Analysis of Research Data Statistical Analysis of Research Data

Purpose of Statistical Analysis To summarize To explore the meaning of deviations in data To compare or contrast descriptively To infer that findings from the sample are indicative of the entire population To examine causality To predict To test the proposed relationships in a theoretical model To infer from the sample to a theoretical model

Elementary Descriptive Statistics Univariate descriptive statistics – these describe and synthesize data from empirical observations Averages and percentages Frequency distributions Cumulative frequencies Bar graph – nominal or qualitative Frequency histograms – ordinal, interval or ratio Frequency polygons – ordinal, interval or ratio

Elementary Descriptive Statistics Measures of central tendency Mode – for nominal data – where scores fall most frequently Median – for ordinal data – the point above which and below which 50% of the scores fall Mean – for interval or ratio data – the sum of the scores/the number of scores

Elementary Descriptive Statistics Measures of dispersion or variability – the degree to which subjects in the sample are similar to each other with respect to the critical aspect – the extent of inter-subject differences Range – for nominal data – the highest number minus the lowest number Inter-quartile range – for ordinal data - the range between the middle two quarters where 50% of scores lie Semi-interquartile range – for ordinal data – half of the range of scores in which the middle 50% of the scores lie Variance – for interval data - how much the score varies from the mean (the average of the sum of squares) Standard deviation – for interval data – the average of the deviations from the mean (the square root of the variance)

Standard Deviations in a Normal Distribution

Two Distributions of Different Variability

Elementary Descriptive Statistics Shapes of distributions Symmetrical – two halves fold over on themselves – the normal curve. In order to compare the scores in one sample with the scores in another, mechanisms were developed to transform raw scores into standard scores Non-symmetrical – skewed Positively – long tail to right (personal income) Negatively –long tail to left (age at death)

Examples of Skewed Distributions

Elementary Descriptive Statistics Shapes of distributions Modality – most distributions are unimodal, but if they have more than one mode, they are most often bimodal – having two peaks Kurtosis – the degree of peakedness of the curve Platykurtic Mesokurtic Leptokurtic

Elementary Descriptive Statistics Bivariate Descriptive Statistics – where there are two variables Contingency tables – a two dimensional frequency distribution – the frequency of two variables are cross tabulated Correlation – can determine the direction of a relationship between two variables – can display data graphically by use of the scatter diagram

BIVARIATE GENDER OBAMA McCAIN TOTAL Male 43 (33.6) 85 (66.4) 128 (100) Female 215 (52.2) 197 (47.8) 412 (100) 258 (52.2) 282 (47.8) 540 (100)

Various Relationships Graphed on Scatter Plots

Elementary Descriptive Statistics In a positive relationship, the scores vary together in the same direction and the slope of the line is from the 0 point of each variable to the upper right corner In a negative relationship, the scores vary inversely, in opposite directions with the 0 of the independent variable being at the upper left corner or high level of the dependent variable and dropping to the lower right corner of the graph The most common correlational statistical technique is Pearson’s Product Moment Correlation Coefficient. The outcome of the formula is an “r” value which indicates the degree of relationship between the two variables with +1 equaling perfect positive correlation and –1 equaling perfect inverse correlation