

Thus, all the boxes of tissues is the population of interest to the tissue manufacturer, all the employees is the population of interest to the company considering dress down day, and all the golf balls is the population of interest to the golf ball manufacturer.


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In each case, the decision maker must learn something about how the population of interest behaves in order to make a recommendation.

The population is everything you wish to study.

Often, when we study a population, we are really interested in knowing about different characteristics of each member of the population. These characteristics are known as variables.


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For each member of the population we may be interested in knowing about one, two, or even more different variables.

A variable is characteristic of the population.


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2.2.2 The Sample A sample is a piece of the population.

If we think about the population as the big oval (all the pies) shown in Figure 2.1, then a sample from this population might be the small oval shown in the figure.


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2.2.2 The Sample This is clearly not the only sample that

could be taken from this population; many different samples may be picked.

Some of the samples may overlap and some may be bigger or smaller than the one shown.


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2.2.3 Why Pick a Sample at All? A natural question to ask is, why should we

bother to examine a sample when what we really want to know about is the population?

Most of the time, we cannot study the entire population and must use a sample as a guide.


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2.2.3 Why Pick a Sample at All? The main reasons are fairly clear when you think

about it for a minute: – It would take too much time to study the entire

population. – It would take too much money to study the entire

population. – It might not be possible to identify all the members

of the population. – If we test the entire population, we might not have

anything left to sell.


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2.2.4 The Difference between the Population and the Sample: An Introduction to Sampling Error By studying the behavior of a sample we

can get a good idea of the behavior of the population. It will not be a perfect picture of the population, but it will be good enough to guide us in our decision making. It is just like eating a piece of the apple pie.


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Imagine that you took a picture of the population but let it develop for only a few seconds. You would be able to get an overall sense of what the population looked like but you would not have all of the details.

This is like taking a sample and using the sample to determine how the population behaves.


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Finally, if you let the picture develop completely, then you would have complete information. This is equivalent to studying the entire population. If you study the entire population, then you have taken what is called a census.

A census is a study of the entire population.


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Since the sample is an imperfect snapshot of the population, you know that there will be differences between the sample and the population.

Unless we study the entire population, we cannot eliminate what is known as sampling error.


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Sampling error is the difference between the behavior of the entire population and a sample of that population.

The size of the sampling error is determined by two factors.

– The first of these factors is the size of the sample. Clearly, the larger the sample you take, the more similar the sample will be to the population, thus decreasing the sampling error.


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– The second factor that influences the size of the sampling error is the amount of variation that exists in the population.

The amount of variation refers to how different the members of the population are from each other with regard to the variable being studied.


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For example, suppose your population of interest is all students taking classes at your university or college. The variable of interest is the age of these students. If all students were exactly the same age then you would say that there was no variation in the age of the members of the population.

In this extreme case you would need to take a sample of only one student to have perfect information about the age of members of the population.


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You will almost never be studying a characteristic that has no variability.

Suppose that the ages of the students have a small amount of variability. Let's say the student ages range from 18 to 22 years old. Clearly, you have to take a sample of more than one student to understand how the ages vary in this population.


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Now suppose the ages of the students range from 17 to 60 years.

You have to take a much larger sample to understand how the ages vary in this population.

As the amount of variability in the population increases, the sampling error also increases.


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2.3 THE DIFFERENCE BETWEEN A PARAMETER AND 2.3 THE DIFFERENCE BETWEEN A PARAMETER AND A STATISTICSA STATISTICS

2.3.1 Parameters: Numerical Descriptors of the Population

There are many different ways to describe the behavior of a population.

One of them is to use numerical values to paint a picture of the population.

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For instance, if you are told that all the values in the population fall between 0 and 10, you form a mental image of the population that is quite different from the picture that is conjured up if you learn that all the values in the population fall between 0 and 1000.

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The traditional numerical measures that are used to describe the population. They are all examples of what are known as parameters.

A parameter is a number that describes a characteristic of the population.


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2.3.2 Statistics: Numerical Descriptors of the SampleWe need numbers to describe the

behavior of the sample for two reasons:

– To paint a picture of the sample and,

– To help us estimate the corresponding population parameter.


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A statistic is nothing more than a number that describes the behavior of the sample.

A statistic is a number that describes the behavior of a sample


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2.4 FACTORS THAT INFLUENCE SAMPLE SIZE: SOME 2.4 FACTORS THAT INFLUENCE SAMPLE SIZE: SOME SAMPLING AND SAMPLE SIZE CONSIDERATIONSSAMPLING AND SAMPLE SIZE CONSIDERATIONS

The next question is how big does the sample need to be? Do we need to eat half of one pie to know how all the pies in a batch taste? What factors will influence our decision in this matter?

2.4.1 Size of the PopulationThe first factor is the size of the population.

How many tubes of glue are manufactured each day? How many employees are there in the company? How many golf balls are manufactured each hour?

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Intuitively it should seem that it would take a different sample size to learn about a population of 1000 compared to a population of 100,000.

The population size is a factor, but it turns out not to be as important as some of the other factors we have identified.

The size of the population is number of members of the population that will be referred to as N.


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2.4.2 Extent of Resources AvailableThe next factor is the amount of time,

money and other resources that you have available.

If you need a decision in six months rather than next week, the amount of data you could possibly collect and analyze is different.


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2.4.3 Amount of Error That Can Be ToleratedWhenever you use a sample to draw

conclusions about the population you will have some sampling error. And the bigger the sample we pick, the less error we will have in our conclusions.

Returning to the ovals in Figure 2.1, you can see that if you take a bigger sample, then you have captured more of the population.


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Therefore, the conclusions you draw about the population based on the sample will be more accurate.

This is shown by the increasing sample sizes in Figure 2.2.


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If we continue taking increasingly bigger sample sizes, eventually we will end up taking a census, or studying the entire population. This is represented by the largest oval.

When this happens, the sample size is as big as the population size, N, and there is no sampling error.


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2.4.4 Amount of Variation in the PopulationEven if we can tolerate only a small

amount of error we still may not need a really large sample size. One other factor is very important: the amount of variability that exists in the population.

Suppose for a moment that I wish to study all the students in your college.


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I wish to be very accurate in my conclusions but all of the students feel exactly the same about the issue I am studying.

How many students do I need to talk to in order to get very precise information about the population?


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The correct answer is that only 1 student is needed. A sample size of 1 is adequate despite the need to be very accurate and despite the large size of the population.

This is because there is no variation in the population; that is, everyone feels the same way about the issue.

A natural question to ask is, how will we know the amount of variability of the population to use in determining the sample size?


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We could estimate the amount of variability using information about the amount of variability in a population similar to the one we are studying

We could also use previous studies of the same population to give us an idea of the amount of variability.

Or we could do a pilot study.


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2.4.5 Summary of Factors Influencing Sample SizeWe have identified the following factors that

are important in determining the size of the sample needed:

– The amount of variation in the population– The amount of error that can be tolerated– The amount of resources available for the

project– The size of the population, N


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No matter what statistics book you pick up you will always see the sample size referred to as n.

Remember that the size of the population is labeled N.


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2.5 SELECTING THE SAMPLE2.5 SELECTING THE SAMPLE

2.5.1 Selecting an Unbiased SampleIdeally, we would like the sample to be a

mini version of the population.

Remember that we will be using the sample to understand the population.

The sample should thus contain all the key features found in the population.

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2.5.2 Selecting a Simple Random SampleThere are many ways to pick a sample but

for the purposes of an introductory course we are going to use what is known as a simple random sample (srs).

This means that each member of the population has an equal chance of being selected as a member of the sample.


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A simple random sample is a sample that has been selected in such a way that all members of the population have an equal chance of being picked.

Simple random sampling is the most obvious way to select a sample.


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2.5.3 The Sampling FrameA sampling frame is a list of all members of

the population.

2.5.4 Using a Table of Random Numbers to Select the SampleA table of random numbers is a table that

consists of a list of numbers randomly generated and listed in the order in which they were generated.

Such a table is shown in Figure 2.3.


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2.6 SOURCES AND TYPES OF DATA2.6 SOURCES AND TYPES OF DATA

There are two major sources of data, primary sources and secondary sources.

Data that come from a primary source are called primary data and data that come from a secondary source ate called secondary data.

Primary data are data that are obtained and used by the organization or individual that actually collected them.

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Secondary data are compiled data that are taken from several primary sources and synthesized or summarized in some way.

Some examples of primary sources of data are the U.S. Bureau of the Census, or an individual or company who distributes a survey on job satisfaction within an organization or industry.

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Primary data are often referred to as raw data and are most of ten collected by the person or organization using the data.

Secondary data are data collected from various sources and then either summarized or combined with other data in some way.


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Some examples of secondary data sources are the Statistical Abstract of the United States or the World Bank.

Secondary data are data that were not collected by the individual who is using them, but were obtained from another source.

Once we have the data, the statistical analysis that we do depends on the type of data we have. There are two major types of variables, qualitative and quantitative.


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2.6.1 Qualitative DataQualitative data, also known as nominal or

categorical data, are the simplest form of data.

Examples of qualitative data are variables such as gender (male or female) or the expected grade in a course (A, B, C, D, or F). Each item in the sample falls into one of a finite number of possible categories.

Qualitative data describe a particular characteristic of a sample item and they are most often nonnumerical in nature.


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Sometimes numbers are used to classify qualitative data.

For example, in surveys that ask for gender we often find that a 0 is used to denote "male" and a 1 is used to denote "female." When this is the case the data are referred to as nominal data.

The numbers are used simply to represent different categories and have no real meaning as numbers.


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Data that are created by assigning numbers to different categories when the numbers have no real meaning are called nominal data.

The order in which numbers are assigned to qualitative data may have some meaning. When numbers are used to name ordered categories the data are called ordinal.


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Another example of ordinal data would be when a characteristic of a sample item, such as income, is classified as 1= low, 2= medium, and 3= high.

Here, the numbers have a relative ordering, but there is no way to compare 1 to 2 to 3 numerically.

Data that are created by assigning numbers to categories where the order of assignment has meaning are called ordinal data.


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2.6.2 Quantitative DataData that are inherently numerical in

form are called quantitative data. This type of data falls into two different categories: discrete and continuous.

Discrete data are data that can take on only certain values. These values are often integers or whole numbers.


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Continuous data are data that can take on any one of an infinite number of possible values over an interval on the number line.

These values are most often the result of measurement.


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Some researchers make a further distinction with continuous data by classifying the data as interval or ratio data.

Interval data involve a variable that does not have an absolute zero in its scale. That is, there is no common place that is recognized as the beginning of the scale.

Interval data are data that can be compared only by looking at the difference between two values, or the interval between them.


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The most common example is temperature. If one temperature measurement is 40° and another is 80°, you can say that one is 40° more than the other, but it does not make sense to say that the 80° measurement is twice as hot (80/40= 2) as the 40° measurement.

In practice, almost all data are ratio data and are compared either by looking at the difference between the measurements or by taking their ratios.


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Ratio data are data that can be compared by looking at either the difference between two values or the ratio of two values.


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2.7 THE DIFFERENCE BETWEEN DESCRIPTIVE 2.7 THE DIFFERENCE BETWEEN DESCRIPTIVE STATISTICS AND INFERENTIAL STATISTICSSTATISTICS AND INFERENTIAL STATISTICS

2.7.1 Descriptive StatisticsThe tools of descriptive statistics are usually

the first ones encountered in any data analysis.

Tools of descriptive statistics allow you to describe the sample and allow you to summarize the data.

There are graphical or visual descriptive tools, which generally include bar charts, pie charts, and histograms.

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Graphical tools help you to see how the data behave and to summarize the data visually.

These tools are used all the time. Some examples are shown in Figure 2.4.


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There are also numerical descriptive tools, which allow you to summarize the data numerically.

Typically, a numerical summary would provide you with such statistics as the average, the median, the mode, and the largest and smallest data values.

Descriptive tools would be adequate if all we wanted to do was describe the sample.


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2.7.2 Inferential StatisticsAn inference is a deduction or a

conclusion.

The techniques of inferential statistics allow us to draw inferences or conclusions about the population from the sample.

We need to draw conclusions about a population based on the observed sample and the theory of probability.


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We will use probability theory to calculate the likelihood of observing or selecting a particular sample from a population Figure 2.5.


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2.8 BASIC SUMMATION NOTATION2.8 BASIC SUMMATION NOTATION

It is useful to introduce some shorthand notation for writing statistical formulas. This notation is called the sigma notation.

Sigma notation is shorthand notation used to write formulas. It is so named because it uses the Greek capital letter sigma, written as .

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2.8.1 Summing the DataIf we have a sample of size n, we can refer

to the individual data values as x1, x2, x3, ………xn, where x1 represents the first data value, x2 the second data value, and so on.

Many of the formulas you encounter will require that you sum the data values. We could write the sum as x1 + x2 + x3 +….. + xn but this will get cumbersome and be awkward to use.


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2.8.1 Summing the Data Using the sigma notation, we can write the sum x1 + x2 + x3

+….. + xn as

The sigma notation on the right of the equation is read as "the sum of xi as i goes from 1 to n."

Upper Limit of Index

General Term

IndexLower Limit of Index

n

iin xxxxx

1321 .........


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The letter under the is called the index and it really doesn't matter what letter you use since you use the same letter in describing the general term.

The value of the index indicated on the bottom of the “” is the starting value for the index and is usually 1, meaning that we are going to start with x1.


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The value on the top of the “” is the ending value for the index and is usually n, the sample size.

When the index goes from i = 1 to n, this indicates that all of the data values in the sample are being used.


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2.8.2 Summing DifferencesWhen we analyze data we often need to

look at how far away data values are from some number.


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1 RES 341 RESEARCH AND EVALUATION WORKSHOP 2 By Dr. Serhat Eren University OF PHOENIX.

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