Statistics assignments test three different skills at once: choosing the right method, carrying it out correctly, and explaining what the result means in plain language.
Students often manage one or two and lose marks on the third, usually the interpretation. A correct p-value with the wrong conclusion still loses most of the marks.
This guide covers the topics that appear in most introductory and intermediate statistics courses, with worked examples, decision tables and reporting templates. Use it to check your approach before you submit, or to see where help would be most useful.
A method for any statistics question
Identify what the question asks (describe, estimate, compare, or relate), check the type of each variable, choose the method that matches, check its assumptions, and calculate. Then interpret: state the result in the context of the question, include the test statistic, degrees of freedom, p-value and an effect size or confidence interval, and avoid claiming more than the data shows.
Topics Students Ask About Most
| Topic | Typical tasks | Common sticking point |
|---|---|---|
| Descriptive statistics | Mean, median, standard deviation, frequency tables, histograms | Choosing median over mean for skewed data |
| Probability | Rules of probability, conditional probability, Bayes' theorem | Confusing P(A|B) with P(B|A) |
| Distributions | Normal, binomial, z-scores | Using the right table or function |
| Sampling and confidence intervals | Standard error, margin of error, CI for a mean or proportion | Interpreting what 95% confidence means |
| Hypothesis testing | Null and alternative hypotheses, p-values, Type I and II errors | Wording the conclusion correctly |
| Comparing groups | t-tests, ANOVA, non-parametric alternatives | Independent vs paired designs |
| Categorical data | Chi-square tests | Expected counts and small samples |
| Relationships | Correlation, simple and multiple regression | Correlation is not causation; interpreting coefficients |
Choosing the Right Method
| The question asks you to... | Data | Method |
|---|---|---|
| Summarize one variable | Continuous | Mean and SD (or median and IQR if skewed), histogram |
| Summarize one variable | Categorical | Counts and percentages, bar chart |
| Estimate a population value | Mean or proportion | Confidence interval |
| Compare two independent groups | Continuous outcome | Independent t-test (Mann-Whitney U if assumptions fail) |
| Compare before and after | Same subjects twice | Paired t-test (Wilcoxon signed-rank) |
| Compare three or more groups | Continuous outcome | One-way ANOVA with post hoc tests (Kruskal-Wallis) |
| Test association of two categorical variables | Counts | Chi-square test of independence |
| Measure a linear relationship | Two continuous variables | Pearson correlation (Spearman for ranks) |
| Predict one variable from others | Continuous outcome | Linear regression |
Hypothesis Testing Without the Confusion
Every hypothesis test follows the same five steps, whatever the method.
- State the hypotheses. H0: no difference or no relationship. H1: there is a difference or relationship (one- or two-tailed, as the question specifies).
- Choose the significance level, usually α = .05.
- Check assumptions and calculate the test statistic.
- Find the p-value (or compare the statistic with the critical value).
- Conclude in context. If p < α, reject H0; otherwise, fail to reject it. Then say what that means for the question.
Two wording rules save marks. Never say you "accept" or "prove" the null hypothesis; you fail to reject it. And a p-value is not the probability that the null hypothesis is true; it is the probability of results at least as extreme as yours if the null hypothesis were true.
One-Tailed or Two-Tailed?
Use a two-tailed test unless the question clearly states a direction before looking at the data ("test whether the new method increases scores").
A one-tailed test has more power in the stated direction but cannot detect an effect in the opposite direction, and choosing it after seeing the data is not legitimate. When in doubt, and in most research, two-tailed is the safer and expected choice.
Type I and Type II Errors
| H0 is actually true | H0 is actually false | |
|---|---|---|
| Reject H0 | Type I error (false positive), probability α | Correct decision (power = 1 − β) |
| Fail to reject H0 | Correct decision | Type II error (false negative), probability β |
Lowering α to reduce false positives increases the risk of false negatives unless you increase the sample size. Questions often ask you to describe each error in the context of the scenario, for example "concluding the drug works when it does not" for a Type I error.
Worked Example: Independent T-Test
Question: A lecturer compares exam scores of students who attended revision sessions (n = 25, M = 68.4, SD = 9.2) with those who did not (n = 25, M = 62.1, SD = 10.5). Is the difference significant at α = .05?
Hypotheses: H0: μ1 = μ2; H1: μ1 ≠ μ2.
Calculation: pooled SD ≈ 9.87; standard error ≈ 9.87 × √(1/25 + 1/25) ≈ 2.79; t = (68.4 − 62.1) / 2.79 ≈ 2.26, with df = 48. The two-tailed p-value is about .03.
Effect size: d = 6.3 / 9.87 ≈ 0.64, a medium effect.
Conclusion: Students who attended revision sessions scored significantly higher than those who did not, t(48) = 2.26, p = .03, d = 0.64. Because students chose whether to attend, the result shows an association, not that the sessions caused higher scores.
The final sentence is where many answers lose marks. Observational data, where groups were not randomly assigned, cannot establish causation.
Worked Example: Conditional Probability and Bayes' Theorem
Question: A screening test detects a condition 95% of the time when it is present (sensitivity) and gives a false positive 5% of the time when it is absent. The condition affects 2% of the population. If a person tests positive, what is the probability they have the condition?
Set up: P(D) = 0.02, P(+|D) = 0.95, P(+|not D) = 0.05.
Total probability of a positive test: P(+) = 0.95 × 0.02 + 0.05 × 0.98 = 0.019 + 0.049 = 0.068.
Bayes' theorem: P(D|+) = 0.019 / 0.068 ≈ 0.28.
Interpretation: Only about 28% of people who test positive actually have the condition, because it is rare and false positives from the large healthy group outnumber true positives.
This result surprises most students, and it is why examiners like the question. Drawing a tree diagram or a table of 10,000 imaginary people makes it intuitive: 200 have the condition, 190 test positive; 9,800 do not, and 490 of those test positive.
Confidence Intervals, Interpreted Correctly
A 95% confidence interval gives a range of plausible values for a population parameter. If you repeated the sampling many times, about 95% of the intervals built this way would contain the true value.
| Wording | Correct? |
|---|---|
| We are 95% confident that the mean commute time for the population is between 31 and 38 minutes. | Yes (standard course wording) |
| There is a 95% probability that the true mean is between 31 and 38 minutes. | No, the true mean is fixed; the interval varies |
| 95% of commuters travel between 31 and 38 minutes. | No, the interval is about the mean, not individuals |
Worked Example: Chi-Square Test of Independence
Question: Is there an association between study mode (online or on campus) and passing the module? Of 80 online students, 56 passed; of 120 on-campus students, 102 passed.
Expected counts: overall pass rate = 158/200 = 0.79. Expected passes: online 63.2, on campus 94.8; expected fails: online 16.8, on campus 25.2. All expected counts are above 5, so the test is appropriate.
Statistic: χ² = Σ (O − E)² / E ≈ 0.82 + 0.55 + 3.09 + 2.06 ≈ 6.51, df = 1, p ≈ .011.
Conclusion: There is a significant association between study mode and passing, χ²(1, N = 200) = 6.51, p = .011. On-campus students passed at a higher rate (85%) than online students (70%). The test shows association only; students chose their mode, so other differences between the groups may explain the result.
Correlation and Regression
Correlation (r) measures the strength and direction of a linear relationship, from −1 to +1. Regression goes further: it gives an equation to predict one variable from another and tells you how much the outcome changes for each unit change in the predictor.
In simple regression, the slope is the expected change in Y for a one-unit increase in X, and R² is the proportion of variation in Y explained by the model.
In multiple regression, each coefficient is interpreted holding the other predictors constant. Always plot the data first: a correlation near zero can hide a strong curved relationship, and one outlier can create a misleading correlation. Our regression analysis guide covers assumptions and interpretation in depth.
Excel, SPSS, R or by Hand?
| Tool | Good for | Note |
|---|---|---|
| By hand or calculator | Exams and early courses where working must be shown | Show each step and formula |
| Excel | Business and introductory courses; Data Analysis ToolPak | Limited assumption checks |
| SPSS | Social sciences, psychology, nursing, education | Report from output tables, not screenshots |
| R or Python | Statistics, data science, reproducible work | Submit code if required, with comments |
| Minitab, Stata, JMP | Engineering, economics, quality improvement | Follow your course's conventions |
Reporting Results the Way Markers Expect
| Test | APA-style template |
|---|---|
| Independent t-test | t(df) = value, p = value, d = value |
| One-way ANOVA | F(df between, df within) = value, p = value, η² = value |
| Chi-square | χ²(df, N = value) = value, p = value, Cramér's V = value |
| Correlation | r(df) = value, p = value |
| Regression | R² = value, F(df1, df2) = value, p = value; b = value, SE = value, p = value for each predictor |
Round to two decimal places for most statistics, report exact p-values to two or three decimals (p < .001 for very small values), and italicize statistical symbols if your course follows APA style.
Mistakes That Cost Marks
- Using an independent t-test for paired data, or the reverse.
- Ignoring assumptions such as normality or equal variances.
- Reporting "p = .000" instead of p < .001.
- Concluding causation from correlational or observational data.
- Answering with numbers only and no interpretation in context.
- Rounding too early in calculations, which changes the final answer.
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Frequently Asked Questions
Yes. Tell us which software your course uses. You receive the output or code along with the interpretation, so you can see exactly how each result was produced.
Yes. For calculation questions, every step is shown with the formula used. For software-based questions, you receive the output, syntax or code, and an explanation.
Interpretation is included. Results are explained in the context of the question, with correct statistical wording and APA-style reporting where required.
Yes, including conditional probability, Bayes' theorem, combinations and permutations, and probability distributions such as binomial and normal.
A t-test compares the means of two groups; ANOVA compares three or more. Running several t-tests instead of an ANOVA increases the chance of a false positive.
Yes. Send your attempted solutions and a tutor will check each answer, explain any mistakes and show the correct approach.
Look at a histogram and a Q-Q plot first, then check skewness and a formal test such as Shapiro-Wilk. In large samples formal tests flag trivial departures, so rely more on the plots and on the robustness of the test you are using.
It means the result would be unlikely if the null hypothesis were true, judged against your chosen significance level (usually .05). It does not mean the effect is large or important, which is why effect sizes are reported alongside p-values.