APPROACHES TO QUANTITATIVE DATA ANALYSIS © LOUIS COHEN, LAWRENCE MANION AND KEITH MORRISON © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

STRUCTURE OF THE CHAPTER Scales of data Parametric and non-parametric data Descriptive and inferential statistics Kinds of variables Hypotheses One-tailed and two-tailed tests Distributions © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

FOUR SCALES OF DATA Nominal Ordinal Interval Ratio It is incorrect to apply statistics which can only be used at a higher scale of data to data at a lower scale. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

PARAMETRIC AND NON-PARAMETRIC STATISTICS Characteristics of, or factors in, the population are known Non-parametric statistics Characteristics of, or factors in, the population are unknown © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

DESCRIPTIVE AND INFERENTIAL STATISTICS Descriptive statistics To summarize features of the sample or simple responses of the sample (e.g. frequencies or correlations). No attempt is made to infer or predict population parameters. Inferential statistics to infer or predict population parameters or outcomes from simple measures, e.g. from sampling and from statistical techniques. Based on probability. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

DESCRIPTIVE STATISTICS The mode (the score obtained by the greatest number of people). The mean (the average score). The median (the score obtained by the middle person in a ranked group of people, i.e. it has an equal number of scores above it and below it). Minimum and maximum scores. The range (the distance between the highest and the lowest scores). The variance (a measure of how far scores are from the mean: the average of the squared deviations of individual scores from the mean). © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

DESCRIPTIVE STATISTICS The standard deviation (a measure of the dispersal or range of scores: the square root of the variance). The standard error (the standard deviation of sample means). The skewness (how far the data are asymmetrical in relation to a ‘normal’ curve of distribution). Kurtosis (how steep or flat is the shape of a graph or distribution of data; a measure of how peaked a distribution is and how steep is the slope or spread of data around the peak). © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

INFERENTIAL STATISTICS Can use descriptive statistics Correlations Regression Multiple regression Difference testing Factor analysis Structural equation modelling © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

SIMPLE STATISTICS Frequencies (raw scores and percentages) Look for skewness, intensity, distributions and spread (kurtosis) Mode For nominal and ordinal data Mean For interval and ratio data Standard deviation © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

High standard deviation 9 8 Mean 7 | 6 5 4 3 2 1 X 10 11 12 13 14 15 16 17 18 19 20 2 3 4 20 Mean = 6 High standard deviation © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

Moderately high standard deviation 9 8 Mean 7 | 6 5 4 3 2 1 X 10 11 12 13 14 15 16 17 18 19 20 1 2 6 10 11 Mean = 6 Moderately high standard deviation © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

Low standard deviation 9 8 Mean 7 | 6 5 4 3 X 2 1 10 11 12 13 14 15 16 17 18 19 20 5 6 6 6 7 Mean = 6 Low standard deviation © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

STANDARD DEVIATION The standard deviation is a standardized measure of the dispersal of the scores, i.e. how far away from the mean/average each score is. It is calculated, in its most simplified form, as: or d2 = the deviation of the score from the mean (average), squared  = the sum of N = the number of cases A low standard deviation indicates that the scores cluster together, whilst a high standard deviation indicates that the scores are widely dispersed. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

DEPENDENT AND INDEPENDENT VARIABLES An independent variable A variable which causes, influences or affects, in part or in total, a particular outcome. A dependent variable The variable, that which is caused or effected by, in total or in part, the independent variable. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

DEPENDENT AND INDEPENDENT VARIABLES Be careful in assuming which is or is not the dependent or independent variable, as the direction of causality may not be one-way or in the direction assumed. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

MODERATOR AND MEDIATOR VARIABLES Moderator variable Affects the strength and/or the direction of a relationship between two other variables, e.g. between an independent and a dependent variable, i.e. whose values influence the values of another variable. Mediator variable Explains the relationship between an independent and dependent variable, or between two other variables. A mediator variable (B) receives the effect of one independent variable (A) and this affects the outcome variable (C). © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

CATEGORICAL, DISCRETE AND CONTINUOUS VARIABLES A categorical variable A variable which has categories of values, e.g. the variable ‘sex’ has two values: male and female. Categorical variables match categorical data. A discrete variable A discrete variable has a finite number of values of the same item, with no intervals or fractions of the value, e.g. a person cannot have half an illness or half a mealtime. A continuous variable A continuous variable can vary in quantity, e.g. money in the bank, monthly earnings. There are equal intervals, and, usually, a true zero, e.g. it is possible to have no money in the bank. Continuous variables match interval and ratio data. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

SIX KEY INITIAL QUESTIONS 1 What kind (scales) of data are there? 2 Are the data parametric or non-parametric? 3 Are descriptive or inferential statistics required? 4 Do dependent and independent variables need to be identified? 5 Do the research and data analysis need to take account of moderating and mediating variables? 6 Are the relationships considered to be linear or non-linear? © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

KINDS OF ANALYSIS Univariate analysis Looks for differences amongst cases within one variable. Bivariate analysis Looks for a relationship between two variables. Multivariate analysis Looks for a relationship between two or more variables. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

HYPOTHESES Null hypothesis (H0) Alternative hypothesis (H1) The null hypothesis is the stronger hypothesis, requiring rigorous evidence not to support it. One should commence with the former and cast the research in the form of a null hypothesis, and only turn to the latter in the case of finding the null hypothesis not supported. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

HYPOTHESES Direction of hypothesis: states the kind of difference or relationship between two conditions or two groups of participants. One-tailed (directional), e.g.: ‘people who study in silent surroundings achieve better than those who study in noisy surroundings’. (‘Better’ indicates the direction.) Two-tailed (no direction), e.g.: ‘there is a difference between people who study in silent surroundings and those who study in noisy surroundings’. (There is no indication of which is the better.) © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

HYPOTHESIS TESTING 1 Commence with a null hypothesis 2 Set the level of significance () to be used to support or not to support the null hypothesis (the alpha () level); the alpha level is determined by the researcher 3 Compute the data 4 Determine whether the null hypothesis is supported or not supported 5 Avoid Type I and Type II errors © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

ONE-TAILED AND TWO-TAILED TESTS A one-tailed test makes assumptions about the population and the direction of the outcome, e.g. Group A will score more highly than another on a test. A two-tailed test makes no assumptions about the population and the direction of the outcome, e.g. there will be a difference in the test scores. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

CONFIDENCE INTERVALS A confidence interval relates to a range of values for the results obtained. Researchers need to know how confident they can be that their particular result falls within that acceptable range, i.e. that the range will include the result in question. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

DISTRIBUTIONS Many statistics assume a normal curve of distribution. Researchers should test for the nature of the distributions to see if they conform to the normal curve, as this affects the choice of statistics. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

THE NORMAL CURVE OF DISTRIBUTION © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

THE NORMAL CURVE OF DISTRIBUTION A smooth, perfectly symmetrical, bell-shaped curve. It is symmetrical about the mean and its tails are assumed to meet the x-axis at infinity. Statistical calculations often assume that the population is distributed normally and then compare the data collected from the sample to the population, allowing inferences to be made about the population. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

THE NORMAL CURVE OF DISTRIBUTION Assumes that: 68.3 per cent of people fall within 1 standard deviation of the mean; 27.1 per cent are between 1 standard deviation and 2 standard deviations away from the mean; 4.3 per cent are between 2 and 3 standard deviations away from the mean; 0.3 per cent are more than 3 standard deviations away from the mean. © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

SKEWNESS The curve is not symmetrical or bell-shaped © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors

KURTOSIS (STEEPNESS OF THE CURVE) © 2018 Louis Cohen, Lawrence Manion and Keith Morrison; individual chapters, the contributors