# Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Save this PDF as:

Size: px
Start display at page:

Download "Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:"

## Transcription

1 Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 1: DESCRIPTIVE STATISTICS - FREQUENCY DISTRIBUTIONS AND AVERAGES: Inferential and Descriptive Statistics: There are four main stages in doing research: designing a study, collecting the data, obtaining descriptive statistics and perhaps performing some inferential statistics. Statistics are a set of tools for obtaining insight into a psychological phenomenon. Descriptive statistics summarise the data, making clear any trends, patterns etc. which may be lurking within them; they consist of visual displays such as graphs, and summary statistics such as means. Inferential statistics attempt to make inferences about the parent population on the basis of the limited samples actually obtained. The term inferential statistics is usually reserved for the various statistical tests used for comparing two or more groups of subjects within an experiment, etc. (Although in effect, most of statistics in psychology involves a process of trying to make inferences about humanity on the basis of a few undergraduate subjects!) Samples and Populations: In statistics, the terms sample and population have special meanings. A sample is a limited subset from a population, and a population is the entire set of things under consideration. What constitutes a population will vary according to what it is you are trying to make extrapolations to: usually the population to which we are trying to generalise from our sample consists of the entire human race. However, if you are interested only in making claims about the behaviour of three-toed Amazonian sloths, then you might be able to get access to the entire population of sloths if there weren't very many of them in existence. Normally, we have to obtain data from a limited sample of subjects because it is impossible to obtain data from the entire population to which they belong. Types of Data: It's important to know what kind of data you are dealing with, as certain statistics can only be used validly with certain kinds of data. (a) Nominal (categorical) data: Numbers are used only as names for categories. Therefore they are not really acting as numbers, but just serving as labels. If the numbers can be replaced by letters or names, then you are dealing with nominal data. So, if you had four groups, you could label them "1", "", "3" and "4", but it would mean just the same as if they were entitled "A", "B", C, and "D". You cannot do any statistics or arithmetic on nominal data, in just the same way as you cannot add together A and B. The numbers on footballers' jerseys are the classic example of nominal data: adding together players at positions 1 and does not give you the player at position 3! (b) Ordinal (rank) data: Things can be ordered in terms of some property such as size, length, speed, time etc. However, successive points on the scale are not necessarily spaced equally apart. To rank Grand Prix racers in terms of whether they finish first, second, third, and so on, is to measure them on an ordinal scale: all that you know for certain is the order of finishing - the difference between first and secondplaced finishers may be quite different from the difference between those placed second and third. In psychology, most attitude scales (e.g., seven-point scales which run from "1" for "completely disagree" through to "7" for "completely agree") are making measurements on an ordinal scale. (c) Interval data: These are measured on a scale on which measurements are spaced at equal intervals, but on which there is no true zero point (although there may be a point on the scale which is arbitrarily named "zero"). The classic examples are the Fahrenheit and Centigrade temperature scales. On the Centigrade scale, a rise of "one degree" represents an increase in heat by the same amount, whether one is talking about the shift from 5 degrees to 6 degrees, or from 90 degrees to 91 degrees. However, because the zero point on the scale is arbitrary, one cannot make any statements about ratios of quantities on the scale: thus it is not permissible to say that 6 degrees is twice 13 degrees or half of 5 degrees. (d) Ratio data: This scale is the same as the interval scale, except that there is a true zero on the scale, representing a compete absence of the thing in question which is being measured. Measurements of height, weight, length, time are all measurements on ratio scales. With this kind of scale, the intervals are equally spaced and we can make meaningful statements about the ratios of quantities measured. Thus, an increase in length between 9 inches and 10 inches is the same as an increase in length between 90 inches and 91 inches; and additionally, we can say that 6 inches is twice 3 inches, and

2 Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page : half of 1 inches. The concept of levels of measurement may seem rather abstract and confusing now, but it will prove important in helping us to decide which statistics are appropriate for our data. For example, some statistical tests can only be used with nominal data, and others can only be used with interval or ratio data. Summarising the data, by using Frequency Distributions: The normal consequence of doing a psychology experiment is to end up with a score or set of scores for each of the individuals that took part in the experiment. The first thing we need to do is to summarise these data in some way, so that we can see the "wood" for the "trees" - in other words, so that we are more likely to spot any trends or effects that are lurking within all those numbers. Imagine we were interested in how well psychology students coped with doing a statistics exam. For each student, we might have their exam score. We might want the answers to various questions: for example, how well people were doing generally; what the highest and lowest scores were; within what range the bulk of the scores fell; etc. Presented simply as a table of unprocessed numbers, answering questions like this would be difficult. One of the goals of statistics is to make it easier to see what's going on in your data. So, the first step might be to construct a frequency distribution of the scores: we could simply count how many people obtained each of the scores that it was possible to obtain. This is an improvement over our original unordered data-set, but things could be better: we are still bogged down in the details of the data. To get a better picture of the overall trends that are present, we could produce a grouped frequency distribution of the scores: we could construct a number of categories, each of which corresponded to a range of scores. We could then count how many scores fell into each of these categories, or class intervals. Thus, instead of recording that four subjects scored 5, two subjects scored 53 and one subject scored 54, we could summarise these data by noting that seven subjects scored between 5 and 54. The number of class intervals is fairly arbitrary, but most statisticians suggest between 10 and 0 gives the best results: too few, and you lose too much information about what's going on in your set of numbers. Too many categories, and you are back to where you started, unable to see the overall picture because you are bogged down in the excessive details. Sometimes we might want to compare frequency distributions which are based on different totals. Imagine that 50 people take a statistics exam one year, and 100 take it the next year, but in both years 5 people fail. 5 failures out of 50 is a lot more serious than 5 failures out of 100. In the former case, half of the students have failed, whereas in the latter case only a quarter have failed! How can we compare frequency distributions when they are based on different totals? The solution is to turn our "raw" frequencies into relative frequencies. To turn a raw frequency into a relative frequency, divide the raw frequency by the total number of cases, and then multiply by 100. Thus (5/50)*100 = 50%, and (5/100)*100 = 5%. By converting frequencies to relative frequencies in this way, we can more easily compare frequency distributions based on different totals. If you display data in the form of relative frequencies (i.e., as percentages of the total), ideally you should also show the raw frequencies, so that readers are aware of how many cases your percentages are based on. At the very least you should show the total number of cases, so that readers can work out for themselves how large a sample your percentages are based on. The reason for this is that it is easy to use relative frequencies to make your data look more impressive than they really are. For example, claiming that "75% of those interviewed said they would vote for Tony Blair" is impressive if we are talking about 75% of a large sample, but less impressive if 75% merely represents 3 out of a total of 4 people! Summarising data with single numbers: measures of central tendency and dispersion: Often we want to summarise our results in a more succinct form than is possible with tables and graphs; we want to talk about the "average" performance on a test. Measures of central tendency: These are single numbers which attempt to convey some impression of what constitutes "typical" performance. The three most commonly encountered are the mean, median and mode. Each has advantages and disadvantages as a summary description of our data.

3 Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 3: (a) The Mode: This is the score which occurs most frequently in a set of scores. Thus, for the set of scores 5, 6, 11,,, 96, 98 the mode is, since there are more instances of than there are of any of the other scores. Advantages of the mode: (i) It is simple to calculate, and intuitively easy to understand. (ii) It is the only measure of central tendency which can be used with nominal data. Disadvantages of the mode: Unfortunately, the disadvantages outweigh these benefits for any serious statistical use. (i) The mode may easily be unrepresentative of the bulk of the data, and so produce a misleading picture. Imagine if we had the following set of scores: 3, 4, 4, 5, 6, 7, 8, 8, 96, 96, 96. Here the mode is 96 - but most of the scores are low numbers, and so 96 is unrepresentative of them. (ii) There may be more than one mode in a set of scores. For example, in the set of scores 3,3,3,4,4,4,6,6,6, there are three modes! (iii) If data are grouped into class intervals, the mode is very sensitive to the size and number of class intervals used; it can easily be made to "jump around" by varying the limits of the class intervals. (b) The Median: This is the middle score of a set of scores, when those scores are arranged in order of size. Thus the median of 3, 4, 4, 5, 6, 7, 8, 8, 96, 96, 96 is 7. If you have an even number of scores, there is no middle score; in that case, the median is the average of the middle two scores. Thus, if we had the scores 3, 4, 4, 5, 6, 7, 8, 8, 96, 96, 96, 96, the median would be (7+8)/ = 7.5. Advantages of the median: (i) It is resistant to the distorting effects of extreme high or low scores: notice that the median in the examples above was relatively unaffected by the few very high scores. (ii) It can be used with ordinal, interval or ratio data; it cannot be used with nominal data, since categories have no numerical order. Disadvantages of the median: (i) It only takes account of the relative ranks of scores with respect to each other; it takes no account of the actual numerical values of the numbers themselves, which is rather wasteful if the data are on an interval or ratio scale. (ii) It is more susceptible to sampling fluctuations than the mean (see below). (iii) It is less mathematically useful than the mean. (c) The Mean: To find the mean of a set of scores, add them all together and then divide this total by the number of scores. In statistics notation: = N Advantages of the mean: (i) It is the only measure of central tendency that uses the information from every single score. (ii) It has certain mathematical advantages; it is very common in statistical formulae, in one form or another. (iii) It is the measure which is most resistant to sampling fluctuation. If you take repeated samples of scores from a population of scores, you will find that each sample is somewhat different from the next. If you work out each sample's mean, median and mode, it turns out that the mean is the measure which will vary the least from sample to sample. Since we usually want to make extrapolations from samples to populations, this is important. Disadvantages of the mean: (i) The mean is susceptible to distortion from extreme scores (e.g., "mean income" in the U.K. is a highly misleading statistic, because the few millionaires that contribute to this mean have a disproportionate effect, biasing the mean upwards from what it would otherwise be).just one or two high or low scores can seriously distort the mean. (ii) The mean can only be used with interval or ratio data; it cannot be used with ordinal or nominal data.

4 Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 4: Measures of Dispersion or Spread: A complete description of a set of data requires a measure of how much the scores vary around our chosen measure of central tendency. (a) The Range: This is the difference between the highest and lowest scores. (i.e., range = highest - lowest). Advantages of the range: It is quick and easy to calculate, and easy to understand. Disadvantages of the range: (i) It is unduly susceptible to extreme scores at either end. Thus the range of the set of scores 3, 4, 4, 5, 100 is (100-3) = 97. Replace the 100 with a 5, and the range is now (5-3)=! variation in just one score can make an enormous difference to the range. (ii) Because the range uses only the two extreme scores, it conveys no information about the spread of scores between these two points. Very different distributions of scores might have the same range. Thus,,,,, 0 and, 0, 0, 0, 0, 0 both have exactly the same range (18), but they are clearly very different sets of scores! (b) The Semi-Interquartile Range: To calculate the SIQR, arrange the scores in order of magnitude (as with the median). Find the score which is a quarter of the way from the bottom, and the score which is a quarter of the way from the top. These two scores are called the lower and upper quartiles respectively. Subtract the lower quartile from the upper quartile, and the result is the semi-interquartile range. Advantages of the SIQR: It is less susceptible than the range to distortion from extreme scores. Disadvantages of the SIQR: It still uses the information from only a couple of the scores in the distribution of scores. (c) The Standard Deviation: This is effectively an "average difference from the mean", which commonly accompanies the mean. The bigger the s.d., the more the scores differ from the mean and between themselves, and the less satisfactory the mean is as a summary of the data. To calculate the standard deviation, you do the following: (i) find the mean of all the scores; (ii) subtract the mean from each score (keeping a note of the sign of the bit that remains each time); (iii) square each of these "deviation scores"; (iv) find the mean of these scores (i.e., add them all up and divide by the total number of scores). This mean is called the variance. (v) Find the square root of the variance; this is called the standard deviation. Two steps in this sequence (iii and v) require explanation. The reason we square the differences in step (iii) is in order to get rid of the minus signs (anything squared, whether positive or negative, becomes a positive number). Some differences from the mean will be positive, and some will be negative. If we didn't get rid of the differences' signs, and simply added them all together, the pluses and minuses would all cancel out and we would always get zero! This squaring is therefore just a piece of arithmetical trickery to get rid of the minus signs. The reason we take the square root of the variance in step (v) is that the variance is okay, but due to the squaring trick, it's not in the same units as our original mean; it is nicer to have a measure of the spread of the scores which is in the same units as our mean, and this is where the standard deviation comes in. This is what the formula looks like in statistical notation: s = n (but see below, in the section entitled "complications.." for further details about this formula). Advantages of the standard deviation: Like the mean, the s.d. uses the information from every score. Disadvantages of the standard deviation: It can only be used with interval or ratio data.

5 Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 5: Complications with using means and standard deviations: The distinction between samples and populations needs to be taken into account when considering means and standard deviations. There are three conditions which can occur: (a) when the mean and s.d. are intended to describe only the sample on which they are based (i.e., they are used purely as descriptive sample statistics); (b) when the mean and s.d. are intended to describe the entire population (and every member of the population can be measured - very rare in psychology); (c) when the mean and s.d. are obtained from a sample, but one wishes to use them as estimates of the mean and s.d. of the parent population (probably the most common situation in psychology). It can be shown that the best estimate of the population mean is the sample mean; the sample mean is said to be an unbiased estimate of the population mean. Consequently, exactly the same formula can be used for the mean, in all three of the circumstances just listed. Unfortunately, the same is not true for the standard deviation; it can be shown that the sample s.d. tends to underestimate the size of the population s.d. So, if you were to try to use your limited sample from a population in order to make inferences about the characteristics of that population, your sample s.d. would tend to be a bit smaller than the "true" s.d. of the parent population. To correct for this underestimation, we simply add a small bodge to the s.d. formula: instead of dividing the sum of the squared deviations by n, the number of scores, we divide it by n-1 (the number of scores minus one). We thus divide the top part of the formula by a smaller number, and hence end up with a slightly larger s.d. than we would otherwise have obtained. In short, if you want to work out the s.d. for conditions (a) or (b), divide by n; but if you want to make inferences from your sample to the parent population from which it is believed to have originated (i.e., condition c), then you need to use the n-1 formula. (a) sample s.d. (used purely as a decscription of your particular sample): s = n (b) sample s.d. (used as an estimate of the population s.d.): s = n 1 (c) population s.d. (if you could get scores from every member of the entire population): δ = N Note that the first and last formulae are identical; only the notation differs, to show that in one case you are referring to a sample, and in the other case you are referring to a population. On Casio

6 Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 6: calculators that are capable of working out standard deviations, you will usually find two keys: one is labelled σn and the other is labelled σn 1. The first corresponds to the first formula above, and will give you the standard deviation of the scores that you have entered, purely as a description of that particular sample; the second is equivalent to the second formula above, and will therefore give you a standard deviation which would serve as an estimate of the standard deviation of the population of scores form which the sample is thought to have been taken. (If you want the calculator to work out the third formula, the population standard deviation itself, use the σn key). Some useful terminology: Mathematical notation is a language with which many of you are unfamiliar, and which most of you find off-putting. Although it will be kept to a minimum in this course, you will find knowledge of the following symbols useful: Σ = an uppercase "sigma". This symbol means "add up the following". δ = a lowercase "sigma". This symbol is often used to represent the population standard deviation. µ = "mu". This symbol is often used to represent the population mean. = "x bar", usually used to denote a sample mean (the mean of the scores in sample, as opposed to Y, Z or any other samples of scores you might have). s = standard deviation of your sample of scores. N is the number of scores in a population. n is the number of scores in a sample. The following two formulae for calculating the mean look different because of the notation, but they involve the same operations - it's just that one is for a sample mean and the other is for a population mean: population mean : µ = N sample mean: = n You might find it useful to know that arithmetic operations are performed in the order: stuff in Brackets first; then Division; Multiplication; Addition; Subtraction. Note that Σ and (Σ) mean completely different things. The former means "square each value of first, and then add together all of these squared values". The latter means "add together all of the values, and then square this total" (i.e. you do the thing in the brackets first). means "take each value of in turn; subtract the mean of from it, to get a difference score; square this difference score; and lastly, add together all of these squared difference scores". You can see that one advantage of statistical notation is that it has the potential to save a lot of paper!

### Descriptive Statistics and Measurement Scales

Descriptive Statistics 1 Descriptive Statistics and Measurement Scales Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1: THE NORMAL CURVE AND "Z" SCORES: The Normal Curve: The "Normal" curve is a mathematical abstraction which conveniently

### Chapter 3: Central Tendency

Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents

### MEASURES OF DISPERSION

MEASURES OF DISPERSION Measures of Dispersion While measures of central tendency indicate what value of a variable is (in one sense or other) average or central or typical in a set of data, measures of

### Northumberland Knowledge

Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about

### ( ) ( ) Central Tendency. Central Tendency

1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution

### Data Analysis: Describing Data - Descriptive Statistics

WHAT IT IS Return to Table of ontents Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data. Descriptive statistics are most

### Means, standard deviations and. and standard errors

CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

### Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences

Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html

### DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

### Descriptive Statistics: Measures of Central Tendency and Crosstabulation. 789mct_dispersion_asmp.pdf

789mct_dispersion_asmp.pdf Michael Hallstone, Ph.D. hallston@hawaii.edu Lectures 7-9: Measures of Central Tendency, Dispersion, and Assumptions Lecture 7: Descriptive Statistics: Measures of Central Tendency

### Levels of measurement in psychological research:

Research Skills: Levels of Measurement. Graham Hole, February 2011 Page 1 Levels of measurement in psychological research: Psychology is a science. As such it generally involves objective measurement of

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

### Why do we measure central tendency? Basic Concepts in Statistical Analysis

Why do we measure central tendency? Basic Concepts in Statistical Analysis Chapter 4 Too many numbers Simplification of data Descriptive purposes What is central tendency? Measure of central tendency A

### MEASURES OF VARIATION

NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

### Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

### Hypothesis Testing with z Tests

CHAPTER SEVEN Hypothesis Testing with z Tests NOTE TO INSTRUCTOR This chapter is critical to an understanding of hypothesis testing, which students will use frequently in the coming chapters. Some of the

### We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

### Introduction to Statistics for Computer Science Projects

Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

### 2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56

2. Describing Data We consider 1. Graphical methods 2. Numerical methods 1 / 56 General Use of Graphical and Numerical Methods Graphical methods can be used to visually and qualitatively present data and

### Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS

Content DESCRIPTIVE STATISTICS Dr Najib Majdi bin Yaacob MD, MPH, DrPH (Epidemiology) USM Unit of Biostatistics & Research Methodology School of Medical Sciences Universiti Sains Malaysia. Introduction

### Univariate Descriptive Statistics

Univariate Descriptive Statistics Displays: pie charts, bar graphs, box plots, histograms, density estimates, dot plots, stemleaf plots, tables, lists. Example: sea urchin sizes Boxplot Histogram Urchin

### Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

### Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion

Statistics Basics Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Part 1: Sampling, Frequency Distributions, and Graphs The method of collecting, organizing,

### A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

### Summarizing Data: Measures of Variation

Summarizing Data: Measures of Variation One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### Report of for Chapter 2 pretest

Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every

### F. Farrokhyar, MPhil, PhD, PDoc

Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

### Introduction to Descriptive Statistics

Mathematics Learning Centre Introduction to Descriptive Statistics Jackie Nicholas c 1999 University of Sydney Acknowledgements Parts of this booklet were previously published in a booklet of the same

### Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

### Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

### 13.2 Measures of Central Tendency

13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

### Central Tendency. n Measures of Central Tendency: n Mean. n Median. n Mode

Central Tendency Central Tendency n A single summary score that best describes the central location of an entire distribution of scores. n Measures of Central Tendency: n Mean n The sum of all scores divided

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Introduction; Descriptive & Univariate Statistics

Introduction; Descriptive & Univariate Statistics I. KEY COCEPTS A. Population. Definitions:. The entire set of members in a group. EXAMPLES: All U.S. citizens; all otre Dame Students. 2. All values of

### LEARNING OBJECTIVES SCALES OF MEASUREMENT: A REVIEW SCALES OF MEASUREMENT: A REVIEW DESCRIBING RESULTS DESCRIBING RESULTS 8/14/2016

UNDERSTANDING RESEARCH RESULTS: DESCRIPTION AND CORRELATION LEARNING OBJECTIVES Contrast three ways of describing results: Comparing group percentages Correlating scores Comparing group means Describe

### Session 1.6 Measures of Central Tendency

Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices

### Frequency distributions, central tendency & variability. Displaying data

Frequency distributions, central tendency & variability Displaying data Software SPSS Excel/Numbers/Google sheets Social Science Statistics website (socscistatistics.com) Creating and SPSS file Open the

### 10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

### Distributions: Population, Sample and Sampling Distributions

119 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 9 Distributions: Population, Sample and Sampling Distributions In the three preceding chapters

### 3: Summary Statistics

3: Summary Statistics Notation Let s start by introducing some notation. Consider the following small data set: 4 5 30 50 8 7 4 5 The symbol n represents the sample size (n = 0). The capital letter X denotes

### Section 3.1 Measures of Central Tendency: Mode, Median, and Mean

Section 3.1 Measures of Central Tendency: Mode, Median, and Mean One number can be used to describe the entire sample or population. Such a number is called an average. There are many ways to compute averages,

### Measures of Central Tendency and Variability: Summarizing your Data for Others

Measures of Central Tendency and Variability: Summarizing your Data for Others 1 I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). 1. Mode :

### Summarizing Scores with Measures of Central Tendency: The Mean, Median, and Mode

Summarizing Scores with Measures of Central Tendency: The Mean, Median, and Mode Outline of the Course III. Descriptive Statistics A. Measures of Central Tendency (Chapter 3) 1. Mean 2. Median 3. Mode

### Data reduction and descriptive statistics

Data reduction and descriptive statistics dr. Reinout Heijungs Department of Econometrics and Operations Research VU University Amsterdam August 2014 1 Introduction Economics, marketing, finance, and most

### Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 1: CHI-SQUARE TESTS: When to use a Chi-Square test: Usually in psychological research, we aim to obtain one or more scores

### What are Data? The Research Question (Randomised Controlled Trials (RCTs)) The Research Question (Non RCTs)

What are Data? Quantitative Data o Sets of measurements of objective descriptions of physical and behavioural events; susceptible to statistical analysis Qualitative data o Descriptive, views, actions

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing.

3.0 - Chapter Introduction In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing. Categories of Statistics. Statistics is

### Histogram. Graphs, and measures of central tendency and spread. Alternative: density (or relative frequency ) plot /13/2004

Graphs, and measures of central tendency and spread 9.07 9/13/004 Histogram If discrete or categorical, bars don t touch. If continuous, can touch, should if there are lots of bins. Sum of bin heights

### Exploratory Data Analysis. Psychology 3256

Exploratory Data Analysis Psychology 3256 1 Introduction If you are going to find out anything about a data set you must first understand the data Basically getting a feel for you numbers Easier to find

### GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

### CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME!

CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME! You may have come across the terms powers, indices, exponents and logarithms. But what do they mean? The terms power(s),

### Chapter 2. Objectives. Tabulate Qualitative Data. Frequency Table. Descriptive Statistics: Organizing, Displaying and Summarizing Data.

Objectives Chapter Descriptive Statistics: Organizing, Displaying and Summarizing Data Student should be able to Organize data Tabulate data into frequency/relative frequency tables Display data graphically

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### 4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: "What do the data look like?"

Data Analysis Plan The appropriate methods of data analysis are determined by your data types and variables of interest, the actual distribution of the variables, and the number of cases. Different analyses

### Chapter 15 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately

### Understanding Variability

3 Vive la Différence Understanding Variability Difficulty Scale (moderately easy, but not a cinch) How much Excel? (a ton) What you ll learn about in this chapter Why variability is valuable as a descriptive

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Calculating Central Tendency & An Introduction to Variability

Calculating Central Tendency & An Introduction to Variability Outcome (learning objective) Students will use mean, median and mode; calculate range, quartiles, interquartiles, variance and standard deviation

### Recitation, Week 3: Basic Descriptive Statistics and Measures of Central Tendency:

Recitation, Week 3: Basic Descriptive Statistics and Measures of Central Tendency: 1. What does Healey mean by data reduction? a. Data reduction involves using a few numbers to summarize the distribution

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### Nominal Scaling. Measures of Central Tendency, Spread, and Shape. Interval Scaling. Ordinal Scaling

Nominal Scaling Measures of, Spread, and Shape Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education and Human Development Department of Teaching and Learning The lowest level of

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Sheffield Hallam University. Faculty of Health and Wellbeing Professional Development 1 Quantitative Analysis. Glossary

Sheffield Hallam University Faculty of Health and Wellbeing Professional Development 1 Quantitative Analysis Glossary 2 Using the Glossary This does not set out to tell you everything about the topics

### Statistics. Measurement. Scales of Measurement 7/18/2012

Statistics Measurement Measurement is defined as a set of rules for assigning numbers to represent objects, traits, attributes, or behaviors A variableis something that varies (eye color), a constant does

### Topic 9 ~ Measures of Spread

AP Statistics Topic 9 ~ Measures of Spread Activity 9 : Baseball Lineups The table to the right contains data on the ages of the two teams involved in game of the 200 National League Division Series. Is

### Homework 3. Part 1. Name: Score: / null

Name: Score: / Homework 3 Part 1 null 1 For the following sample of scores, the standard deviation is. Scores: 7, 2, 4, 6, 4, 7, 3, 7 Answer Key: 2 2 For any set of data, the sum of the deviation scores

### Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

### 32 Measures of Central Tendency and Dispersion

32 Measures of Central Tendency and Dispersion In this section we discuss two important aspects of data which are its center and its spread. The mean, median, and the mode are measures of central tendency

### 1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics)

1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics) As well as displaying data graphically we will often wish to summarise it numerically particularly if we wish to compare two or more data sets.

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

### DESCRIPTIVE STATISTICS & DATA PRESENTATION*

Level 1 Level 2 Level 3 Level 4 0 0 0 0 evel 1 evel 2 evel 3 Level 4 DESCRIPTIVE STATISTICS & DATA PRESENTATION* Created for Psychology 41, Research Methods by Barbara Sommer, PhD Psychology Department

### Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion

Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

### Psychology 312: Lecture 6 Scales of Measurement. Slide #1. Scales of Measurement Reliability, validity, and scales of measurement.

Psychology 312: Lecture 6 Scales of Measurement Slide #1 Scales of Measurement Reliability, validity, and scales of measurement. In this lecture we will discuss scales of measurement. Slide #2 Outline

### Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs.

ECO 51 Research Methods 1. Data and Descriptive Statistics (Review) Data A variable - a characteristic of population or sample that is of interest for us. Data - the actual values of variables Quantitative

### Lesson 4 Measures of Central Tendency

Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central

### Concepts of Variables. Levels of Measurement. The Four Levels of Measurement. Nominal Scale. Greg C Elvers, Ph.D.

Concepts of Variables Greg C Elvers, Ph.D. 1 Levels of Measurement When we observe and record a variable, it has characteristics that influence the type of statistical analysis that we can perform on it

### 9 Descriptive and Multivariate Statistics

9 Descriptive and Multivariate Statistics Jamie Price Donald W. Chamberlayne * S tatistics is the science of collecting and organizing data and then drawing conclusions based on data. There are essentially

### Chi Square Analysis. When do we use chi square?

Chi Square Analysis When do we use chi square? More often than not in psychological research, we find ourselves collecting scores from participants. These data are usually continuous measures, and might

### Describing Data. We find the position of the central observation using the formula: position number =

HOSP 1207 (Business Stats) Learning Centre Describing Data This worksheet focuses on describing data through measuring its central tendency and variability. These measurements will give us an idea of what

### Statistics GCSE Higher Revision Sheet

Statistics GCSE Higher Revision Sheet This document attempts to sum up the contents of the Higher Tier Statistics GCSE. There is one exam, two hours long. A calculator is allowed. It is worth 75% of the

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### SOST 201 September 18-20, 2006. Measurement of Variables 2

1 Social Studies 201 September 18-20, 2006 Measurement of variables See text, chapter 3, pp. 61-86. These notes and Chapter 3 of the text examine ways of measuring variables in order to describe members

### Using SPSS 20, Handout 3: Producing graphs:

Research Skills 1: Using SPSS 20: Handout 3, Producing graphs: Page 1: Using SPSS 20, Handout 3: Producing graphs: In this handout I'm going to show you how to use SPSS to produce various types of graph.

### Each exam covers lectures from since the previous exam and up to the exam date.

Sociology 301 Exam Review Liying Luo 03.22 Exam Review: Logistics Exams must be taken at the scheduled date and time unless 1. You provide verifiable documents of unforeseen illness or family emergency,

### 1) Overview 2) Measurement and Scaling 3) Primary Scales of Measurement i. Nominal Scale ii. Ordinal Scale iii. Interval Scale iv.

1) Overview 2) Measurement and Scaling 3) Primary Scales of Measurement i. Nominal Scale ii. Ordinal Scale iii. Interval Scale iv. Ratio Scale 4) A Comparison of Scaling Techniques Comparative Scaling

### MAT 142 College Mathematics Module #3

MAT 142 College Mathematics Module #3 Statistics Terri Miller Spring 2009 revised March 24, 2009 1.1. Basic Terms. 1. Population, Sample, and Data A population is the set of all objects under study, a

### Association Between Variables

Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

### Table 2-1. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No.

Chapter 2. DATA EXPLORATION AND SUMMARIZATION 2.1 Frequency Distributions Commonly, people refer to a population as the number of individuals in a city or county, for example, all the people in California.

### AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

### Remember this? We know the percentages that fall within the various portions of the normal distribution of z scores

More on z scores, percentiles, and the central limit theorem z scores and percentiles For every raw score there is a corresponding z score As long as you know the mean and SD of your population/sample