Quantitative Method

Concept

Quantitative research expresses phenomena in numbers and analyses them statistically. It is deductive (it starts with theory and hypotheses), uses structured instruments such as tests and questionnaires, draws large and preferably random samples, aims at objectivity and generalisation, and can be replicated. Its designs include descriptive surveys, correlational studies and experimental or quasi-experimental studies. Its strength is precision and the ability to test hypotheses; its weakness is limited attention to context and meaning.

Concept

S. S. Stevens (1946) classified measurement into four levels, remembered as NOIR. A nominal scale only classifies into categories (gender, religion); the mode and chi-square are suitable. An ordinal scale ranks (first, second, third) without equal gaps; the median, percentiles and rank correlation fit. An interval scale has equal intervals but an arbitrary zero (Celsius temperature, IQ, calendar year), so differences are meaningful but ratios are not; the mean, standard deviation and Pearson’s r are suitable. A ratio scale has equal intervals and a true zero (height, weight, income, age), so ratios are meaningful and every statistic can be used.

Concept

Measures of central tendency are the mean, median and mode; measures of dispersion are the range, quartile deviation, mean deviation, variance and standard deviation. In a perfectly normal distribution the curve is bell-shaped and symmetrical, mean = median = mode, and about 68%, 95% and 99.7% of cases lie within one, two and three standard deviations of the mean. In a positively skewed distribution mean > median > mode, and in a negatively skewed one mean \< median \< mode. Karl Pearson’s empirical relation for moderate skew is mode = 3 median − 2 mean. The correlation coefficient r ranges from −1 to +1; its square, r², is the proportion of variance shared. Spearman’s rho is used for ranks.

Concept

A Type I error (alpha) is rejecting a null hypothesis that is true, a false positive. A Type II error (beta) is failing to reject a null hypothesis that is false, a false negative. Power (1 − beta) is the probability of correctly rejecting a false null. The significance level (commonly 0.05) is the maximum Type I risk the researcher accepts. The p-value is the probability of obtaining a result at least as extreme as the one observed if the null hypothesis were true; if p is less than alpha, H0 is rejected. For a fixed sample size, lowering alpha raises beta.

Concept

Parametric tests assume interval or ratio data, approximate normality and equal variances. The t-test (William Gosset, writing as “Student”, 1908) compares means of one sample, two independent groups or paired observations, especially with small samples; the z-test is used with large samples. ANOVA (Fisher) compares three or more means through the F-ratio of between-group to within-group variance. The chi-square test (Pearson) analyses frequencies in categories, for goodness of fit or independence. Non-parametric alternatives include the Mann–Whitney U test (for the independent t-test), the Wilcoxon signed-rank test (for the paired t-test) and the Kruskal–Wallis test (for one-way ANOVA).