This material is taught at both Standard Level and Higher Level. Only Higher Level assesses data analysis and interpretation.

Let the study and the data guide the analysis

Data analysis is a research process in which researchers organise and examine collected information to find patterns relevant to a research aim. Data interpretation is a reasoning process that explains those patterns in relation to the aim and the study's limitations. Analysis tells us what the results show. Interpretation tells us what we can reasonably conclude from them.

The choice of analysis depends on the nature of the data as well as the design of the study. You can't treat numerical scores as interchangeable with category counts or interview transcripts.

Quantitative data are numerical measurements or counts. Qualitative data take the form of descriptive material, such as participants' words. You can summarise a task-completion time numerically; to analyse a participant's account of experiencing the task, you need to preserve its meaning.

Numbers don't all do the same job, even in quantitative data. A number that labels a response category is not a measured quantity. Ranked responses tell us the order, but the gaps between ranks need not be equal. Measurements such as time allow meaningful comparisons of numerical differences. So a column containing numbers is not necessarily a column for which you should calculate a mean.

Study design matters too. Two separate groups' scores call for a different approach from scores collected from the same participants in two conditions. Paired observations are connected. Treating them as independent throws away that connection. If a study asks whether two measurements vary together, the analysis needs to examine a relationship rather than compare group averages.

Reading tables, graphs and results

A representation should reveal the relevant pattern without changing the meaning of the data. First, check what was measured and in which units. Identify the groups or conditions, and check whether you are looking at individual observations or summaries. Describe the pattern before trying to explain it.

A frequency table records how often each value, interval or category occurs in a tabular display. Use it when exact counts matter. A larger count may simply come from a larger group, though. When group sizes differ, proportions can be more informative.

A bar graph uses separate bars to represent values for distinct categories or groups. It can compare frequencies or group summaries. By contrast, a histogram uses adjoining bars to show how numerical observations are distributed across intervals. Use a bar graph for separate categories and a histogram for intervals of a numerical measurement.

A distribution is a pattern showing how observations spread across possible values. In a normal distribution, observations cluster around the centre of a symmetrical, bell-shaped distribution. A skewed distribution is asymmetrical, with a longer tail on one side. Look at the tail to determine the direction of skew, not at the side with the tallest bars.

A line graph joins successive plotted values to show change across an ordered variable, commonly time. It suits a study following a group's performance over successive sessions. If you connect unrelated response categories, however, you imply an order or continuity that the data don't have.

A scatterplot plots paired numerical observations to show the relationship between two variables. Each point stands for one pair of observations, often from one participant. Check the direction and strength of the relationship. Look too for unusual observations and whether the pattern is roughly straight or curved.

An outlier is an observation unusually far from the main pattern in the data. It may reflect a genuine individual difference or a recording error; an unusual circumstance is another possibility. Investigate it. Don't automatically delete the participant.

Keep observations separate from explanations when interpreting a display. “The second group has a higher average score” describes a result. “The intervention caused improvement” claims causation and needs support from the study design. Check the axis scales as well: restricting the vertical scale can make a small difference look dramatic.

Descriptive and inferential statistics do different jobs

Descriptive statistics are numerical summaries of the characteristics of the observed data. They describe features such as a typical score and how widely scores are spread.

Inferential statistics are statistical procedures that use sample data to evaluate evidence about differences or relationships beyond the observed sample. They help assess whether a pattern is compatible with a specified statistical explanation. This assessment relies on assumptions about the data and sampling.

Keep the two jobs separate. A descriptive difference between two sample means does not, by itself, establish a statistically significant difference. Equally, a statistically significant result doesn't tell us whether the difference is large or important in everyday life.

Connect the descriptive pattern and the inferential result to the research aim when interpreting findings. Statistics cannot repair biased sampling or poor measurement. Nor can they repair a design that cannot establish causality.

Thematic analysis: finding patterns in textual data

Thematic analysis is a qualitative analysis method that identifies, develops and interprets patterns of meaning in textual data. Researchers use it to develop themes that help answer the research question, not simply to count frequently used words.

A code is an analytic label attached to a segment of data to identify something relevant to the research aim. A theme is a coherent pattern of shared meaning across the data that addresses an aspect of that aim. Think of the distinction this way: a code labels a particular feature, while a theme develops a broader interpretation.

The stages are usually set out in sequence. In practice, researchers often move backwards and forwards as their understanding develops:

1. Become familiar with the data. Read and reread the material. Check transcripts where necessary and note initial ideas. Interpret comments in context rather than picking out isolated phrases.
2. Generate initial codes. Work systematically across the dataset, labelling relevant segments. Give a passage more than one code if it contains several relevant meanings.
3. Develop candidate themes. Look at how codes connect, then group them into possible patterns relevant to the research aim.
4. Review the themes. Check each candidate theme against both its supporting extracts and the wider dataset. You may need to combine or separate themes, or discard them. Don't hide contradictory accounts just because they complicate the pattern.
5. Define and name the themes. Specify what each theme captures and where its boundaries lie. Make clear how it differs from other themes.
6. Produce the account. Explain the themes in relation to the research aim and use selected extracts as evidence. Provide an interpretation, not a string of quotations.

The researcher's perspective shapes coding and theme development. Careful analysis makes the route from extracts to codes to themes transparent. Judge a theme's importance by its relevance and explanatory value, not just by how many participants mention it.

Central tendency: what counts as a typical score?

Central tendency is a characteristic of a dataset represented by a value that describes its centre or a typical observation. The mean, median and mode do this in different ways.

To calculate the mean, a measure of central tendency, divide the sum of the scores by the number of scores. Every observation contributes, which makes it sensitive to unusually high or low values. It is useful for numerical measurements when the average meaningfully summarises the distribution.

The median is a measure of central tendency given by the middle score once observations are ordered. With an even number of observations, it is the mean of the two middle scores. Extreme values affect it less than they affect the mean, so it can be useful for skewed numerical data or ordered responses.

The mode is a measure of central tendency given by the value or category that occurs most frequently. It can describe categorical data, for which a mean would make no sense. There may be more than one mode in a dataset, or no uniquely most frequent value.

Take five completion times: 12, 13, 14, 14 and 47 seconds. The mean is 20 seconds. Both the median and the mode are 14 seconds. Although the mean is correct, the unusually long time pulls it upwards. Calling a value an “average” without specifying the measure can therefore mislead.

In a symmetrical, single-peaked distribution, the mean, median and mode are typically close together. A long upper tail often puts the mean above the median; a long lower tail often puts it below. These are distribution patterns, not rules that every small sample must obey.

Dispersion: how much do the scores vary?

Dispersion is a characteristic of a dataset that describes how widely its observations are spread. The same central tendency can occur in two groups with substantially different variability.

The range is a measure of dispersion found by subtracting the lowest score from the highest. It's straightforward, but only the two extremes contribute. A single unusual observation can change it substantially.

The standard deviation is a measure of dispersion based on the squared deviations of scores from their mean. Its units are the same as those of the original measurements. A smaller standard deviation shows tighter clustering around the mean; a larger one shows greater spread. It describes how observations vary, not how accurate the measuring instrument is.

The semi-interquartile range is a measure of dispersion equal to half the difference between the upper and lower quartiles. Quartiles are dividing values that split ordered data into four approximately equal parts. By describing the spread of the middle half of the observations, the semi-interquartile range is less sensitive to extreme values than the range.

Mean and standard deviation often work together to summarise numerical data. For a skewed distribution or one containing extreme observations, median and semi-interquartile range can give a more representative description. Choose measures that fit the data, not measures that produce the conclusion you prefer.

A box and whisker plot summarises an ordered numerical distribution using its median, quartiles and whiskers. The middle half of the observations falls within the box. Check the whisker convention: whiskers may show the extremes or follow another stated rule. The groups in the comparison below have the same median but different spreads. Typical performance and consistency are separate findings. Five-number summaries show equal medians but greater dispersion in Group B; whiskers represent minima and maxima.

Inferential tests: matching the analysis to the comparison

A null hypothesis is a statistical statement specifying no population difference or relationship of the kind being tested. Under its assumptions, an inferential test evaluates how compatible the observed evidence is with that statement.

Noticing that there are two groups isn't enough to choose a test. Are the observations independent or paired? Are you working with measurements, ranks or category counts? Check whether the proposed test's assumptions are reasonable too.

The following tests serve different purposes:

• A chi-square test is an inferential test that compares observed category frequencies with the frequencies expected under a null hypothesis. For two groups, it can assess whether their distributions across response categories differ. It analyses counts, not a difference between mean scores.
• An unrelated t-test is an inferential test that evaluates a difference between the means of two independent groups, using numerical data and assumptions about their distributions. Different participants make up the groups, and their observations are independent.
• A related t-test is an inferential test that evaluates the mean difference between paired numerical observations. Observations may be paired because they come from the same participants in two conditions or from matched participants. The distributional assumption concerns the paired differences.
• A Mann–Whitney test is a rank-based inferential test comparing the distributions of observations from two independent groups. It can be appropriate for ordered data or for numerical data that do not suit the assumptions of an unrelated t-test. To interpret it specifically as a test of medians, appropriate assumptions about the distributions' shapes must hold.
• A Wilcoxon signed-rank test is a rank-based inferential test that evaluates paired differences using their directions and ranked magnitudes. Use it for paired data where this analysis and its assumptions are appropriate, rather than for independent groups.

Comparing task scores from two separate groups, for example, requires an independent-groups analysis. A comparison of each participant's score before and after a change requires a paired analysis. Even if the research aims look similar, the structure of the observations changes the test.

When reading a reported test result, identify what was compared and link it to the descriptive findings. A significant test result alone may not tell you which group scored higher. For that direction, look at the group summaries. If the report supplies a critical value rather than a probability, follow the stated decision rule for that particular test. “Larger than the critical value” is not the rule for every test.

Relationships and correlation coefficients

A correlation coefficient is a numerical statistic that summarises the direction and strength of a specified form of relationship between two variables. The usual coefficients are dimensionless and range from $-1$ to $+1$.

With a positive coefficient, higher values of one variable tend to occur with higher values of the other. With a negative coefficient, higher values of one tend to occur with lower values of the other. The closer a value is to either endpoint, the stronger the relationship of the form being measured. Don't confuse sign with strength: $-0.80$ represents a stronger relationship than $+0.30$.

A coefficient near zero indicates little relationship of the form that coefficient assesses. It need not mean there is no relationship. A coefficient measuring a straight-line relationship may poorly represent a strong curved pattern.

Correlation does not establish causation. One variable might influence the other, or the direction of influence might be reversed. Another factor might influence both. A statistically significant correlation doesn't rule out these alternatives.

Statistical significance and effect size

Statistical significance is a property of a test result that meets a specified decision threshold for evidence against the null hypothesis.

A p-value is the probability of obtaining a result at least as extreme as the observed result, calculated assuming that the null hypothesis and the test's assumptions hold. It is not the probability that the null hypothesis is true. Nor is it simply “the probability that the findings happened by chance”.

Suppose the stated significance threshold is 0.05. A p-value of 0.03 meets it; a p-value of 0.12 does not. Under that decision rule, the first result supports rejecting the null hypothesis. For the second, the evidence does not meet the threshold. That does not prove there is no difference or relationship.

Effect size is a quantitative measure of the magnitude of a difference or relationship. It asks “How large is the pattern?” Statistical significance instead addresses evidence against a null hypothesis under a statistical model.

Sample size and variability influence statistical significance, as does effect magnitude. That's why the distinction matters. A large sample can yield a statistically significant result for a small effect. In a small or highly variable sample, a substantial observed effect may fail to reach significance. Don't interpret either result from the p-value alone.

Some effect sizes use the original measurement units. Others are standardised or dimensionless. Their practical importance depends on the research context: a small difference in a brief laboratory task may mean something different from a similarly standardised difference in everyday functioning.

To interpret a result soundly, consider the direction and magnitude of the result alongside the variation in the data. Bring in the inferential evidence and the limits of the study design too. Rather than stopping at the label “significant”, explain what the evidence supports—and what it leaves unresolved.