Statistics Guide: From Data Scales to Your Results Chapter

Nearly every decision in statistics traces back to one question: what scale was this variable measured on? Stevens described four levels of measurement. Nominal variables are unordered categories such as gender, region of origin or degree programme. Ordinal variables are ranked categories such as low, medium and high satisfaction. Interval variables are numbers with equal spacing and no true zero, temperature in Celsius being the classic case. Ratio variables have a meaningful zero, so weight, income and minutes of study all qualify.

Questionnaire data is where most learners get stuck. A single five-point Likert item is technically ordinal, because the gap between agree and strongly agree may differ from the gap between neutral and agree. Many studies add several items into a total score and treat that score as numeric. The choice deserves a written note and a conversation with the supervisor, since it decides whether later analysis works with means or with ranks. When items are combined, item validity and reliability, often measured with Cronbach's alpha, are usually checked first.

The most useful drill at this stage is a two-column table: variable names on the left, scales on the right, plus a one-line reason for each. Take any questionnaire, from a canteen satisfaction survey to your own research instrument, and fill in the table without looking at your notes. Measurement scales are settled when a learner can explain why a student ID number is nominal even though it is written in digits.

Probability is the bridge between describing data and drawing inferences from it. At school level the topic covers sample spaces, events, permutations, combinations and compound probability. At university it grows into distributions: the binomial distribution for counting successes across a fixed number of trials, and the normal distribution for continuous data that forms a symmetrical bell around its mean.

The normal distribution follows a pattern worth understanding properly. Roughly 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and around 99.7 percent within three. A z-score converts a raw value into its distance from the mean measured in standard deviations, which lets learners compare results from two tests marked on different scales. The central limit theorem then explains why so many tests work: means taken from reasonably large samples tend towards a normal distribution even when the underlying data are skewed. A common rough guide puts that sample size at about 30.

Simulation is the quickest way to make this click. In R or a spreadsheet, draw 1,000 random samples of size 5 and then of size 30 from a skewed dataset, compute each sample's mean, and plot a histogram of those means. The change in shape is plain to see. A learner has this part covered once they can explain why conclusions about a population may be drawn from a sample, and how much uncertainty comes with them.

A p-value is the probability of getting data as extreme as those observed, or more extreme, if H0 were true. In SPSS output it appears in the Sig. column. The decision rule is simple: when p falls below the alpha set in advance, usually 0.05 in the social sciences and economics, H0 is rejected. When p is equal to or above alpha, the researcher fails to reject H0. Fields that demand tighter control sometimes use an alpha of 0.01.

This topic hides several misreadings that examiners like to probe. A p-value measures how surprising the data would be under H0, and the probability that H0 itself is true lies outside what the test can tell you. A p-value of 0.04 means the evidence has crossed the 0.05 line, and the size of the effect has to be read from an effect size measure. A non-significant result means the evidence fell short, so the correct wording is failing to reject H0. In 2016 the American Statistical Association published a statement on the limits of p-values, and it is worth reading once before writing up results.

Two kinds of decision error complete the picture. A Type I error rejects an H0 that is actually true, and its probability equals alpha. A Type II error fails to reject an H0 that is actually false, and its probability shrinks as the sample grows. Statistical power equals one minus the probability of a Type II error, and Cohen recommended a power of at least 0.80. That is why a power analysis, often run in G*Power, is commonly used to set the sample size in the methods chapter.

Parametric tests such as the t-test, ANOVA and Pearson correlation assume numeric data that are close to normally distributed, with roughly equal variances across groups. Normality is usually checked with Shapiro-Wilk for small to medium samples, or with Kolmogorov-Smirnov, and a significance value above 0.05 suggests no meaningful departure from normality. Equal variances are checked with Levene's test. In SPSS output for an independent samples t-test, Levene's result tells you which row to read: equal variances assumed or equal variances not assumed.

When the assumptions fail, especially with small samples, each main test has a non-parametric counterpart. Mann-Whitney stands in for the independent samples t-test, Wilcoxon for the paired t-test, Kruskal-Wallis for one-way ANOVA, and Spearman for Pearson correlation. These tests work on ranks, so they cope better with outliers and suit ordinal data such as a single Likert item. A Q-Q plot complements the formal tests: points that hug the diagonal line indicate a roughly normal distribution. With very large samples a normality test can turn significant over tiny deviations, which is why the plot carries weight too.

The drill uses two routes on one dataset. Run a t-test and a Mann-Whitney test on the same data, compare the conclusions, and write a short paragraph on which route you chose and why. Report the assumption checks in the methods chapter or an appendix. A learner is ready once they can read a Q-Q plot and a Levene's output and settle on the right route with confidence.

Software handles the arithmetic while the learner keeps hold of the analytical decisions. JASP and Jamovi are free, menu-driven and produce tidy tables ready for a report. SPSS is widespread at universities and usually needs a licence. R is free and the most flexible of the four, with a steeper learning curve because analyses are written as commands. The guide to learning statistics for a thesis in this hub gives clear advice: pick one tool and get fluent in it before trying another.

A healthy workflow starts with clean data: one row per respondent, one column per variable, short variable names without spaces, and category codes recorded on a separate key sheet. After that the order stays the same for every study: clean the data and check for missing values, run descriptive statistics, check assumptions, run the main test, then export the tables. Keep a record of each step in notes or a syntax file so the analysis can be repeated without rebuilding it from memory. In R, the script file doubles as that complete record.

A 2013 review by Dunlosky and colleagues ranked spaced practice and retrieval practice among the study techniques with the strongest evidence behind them. For statistics this means three short sessions a week, each built around one small analysis and closed by explaining the result out loud without looking at your notes. Start with sample data well before your own data come in. That guide estimates roughly 6 to 10 weeks to build applied statistics skills this way, depending on your starting point and how complex the analysis is. Statistics anxiety, which Onwuegbuzie and Wilson found affects most postgraduate students, also becomes easier to manage when the material is broken into pieces this small.