1. Social Research Methods
Alan Bryman
Social Research Methods
Chapter 15: Quantitative data analysis

2. Introduction
Think about data analysis at an early stage in the research process
Decisions about methods and sample size affect the kind of analysis you are able to do
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3. Types of variable Interval/ratio Ordinal Nominal Dichotomous
regular distances between all categories in range
Ordinal
categories can be ranked, but unequal distances between them
Nominal
qualitatively different categories - cannot be ranked
Dichotomous
only two categories (e.g. gender)
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4. Deciding how to categorize a variable
Figure 15.1
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5. Univariate analysis (analysis of one variable at a time)
Frequency tables
Number of people or cases in each category
Often expressed as percentages of sample
Interval/ratio data needs to be grouped
Diagrams
Bar chart or pie chart (nominal or ordinal variables)
Histogram (interval/ratio variables)
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6. A bar chart (gym study)
Figure 15.2
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7. A pie chart
Figure 15.3
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Main reasons for visiting the gym

8. A histogram
Figure 15.4
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9. Measures of central tendency
Mean
Sum all values in distribution, then divide by total number of values
Median
Middle point within entire range of values
Not distorted by outliers
Mode
Most frequently occurring value
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10. Measures of dispersion
Dispersion is the amount of variation in a sample
Measures of dispersion compare levels of variation in different samples to see if there is more variability in one sample’s variable than in another sample
The range is the difference between the minimum and maximum values in a sample
The standard deviation is the average amount of variation around the mean, reducing the impact of extreme values (outliers)
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11. Bivariate analysis (analysis of two variables at a time)
Explores relationships between variables
Searches for co-variance and correlations
Cannot establish causality
Can sometimes infer the direction of a causal relationship
If one variable is obviously independent of the other
Contingency tables
Connects the frequencies of two variables
Helps you identify any patterns of association
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12. Pearson’s r : relationship between two interval/ratio variables
Coefficient shows the strength and direction of the relationship
Lies between -1 (perfect negative relationship) and +1 (perfect positive relationship)
Relationships must be linear for the method to work, so, plot a scatter diagram first
Coefficient of determination
Found by squaring the value of r
Shows how much of the variation in one variable is due to the other variable?
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13. Analysing the relationships between other, or mixed types of, variables
Spearman’s rho: for the relationship between two ordinal variables, or one ordinal and one interval/ratio variable (values of -1 to +1)
Phi coefficient: for the relationship between two dichotomous variables (values of -1 to +1)
Cramer’s V: for the relationship between two nominal variables, or one nominal and one ordinal variable (values between 0 and 1)
Comparing means: when a nominal variable is identified as the independent variable, the means of the interval/ratio variable are compared for each sub-group of the nominal variable
eta: for the level of association between different types of variables, even when there is no linear relationship between them
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14. Multivariate analysis (three or more variables)
The relationship between two variables might be spurious
Each variable could be related to a separate, third variable
There might be an intervening variable
A third variable might be moderating the relationship
e.g. correlation between age and exercise could be moderated by gender
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15. Example of a spurious relationship
Figure 15.11
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16. Statistical significance
How confident can we be that the findings from a sample can be generalised to the population as a whole?
How risky is it to make this inference?
Only applies to probability samples
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17. Testing procedure for statistical significance
Set up a null hypothesis - suggesting no relationship between examined variables in the population from which the sample was drawn
Decide on an acceptable level of statistical significance
Use a statistical test
If acceptable level attained
-reject null hypothesis
If acceptable level not attained
-accept it
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18. …but we might be wrong to accept or reject the null hypothesis
Type I and Type II errors
Figure 15.12
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19. Tests of statistical significance
The chi-square test
establishes how confident we can be that there is a relationship between the two variables in the population
Correlation and statistical significance
provides information about the likelihood that the coefficient will be found in the population from which the sample was taken
Comparing means and statistical significance – the F statistic
expresses the amount of explained variance in relation to the amount of error variance
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20. The chi-square test
The chi-square (2) test is applied to contingency tables. It establishes how confident we can be that there is a relationship between the two variables in the population. The test calculates, for each cell in the table, an expected frequency or value - one that would occur on the basis of chance alone. The chi-square value is determined by calculating the differences between the actual and expected values for each cell and then summing those differences.
Whether a chi-square value achieves statistical significance depends not just on its magnitude, but also on the number of categories of the two variables being analysed. This latter issue is governed by what is known as the ‘degrees of freedom’ associated with the table.
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21. Correlation and significance
How confident can we be about a relationship between two variables?
Whether a correlation coefficient is statistically significant depends on:
the size of the coefficient (the higher the better)
the size of the sample (the larger the better)
e.g. if coefficient is 0.62 and p<0.05, we can reject the null hypothesis
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22. Comparing means
Statistical significance of relationship between two variables means
Total variation in dependent variable:
error variance (variation within subgroups of IV)
explained variance (variation between subgroups of IV)
F statistic
expresses amount of explained variance in relation to amount of error variance
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