Chapter 8 Example

Frequency Table

Time spent travelling to school – to the nearest 5 minutes (Sample of Y7s) Time Frequency Per cent Valid per cent Cumulative per cent 5.00 4 7.4 7.4 7.4 10.00 10 18.5 18.5 25.9 15.00 20 37.0 37.0 63.0 Valid 20.00 15 27.8 27.8 90.7 25.00 3 5.6 5.6 96.3 35.00 2 3.7 3.7 100.0 Total 54 100.0 100.0

Using Pie Charts Pie chart showing relationship of something to a whole

Children's Food Preferences

Other 28%

Chips 72%

Children's Food Preferences Fruit Ice Cream 2% 2% Biscuits 3%

Pasta 11%

Pizza 10%

Chips 72%

Using Bar Charts and Histograms Bar chart

Mode of Travel to School (Y7s) 14

6 mode of travel

0 walk car bus cycle other Ö

Number of students

0 0204060 80 100 Score on final exam (maximum possible = 100)

Median and Mean The median and mean of these two sets of numbers is clearly 50, but the spread can be seen to differ markedly

48 49 50 51 52 30 40 50 60 70

These are simple and commonly used methods for showing relationships between pairs of variables and are very similar to frequency tables in that they include percentage values, making for easy interpretation of the information conveyed. They can be used to show relationships between any types of variables, but are more usefully employed for the following combinations:

Two nominal variable categories Nominal and ordinal variable categories Nominal and interval/ratio variable categories

Contingency table showing relationship between gender and reported number of times per week engaged in walking to keep ﬁ t

Walking frequency zero x1/wk x2/wk x3/wk x4/wk x5/wk Total G Male Count 15 11 7 1 1 0 35 e % within gender 42.9% 31.4% 20.0% 2.9% 2.9% .0% 100.0% n Female Count 0 0 8 9 4 8 29 d e % within gender .0% .0% 27.6% 31.0% 13.8% 27.6% 100.0% r Total Count 15 11 15 10 5 8 64 % 23.4 17.2 23.4 15.6 7.8 12.5 100.0

This table shows a relationship between gender and walking for the purposes of exercise. It can be seen that in this hypothetical sample, females use walking as a method of exercise more often per week than males. Note that this relationship includes nominal (gender) and interval (equidis- tant frequency) level variables.

Measuring the Strength of a Linear Relationship between Two Variables Pearson’s Correlation Coefficient (Pearson’s r)

Correlation is method of investigating linear relationships between pairs of interval/ratio variables, for example age and time spent reading. Pearson’s Correlation Coefficient (Pearson’s r) is a measure of the strength of the association, if any, between two interval/ratio variables and takes on a value between +1 and –1, with 0 indicating no relationship. A correlation coefficient of +1 (positive correlation) indicates a perfect positive linear relationship between two variables: as one variable increases in value, so does the other, following an exact linear rule. In the case of age and reading a correlation coefficient of +1 would indicate that time spent reading Ö

© Mark O’Hara, Caron Carter, Pam Dewis, Janet Kay and Jonathan Wainwright 2011 O’Hara, M., Carter, C., Dewis, P., Kay, J., and Wainwright, J. (2011) Successful Dissertations. London: Continuum. increases with age. A correlation coefficient of –1 (negative correlation) on the other hand would denote the opposite, that is that time spent reading decreases with age. The closer the coefficient is to one (in both directions) the stronger the correlation so that the nearer to Zero the coefficient, the weaker the relationship. Correlations are often presented using scatter diagrams.

Table showing Pearson Correlation Coefﬁ cient of +1 Age Walking frequency Age Pearson Correlation 1 .969** Sig. (2-tailed) .000 N6464 Walking frequency Pearson Correlation .969** 1 Sig. (2-tailed) .000 N6464 **. Correlation is significant at the 0.01 level (2-tailed).

The footnote indicates that the linear relationship between the two variables in questions is statis- tically significant: that is, it is unlikely to have occurred by chance. When using SPSS to calculate correlation coefficients using Pearson’s r and other similar statistical tests (see below) statistical significance is automatically calculated. Spearman’s Rank Correlation Coefficient (Spearman’s rho)

Spearman’s rho is almost exactly the same as Pearson’s r in that it is used to denote the strength of a linear relationship between two variables taking on a value between +1 and –1. Spearman’s rho, however, is used where the pairs of variables are either both ordinal or where one variable is ordinal and the other interval/ratio. The strength of the relationship appears in a table like the Pearson’s r table above. Phi (Φ)

The phi coefficient is a measure of the degree of a relationship between two dichotomous variables (e.g. gender and a yes response where the possibilities were simply a yes or a no). Like Pearson’s r and Spearman’s rho, phi takes on a value between +1 and –1 and is also displayed in a table like the previous two examples. * SPSS can be used to produce scatter diagrams to give a graphical display of these linear relationships alongside the tables. Cramér’s V

Cramér’s V differs slightly to the tests above in that it is used to analyse the relationship between pairs of nominal variables and is only capable of a positive value. Thus it can only indicate the strength of an association between two variables and not the direction (i.e. it does not show a positive or negative association). The value of Cramér’s V is usually indicated on a contingency table (see above) rather than being presented on its own.