DESCRIPTIVE STATISTICS © LOUIS COHEN, LAWRENCE MANION & KEITH MORRISON

STRUCTURE OF THE CHAPTER Frequencies, percentages and crosstabulations Measures of central tendency and dispersal Taking stock Correlations and measures of association Partial correlations Reliability

FREQUENCIES AND PERCENTAGES Graphical forms of data presentation: –Frequency and percentage tables; –Bar charts (for nominal and ordinal data); –Histograms (for continuous – interval and ratio – data); –Line graphs; –Pie charts; –High and low charts; –Scatterplots; –Stem and leaf displays; –Boxplots (box and whisker plots).

FREQUENCIES AND PERCENTAGES Bar charts for presenting categorical and discrete data, highest and lowest; Avoid using a third dimension (e.g. depth) in a graph when it is unnecessary; a third dimension to a graph must provide additional information; Histograms for presenting continuous data; Line graphs for showing trends, particularly in continuous data, for one or more variables at a time; Multiple line graphs for showing trends in continuous data on several variables in the same graph;

FREQUENCIES AND PERCENTAGES Pie charts and bar charts for showing proportions; Interdependence can be shown through cross- tabulations; Boxplots for showing the distribution of values for several variables in a single chart, together with their range and medians; Stacked bar charts for showing the frequencies of different groups within a specific variable for two or more variables in the same chart; Scatterplots for showing the relationship between two variables or several sets of two or more variables on the same chart.

A crosstabulation is a presentational device. –Rows for nominal data, columns for ordinal data. –Independent variables as row data, dependent variables as column data. CROSSTABULATIONS

BIVARIATE CROSSTABULATION

TRIVARIATE CROSSTABULATION Acceptability of formal, written public examinations Traditionalist Progressivist/ child-centred Formal, written public exams Socially advantaged Socially disadvantaged Socially advantaged Socially disadvantaged In favour65%70%35%20% Against35%30%65%80% Total per cent 100%

MEASURES OF CENTRAL TENDENCY AND DISPERSAL The mode (the score obtained by the greatest number of people); –For categorical (nominal) and ordinal data The mean (the average score); –For continuous data –Used if the data are not skewed –Used if there are no outliers

MEASURES OF CENTRAL TENDENCY AND DISPERSAL 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); –For continuous data –Used of the data are skewed –Used if there are outliers

MEASURES OF CENTRAL TENDENCY AND DISPERSAL Standard deviation (the average distance of each score from the mean, the average difference between each score and the mean, and how much, the scores, as a group, deviate from the mean. –A standardized measure of dispersal. –For interval and ratio data

STANDARD DEVIATION The standard deviation is calculated, in its most simplified form as: or d 2 = 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.

9 8 Mean 7 | 6 | 5 | 4 | 3 | 2 | 1 XXXX|X Mean = 6 High standard deviation

9 8 Mean 7 | 6 | 5 | 4 | 3 | 2 | 1 XXXXX Mean = 6 Moderately high standard deviation

9 8 Mean 7| 6| 5| 4| 3X 2X 1XXX Mean = 6 Low standard deviation

THE RANGE AND INTERQUARTILE RANGE The range: –The difference between the minimum and maximum score. –A measure of dispersal. –Outliers exert a disproportionate effect. The interquartile range: –The difference between the first and the third quartile, the difference between the 25 th and the 75 th percentile, i.e. the middle 50 per cent of scores (the second and third quartiles). –Overcomes problems of outliers/extreme scores.

CORRELATION Measure of association between two variables. Note the direction of the correlation: –Positive: As one variable increases, the other variables increases –Negative: As one variable increases, the other variable decreases –The strongest positive correlation coefficient is +1. –The strongest negative correlation coefficient is -1.

CORRELATION Note the magnitude of the correlation coefficient: − 0.20 to 0.35: slight association − 0.35 to 0.65: sufficient for crude prediction − 0.65 to 0.85: sufficient for accurate prediction − >0.85: strong correlation Ensure that the relationships are linear and not curvilinear (i.e. the line reaches an inflection point)

CURVILINEAR RELATIONSHIP

CORRELATION Foot size Hand size12345 Perfect positive correlation: + 1

CORRELATION Foot size Hand size Perfect negative correlation: + 1

CORRELATION Hand sizeFoot size Positive correlation: <+1

PERFECT POSITIVE CORRELATION

PERFECT NEGATIVE CORRELATION

MIXED CORRELATION

CORRELATIONS Correlations –Spearman correlation for nominal and ordinal data –Pearson correlation for interval and ratio data

BIVARIATE CORRELATIONS Correlations –Spearman correlation for nominal and ordinal data –Pearson correlation for interval and ratio data

MULTIPLE AND PARTIAL CORRELATIONS Multiple correlation: –The degree of association between three or more variables simultaneously. Partial correlation: –The degree of association between two variables after the influence of a third has been controlled or partialled out. –controlling for the effects of a third variable means holding it constant whilst manipulating the other two variables.

RELIABILITY Split-half reliability (correlation between one half of a test and the other matched half) The alpha coefficient

SPLIT-HALF RELIABILITY (Spearman-Brown) Reliability = r = the actual correlation between the two halves of the instrument (e.g. 0.85); Reliability = = = (very high)

CRONBACH ALPHA Reliability as internal consistency: Cronbach’s alpha (the alpha coefficient of reliability). A coefficient of inter-item correlations. It calculates the average of all possible split half reliability coefficients.

INTERPRETING THE RELIABILITY COEFFICIENT Maximum is +1 >.90very highly reliable highly reliable reliable marginally/minimally reliable <.60unacceptably low reliability