Quantitative and qualitative information — Quantitative data express counts or measured quantities; qualitative data preserve descriptions, meanings or categories in a contextual account. The collection method does not completely determine analysis: open responses can be coded into counts, and numerical patterns can prompt qualitative interpretation. State what information is being analysed.
Primary and secondary sources — Primary data are collected for the current investigation; secondary data were originally obtained for another purpose. Secondary evidence can provide scale and historical comparison while imposing limits from the original sampling and measurement. Primary collection gives control over some decisions without guaranteeing quality.
Meta-analysis — A meta-analysis systematically combines comparable study findings, often through effect estimates and their uncertainty. It is not simply a vote counting significant studies. Inclusion criteria, study quality, dependence and publication bias affect the pooled conclusion.
Mean, median and mode — The mean is the sum divided by the number of observations; the median is the central ordered value or average of the two central values; the mode is the most frequent value. Choose and interpret a measure in relation to the distribution and measurement scale rather than assuming one is always best.
A worked central-tendency example — For the hypothetical scores 2, 4, 4 and 10, the sum is 20 and n = 4, so the mean is 5. The middle ordered scores are both 4, giving median 4; mode is also 4. The high score pulls the mean above the other summaries.
Range and spread — Range is maximum minus minimum; for 2, 4, 4, 10 it is 10 − 2 = 8. It uses only the endpoints and can be strongly affected by one unusual value. Two samples can have the same mean and range while differing in their internal distribution.
Standard deviation — Standard deviation summarises dispersion around the mean using all observations and is expressed in the original measurement units. Larger values indicate more spread under the relevant comparison. A small standard deviation does not mean the mean is accurate or that every participant had the same score.
Percentages and proportion — A percentage expresses a part as a fraction of an appropriate total multiplied by 100. If 18 of 24 responses are correct, 18/24 × 100 = 75%. State the denominator: 75% of responses is not necessarily 75% of participants, and a percentage-point change differs from a relative percentage change.
Normal and skewed distributions — A normal distribution is symmetrical and unimodal (having one peak), with mean, median and mode coinciding in the ideal case. Positive skew has a longer high-value tail and commonly pulls the mean above the median; negative skew has a longer low-value tail. A histogram and context are more informative than naming skew from one summary alone.
Normal spread and caution — In the ideal normal model, roughly 68% of values lie within one standard deviation of the mean and roughly 95% within two. These model properties do not make any real dataset normal automatically. Very small samples may show irregular shapes even when the population model is reasonable.
Display and measurement
Tables and transparent units — A table should identify the variables, groups, units, totals and summaries clearly. Use appropriate precision and distinguish raw observations from aggregated scores. A total percentage without its denominator or a mean without its scale can mislead.
Bar charts and histograms — A bar chart compares discrete categories with separate bars. A histogram displays a continuous distribution using adjacent intervals; area represents frequency, so unequal-width classes require frequency density rather than treating every height as a count. Frequency density is frequency divided by class width. For 12 observations in an interval of width 4, density is 3; the bar's area is 4 × 3 = 12, showing why width matters. Label both axes and interval boundaries.
Scattergrams — A scattergram plots paired values of two co-variables and reveals direction, strength, unusual points and possible nonlinearity. A tight upward pattern supports a strong positive association; a tight downward pattern a strong negative one. The pattern does not by itself establish causation.
Correlation coefficients — A coefficient can summarise an association on a scale from −1 to +1 for the specified method. Sign indicates direction, while magnitude indicates strength within that method. A value near zero may conceal a nonlinear relation, so inspect the graph and what the chosen coefficient actually measures.
Nominal measurement — Nominal data classify observations into categories with no intrinsic numerical order. Counting how many people choose each category gives frequencies. Coding categories as 1 and 2 does not turn their labels into an interval scale.
Ordinal measurement — Ordinal data indicate order without establishing equal intervals between values. A ranked preference or many rating-scale items can provide ordered information. The difference between ranks 1 and 2 need not represent the same underlying difference as ranks 2 and 3.
Interval measurement — Interval measurement treats equal numerical differences as equal differences in the measured attribute. It supports arithmetic comparisons used in suitable parametric analyses, subject to other assumptions. Ratio measurement also has equal intervals but adds a meaningful zero, allowing statements such as twice as long: 20 seconds is twice 10 seconds. A zero on an interval scale need not mean absence of the attribute, as 0°C does not mean no temperature. The course's test-selection framework commonly groups interval and ratio measurements for these purposes.
Coding content — A coding scheme can turn material into quantitative frequencies or organise interpretation. Define the unit and category rules, check independent application and preserve context for ambiguous examples. Numbers created through coding still depend on the quality of the original definitions.
Sign test and significance
Inference and a null model — Inferential testing compares an observed result with what a specified null model would make sufficiently unusual. The null concerns no relevant population effect or association. A test does not turn a sample into certain knowledge about a population or correct a biased design automatically.
Probability and the p-value — A p-value concerns the probability, under the null model and assumptions, of obtaining a result at least as extreme as the one observed according to the test. It is not the probability that the null hypothesis is true. Its meaning depends on the design, test and assumptions.
Significance threshold — A threshold such as 0.05 is specified as a decision criterion. If a valid test gives p = 0.03 at that threshold, the result is statistically significant; p = 0.08 is not. The threshold is not a universal boundary between important and unimportant effects.
Type I and Type II errors — A Type I error rejects a true null; a Type II error fails to reject a false null. Reducing the threshold makes a Type I error less likely under the model but can increase missed effects if other features stay the same. For a null hypothesis of no effect of background speech on recall, Type I error means concluding that there is an effect when none exists; Type II means missing a real effect. Sample size, variability and effect size also affect detection.
Sign-test suitability — The sign test can test a difference with related observations when each nontied pair can be classified by the direction of change. It uses signs rather than the size of the differences. The independent units are the participants or pairs; unrelated conditions require a different design analysis.
Calculating signs — Choose a consistent direction, such as second score minus first. Assign plus or minus to each nonzero difference and exclude ties from N. Count the less frequent sign to obtain S under the usual table convention. Reversing the subtraction changes sign labels but not the smaller count.
A worked sign test — In a hypothetical set of 12 pairs, nine differences are positive, one negative and two zero. Excluding ties gives N = 10; the smaller nonzero sign count is S = 1. For a two-tailed 5% test with a supplied critical value of 1, S ≤ 1 is significant, so reject the no-directional-tendency null under the assumptions.A hypothetical worked example using the usual smaller-count convention and the stated supplied critical value. Reversing subtraction does not change S.
Direction and critical values — A one-tailed test requires a justified directional prediction and uses the matching table column; a two-tailed test concerns either direction. Use the correct N and threshold. For the usual sign-test convention, a small S is unusual, so compare S ≤ critical value rather than using the rule for a t statistic.
Interpreting a non-significant result — Failing to reject the null is not proof of no effect. Limited power, noisy measurement or a small sample can leave a real effect undetected. Report the observed pattern and uncertainty without claiming the study established equality.
Selecting and interpreting other tests
A selection sequence — First distinguish an association from a difference, then identify whether observations are related or independent, the level of measurement and relevant distributional assumptions. A test name alone does not justify a choice. Keep the sampling unit and dependence structure explicit.
Spearman's rho — Spearman's rho assesses an appropriate monotonic association using ranks and is suitable for ordinal information in the course's selection framework. A consistently increasing curved relation can be monotonic without being linear. Outliers, tied ranks and the actual design still need consideration.
Pearson's r — Pearson's r summarises a linear association between suitable quantitative variables, with relevant assumptions for significance testing. A strong curved relation can have a misleading linear coefficient. Independence, outliers and measurement need review rather than choosing Pearson merely because numbers appear in the data.
Wilcoxon and Mann-Whitney — Wilcoxon's signed-rank test compares related observations using ranked differences under its assumptions. Mann-Whitney compares independent groups using ranks. Their use differs from the sign test's direction-only treatment; do not describe Wilcoxon as ignoring difference magnitudes entirely.
Related and unrelated t-tests — A related t-test uses within-pair differences for suitable interval-level data; an unrelated t-test compares independent groups under the selected version's assumptions. Relevant normality and variance considerations must be checked rather than assuming every numeric scale permits a parametric test.
Chi-squared — A chi-squared association test compares observed categorical frequencies with expected frequencies under independence. It uses counts in mutually appropriate categories and requires suitable expected-frequency conditions. A table of percentages alone is not the raw frequency information needed, and related responses cannot be treated as independent counts without justification.
Observed and critical statistics — The comparison rule depends on the statistic and table convention. Conventional chi-squared and absolute t values become more unusual when large; conventional sign S, Wilcoxon T and Mann-Whitney U tables use sufficiently small values. Read the supplied table and tails rather than applying one inequality to every test.
Degrees of freedom and tables — For a chi-squared association table with r rows and c columns, df = (r − 1)(c − 1); a 2 by 3 table has df = 2. Other tests use different information such as sample size or paired counts. State the statistic, relevant N or df, threshold, tail convention and decision clearly.
Statistical and practical interpretation — A statistically significant association does not establish causation or a large useful effect. A small effect can become detectable in a large sample, while a practically important effect may remain uncertain in a small one. Integrate the graph, summaries, design, effect and limitations into the final claim.
Legacy requirement: thematic analysis
From data to themes — Thematic analysis identifies and interprets patterns of meaning within qualitative material. Familiarise yourself with the dataset, code relevant features, develop candidate themes, review their fit and define the final account. These stages involve revisiting data rather than a single mechanical pass. A theme is an interpreted pattern relevant to the research question, not necessarily the most frequent word.
Coding and interpretation — A code labels a meaningful feature of a passage; a theme brings together a wider pattern across codes and data. In an invented interview about revision, references to hiding uncertainty and avoiding questions might contribute to a theme of fear of judgment. The interpretation needs supporting extracts and attention to conflicting cases, not a label imposed regardless of evidence.
Inductive and deductive choices — An inductive approach develops patterns primarily from the material, while a deductive approach uses prior concepts to guide analysis. Neither removes interpretation or prior assumptions entirely. Explain the research question, coding decisions and analytic perspective so another reader can understand how claims were produced.
Thematic and content analysis — Content analysis may systematically count pre-defined categories, although it can also include qualitative interpretation. Thematic analysis focuses on patterns of meaning and their contextual interpretation. A table of word frequencies is therefore not by itself a thematic explanation; frequency and significance to participants need not be identical.
Evaluating qualitative analysis — Retain an audit trail linking extracts, codes, themes and conclusions, consider researcher reflexivity and examine alternative or discrepant cases. Independent coding can assess agreement where the chosen approach seeks shared coding, but consensus alone does not prove the only valid interpretation. Avoid generalising beyond the sample and protect identifiable narratives.
Numerical methods and worked calculations
Decimal and standard form — Standard form writes a number as a coefficient from 1 up to but not including 10 multiplied by a power of ten. A hypothetical reaction-time total of 2.4 × 10³ milliseconds is 2400 milliseconds; 7.2 × 10⁻⁴ seconds is 0.00072 seconds. Preserve units when changing notation: 0.35 seconds equals 350 milliseconds, not 0.35 milliseconds.
Ratios, fractions and percentages — A hypothetical group contains 18 category-A responses and 12 category-B responses. The ratio A:B is 18:12 = 3:2; A is 18/30 = 3/5 = 60% of the total. A is also 150% of B, which uses a different denominator. A rise from 20% to 30% is 10 percentage points but a 50% relative increase.
Estimation and order of magnitude — Check an answer's scale before trusting calculator output. A hypothetical total score of about 25 000 across roughly 500 participants suggests a mean near 50, not 0.5 or 5000. Rounding inputs can give a useful check but should not silently replace the exact calculation requested. A factor of ten changes order of magnitude; a small rounding change does not.
Precision and significant figures — Retain enough precision in intermediate steps and round the final value appropriately. The coefficient 0.7462 becomes 0.746 to three significant figures; leading zeroes do not count as significant digits. Report measurement units and avoid implying precision beyond the instrument or data. Rounding before a critical-value comparison can change a borderline decision, so use the supplied instructions.
Calculating standard deviation — Follow the supplied formula and identify its denominator. For a population-style descriptive formula, subtract the mean from each score, square the deviations, sum them, divide by N and take the square root. An estimate using the sample formula divides by N − 1 instead. The two formulas answer related but different questions; do not silently swap them or confuse variance with standard deviation.
Worked standard deviation — For the hypothetical scores 2, 4 and 6, the mean is 4. Deviations are −2, 0 and +2, so squared deviations are 4, 0 and 4, totalling 8. Using the supplied population-style denominator N = 3 gives variance 8/3 and standard deviation √(8/3) ≈ 1.63 score units. If the supplied formula instead uses N − 1, the result is √(8/2) = 2 score units. Show the formula used and keep the original measurement units.
Substitution and rearrangement — Translate a verbal relationship into the appropriate calculation. If mean = total/N, then total = N × mean: 24 participants averaging 6 correct answers contribute 144 correct answers in total. For a chi-squared table, substitute the actual row and column counts into df = (r − 1)(c − 1); knowing df and r can also allow c to be found algebraically. State which symbols represent counts, categories or measurement units.
Expected frequencies and a worked chi-squared substitution — For an association table under independence, an expected cell count is row total × column total / grand total. In a hypothetical 2 × 2 table with rows [20, 10] and [10, 20], each row and column total is 30 and the grand total is 60, so each expected count is 15. In χ² = Σ((O − E)²/E), O is an observed count, E its expected count and Σ means add the contributions from all cells. Each cell differs from its expected 15 by 5 or −5; squaring gives 25. Substitution gives four contributions of 25/15, hence χ² ≈ 6.67 and df = 1. With a supplied 5% critical value 3.84, this is significant under the assumptions. The result concerns association, not causation.
Reading symbols and graph scales — The symbols < and > mean less than and greater than; ≪ and ≫ indicate much less or much greater in an appropriate context. Proportionality (∝) is a stronger relationship than simply both quantities increasing, and ~ must be interpreted using its stated convention. When moving between a table, equation and graph, preserve units, paired observations and scale intervals. A truncated or logarithmic axis needs explicit interpretation rather than treating every visual step as the same raw increase.
Test yourself
50 questions · Random sets of 10. These quick checks support revision; practise longer explanations and justified judgements too.
Mr Kidman's Classroom
