Mathematics within the Psychology Curriculum

Transcription

1 Mathematics within the Psychology Curriculum Statistical Theory and Data Handling Statistical theory and data handling as studied on the GCSE Mathematics syllabus You may have learnt about statistics and data handling on the GCSE Mathematics syllabus. In that course, statistics and data handling can be thought of as a set of mathematical tools which allow you to describe relationships between sets of data, both geometrically and analytically. Using statistical theory and data handling in a Psychology degree course The first semester covers first principles of the scientific method with the assumption of basic numeracy only. This provides a foundation that will be built upon throughout the degree course where more complex inferential statistics are introduced. demonstrate an understanding of classical test theory and appreciate the basic tenants of measurement error; demonstrate an understanding of why replication of method does not normally lead to exact replication of results; understand the statistical concepts of population and sample and be introduced to the theory of inferential statistics; be aware of conceptual issues related to data collection regarding sources of bias; identify different designs of experimental studies (within/between group comparisons, correlation); classify level of measurement as Nominal, Ordinal, Interval and Ratio and appreciate the difference between discrete and continuous variables; identify independent and dependent variables; introduce the concept of hypothesis testing and types of hypothesis (null/alternative and one/two tailed); understand the three main questions that can be applied to data (difference, relationship, association); present and interpret frequency tables and various forms of graphs (line, scatter, histogram, pie, boxplot, stem and leaf); describe patterns from graphs (bimodal, linear, Gaussian, exponential, skew, kurtosis) and identify outliers; and design and evaluate multi-scale questionnaire surveys with an understanding of statistical validity and reliability, with an appreciation of acquiescence (lie scales), reverse-scored items, scale length and bloated specifics. 1

2 GCSE Mathematics Modules T1, T2 and T5 (Foundation Tier) -- Statistical Component GCSE Mathematics Modules T3 and T4 (Higher Tier) -- Statistical Component GCSE Additional Mathematics 1 Statistical Component (Topic 1) GCE AS/A2 Mathematics Statistics Modules: S1 (Topics 1 and 2) and S2 (Topic 4) Descriptive Statistics Descriptive statistics as studied on the GCSE Mathematics syllabus You may have learnt about descriptive statistics on the GCSE Mathematics syllabus. In that course, descriptive statistics can be thought of as a set of mathematical tools which allow you to describe representive characteristics of a set of data, such as the mean value. Using descriptive statistics in a Psychology degree course The ability to summarise and evaluate quantitative data is a cornerstone of the undergraduate Psychology course. A theoretical understanding of the mean and standard deviation is required to understand more complex statistical techniques taught in the second semester of Stage 1 and both semesters of Stage 2 (c.f. Worked example 2 with examples 3 and 4 in this document). Understand descriptive statistical measures such as mean, median, mode, standard deviation, variance, range and semi-interquartile range; understand how to calculate these statistics using functions in Microsoft Excel and PASW (social science statistical analysis software formerly known as SPSS) and also by hand; appreciate the appropriateness and usefulness of the different statistics; and be able to interpret, compare and evaluate the statistics GCSE Mathematics Modules T1,T2, T5 (Foundation Tiers) and T3 (Higher Tier) -- Statistical Components GCSE Additional Mathematics Statistical Component (Topic 3) GCE AS/A2 Mathematics Statistics Module: S1 (Topics 1 and 2) 1 CCEA are currently reviewing their mathematics specifications and the future of the GCSE additional mathematics qualification is currently guarenteed only until 2011, when the results of a consultation is known (see Appendix A) 2

3 Inferential Statistics Inferential statistics as studied on the GCSE Mathematics syllabus You may have learnt a little about inferential statistics on the GCSE Mathematics syllabus, as part of other topics such as time series analysis and correlation analysis for line-of-best-fit estimation. In that course, inferential statistics can be thought of as a set of mathematical tools which allow you to describe relationships between sets of data, such as the slope of a best-fit straight line. Using inferential statistics in a Psychology degree course This is taught in the second semester of Stage 1 and both semesters of Stage 2. Students are expected to manipulate data collected in frequency tables for input into PASW, chose an appropriate form of an analysis to run and interpret the appropriate statistical output. Students are required to present the results in the standard format used by the APA (American Psychological Association). Particular emphasis is placed on the assumptions regarding the use of inferential statistics with expected to know the meaning of each assumption, criteria used to assess them and how to deal with violations. Students are expected to study statistical textbooks; this is assessed in coursework that expects assumption-criteria to be justified using appropriate citations. Assumptions taught include the concepts of: level of measurement, sample-size, outliers, linearity, normality, homoscedasticity, (mutli)colinearity and sphericity. Some output from inferential statistics use the scientific notation and an understanding of this notation is necessary for the degree course. understand the difference between parametric and non-parametric data and analyses; appreciate significance testing, meaning of p-values and the threshold p<.05 in the context of hypothesis testing; understand statistical power, type I and II errors and their relationships with samplesize; be able to manipulate raw data for input into a statistical software package; select and justify an appropriate analysis for a statistical software package; and interpret the meaning of statistical output and report relevant results in the accepted format. These topics should enable the student to calculate, interpret and report: Pearson and Spearman Correlations; Student t-tests to compare means between two groups, both between and within group comparisons; Mann-Whitney U statistic used as a non-parametric between groups t-test; Wilcoxon s test used as a non-parametric within group t-test; Analysis of Variance (ANOVA) to compare means of more than two groups for between, within, and mixed group comparisons; Kruskal-Wallis statistic used as a non-parametric between groups ANOVA; 3

4 Friedman test used as a non-parametric within group ANOVA; Chi-squared test of association between nominal data; and multiple linear regression. GCSE Mathematics Module T6 (Higher Tier) Number and Algebra Component GCSE Additional Mathematics Statistical Component (Topics 6 and 7) GCE AS/A2 Mathematics Statistics Modules: S1 (Topics 1, and 2) and S2 (Topics 4, 5 and 6) N.B. The above analyses are not directly covered in any secondary level syllabi with the exception of regression, correlation and (to a limited extent) the t-test. Learning in the area of inferential statistics will be severely inhibited if have not grasped the basic concepts covered in the areas of statistical theory, data handling and descriptive statistics (e.g. significance testing, mean and standard deviation). Other Mathematics Other mathematics as studied on the GCSE Mathematics syllabus You will have learnt other about mathematical tools on the GCSE Mathematics syllabus, such as geometry, curve sketching, trigonometry and logarithms. Using other mathematics in a Psychology degree course This section details miscellaneous mathematical knowledge outside the domain of statistics that is covered in the psychology degree. Whilst the basic, relevant concepts are taught as part of the degree course, previous exposure and advanced knowledge of these areas undoubtedly provide with the advantage of a deeper understanding based on firstprinciples mathematics. Other topics in the Psychology degree course 1: factor analysis Factor Analysis uses correlation matrices to detect structure in a body of variables which can be thought of geometrically in factor space. It is typically applied to questionnaire items and ability (IQ) tests to identify clusters of related items based on inter-item covariances. Other topics in the Psychology degree course 2: logarithmic transformations Logarithmic transformation of simple exponential equations is used in two parts of the degree: to investigate the speed/accuracy trade-off of aimed movement (Fitt s Law) as part of the psychology of perception and action strand; and to express the binary information processing theory of cognitive ability (Hick s Law) in the human intelligence strand. 4

5 perform a Factor Analysis using PASW, interpret and present relevant output; understand how simple exponential equations may be expressed in logarithmic form; use logarithmic transformation to derive a linear expression in the form y=mx+c; and understand the equation of a straight line and meaning of gradient and intercept parameters. The number and algebra component of GCSE Mathematics module T3 (Higher Tier) covers the equation of a straight line and its parameters. Module T6 (Higher Tier) would be very useful for understanding factor analysis theory, as inter-item correlations can be expressed geometrically, with items represented as vectors with a common origin and constant magnitude. The angular separation between the vectors is Cos -1 (r), where r is the correlation between two items. Even more useful for this purpose would be A-level module C2 topic 3 that teaches the Cos function graph, periodicity and symmetry where GCSE level may see how to think geometrically, A-level will see why they can think geometrically. For the most part, these methods are not fully covered in the GSCE Mathematics Syllabus. The matrices component of the CCEA GCSE Additional Mathematics is useful with respect to factor analysis, providing an introduction to matrix notation and use. The teaching of factor analysis is limited to its theoretical application, how to input data, how to run the analysis on PASW and how to interpret the output. The knowledge of matrices is not required for the undergraduate degree per se. However a knowledge of matrices is required for a level three module (Advanced Statistical Methods) that does explain the calculation of factor analysis and nonetheless allows for a deeper appreciation of the analysis, even at level two of the undergraduate degree. The GCSE Additional Mathematics Syllabus introduces logarithm theory which is sufficient to understand the two instances of use in the course (noted above) from first principles. Logarithm theory is also covered in A-Level module C2 (Topic 4). 5