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Collecting data is only the beginning. A column of raw scores tells a psychologist almost nothing until it is organised, summarised and displayed: we need to capture the typical value, the spread of values, the shape of the distribution, and the right picture of the pattern. This is the work of descriptive statistics, and it is a heavily examined part of Edexcel Paper 3 — not because the arithmetic is hard, but because the exam rewards judgement: choosing the right measure for the level of data, reading a graph correctly, and interpreting what a mean and standard deviation together reveal about a study. This lesson builds that judgement. It covers the three levels of measurement (nominal, ordinal, interval), the measures of central tendency (mean, median, mode) and dispersion (range, standard deviation), how to present data in tables and graphs (bar chart, histogram, scattergram), and the difference between normal and skewed distributions — the descriptive foundation on which the next two lessons (inferential testing and test choice) are built.
Key Definition: Descriptive statistics are methods of organising, summarising and displaying quantitative data — using measures of central tendency, measures of dispersion and graphical displays — so that patterns in the data can be seen and communicated. They describe a sample; they do not, by themselves, test whether a result is significant.
By the end of this lesson you will be able to:
Edexcel 9PS0 — Paper 3: Psychological Skills (Research Methods). This lesson develops the descriptive-statistics and data-presentation content assessed in Section A of Paper 3, where numerical skills are tested in the context of a described study — typically by asking candidates to select, calculate or interpret a statistic or graph for a given dataset. Our sequence leads with the level of measurement because it is the decision that governs every later choice, so the ordering reflects our teaching rationale rather than the specification's own.
| Our lesson covers | Edexcel 9PS0 research-methods area |
|---|---|
| Nominal, ordinal and interval data | Levels / types of measurement |
| Mean, median, mode | Measures of central tendency |
| Range, standard deviation | Measures of dispersion |
| Bar chart, histogram, scattergram, tables | Presentation and display of quantitative data |
| Normal and skewed distributions | Characteristics of distributions |
Assessment Objectives. These items combine AO2 (selecting and calculating the right statistic or graph for a described dataset) with AO3 (evaluating the appropriateness of a chosen measure or display, and interpreting what the numbers imply), on an AO1 base of accurate definitions and formulae. Questions are application-heavy and frequently require a short calculation or the reading of a graph; a purely definitional answer scores poorly.
Connects to…
The level of measurement describes how precisely a variable has been measured, and it is the single most important classification in this topic because it constrains which statistics and which tests are valid. Edexcel requires three levels.
| Level | Description | Example | Key feature |
|---|---|---|---|
| Nominal | Data in named categories; frequency counts | Preferred therapy (CBT / drugs / none); yes/no | Categories only — no order |
| Ordinal | Data that can be ranked, but the intervals between ranks are unequal | Finishing position (1st, 2nd…); Likert rating (1–7) | Ordered, unequal gaps |
| Interval | Equal, measured intervals on a genuine scale | Reaction time (ms); temperature (°C); number of words recalled | Equal gaps between units |
Key Definition: Nominal data are the simplest level: each participant's score is placed into a named category (e.g. male/female, or which of three strategies they used), with no quantitative comparison possible between categories. Only frequencies — how many fall in each category — can be counted.
The crucial insight is that the same underlying construct can often be measured at different levels, and the level chosen has real consequences. Take "anxiety". A researcher could record simply whether each participant is "anxious" or "not anxious" (nominal — only the mode and frequency-based analysis are available); ask them to rank situations from most to least anxiety-provoking, or rate anxiety on a 1–10 scale (ordinal — the median and rank-based tests apply, because the gap between 3 and 4 may not equal the gap between 8 and 9); or measure a physiological correlate such as heart rate in beats per minute (interval — the mean, standard deviation and the most powerful tests become available). Each step up the hierarchy preserves the information of the level below and adds more, which is why researchers prefer to measure at the highest level a variable will allow.
A short classification exercise shows the reasoning in action:
| Variable as measured | Can it be ranked? | Equal intervals? | Level |
|---|---|---|---|
| Favourite type of therapy (CBT / drugs / none) | No | No | Nominal |
| Finishing position in a memory-recall competition | Yes | No (1st→2nd ≠ 2nd→3rd) | Ordinal |
| Rating of mood on a 1–7 Likert scale | Yes | Not guaranteed | Ordinal |
| Number of words recalled (out of 20) | Yes | Yes | Interval |
| Reaction time in milliseconds | Yes | Yes | Interval |
Exam Tip: To classify a level, ask two questions in order. Can the data be ranked? If no → nominal. If yes, are the gaps between values equal, properly measured units? If no (as with a rating scale) → ordinal; if yes → interval. Note the recurring controversy over Likert scales: although the numbers look like equal intervals, the psychological gap between "agree" and "strongly agree" is not guaranteed to equal that between "neutral" and "agree", so rating-scale data are safest treated as ordinal.
A measure of central tendency summarises a whole dataset with a single "typical" or "central" value.
Key Definition: The mean is the arithmetic average, found by dividing the sum of all values by the number of values:
xˉ=n∑x
where xˉ is the mean, ∑x is the sum of all the scores, and n is the number of scores.
| Measure | How it is found | Strengths | Limitations |
|---|---|---|---|
| Mean | xˉ=n∑x | Uses all the data; most sensitive; required for the most powerful tests (interval data) | Distorted by outliers; may give an "impossible" value (e.g. 2.4 children) |
| Median | The middle value when scores are ranked (the mean of the two middle values if n is even) | Not affected by outliers; suits skewed or ordinal data | Ignores most of the data; less sensitive than the mean |
| Mode | The most frequent value | The only measure usable with nominal data; is always a real value in the set | May be unrepresentative; a set can be bimodal or have no mode |
Worked example. Consider the scores: 3, 5, 7, 8, 8, 10, 12, 14, 47.
Here the mean (12.67) is dragged upward by the single outlier of 47 and misrepresents the bulk of the data, so the median (8) is the better summary of what is typical.
A frequent exam task is not merely to calculate a measure of central tendency but to justify which one to report — and the level of measurement is the deciding factor:
A second worked example shows how the choice can change the story the data tell. Suppose seven participants take a memory test scored out of 20, obtaining: 4, 6, 6, 7, 8, 9, 20.
The single high score of 20 inflates the mean to 8.57, well above six of the seven participants, whereas the median (7) and mode (6) sit comfortably among the bulk of the data. Reporting only the mean would overstate typical performance — a concrete illustration of why the choice of measure matters, and why outliers must always be inspected.
Exam Tip: With outliers or a skewed distribution, recommend the median. With normally distributed interval data, prefer the mean, because it uses every data point and supports the most powerful tests. With named categories, only the mode is available.
A measure of dispersion describes how spread out the data are around the centre — information the central tendency alone cannot give.
| Measure | What it is | Strengths | Limitations |
|---|---|---|---|
| Range | Highest − lowest value (sometimes +1) | Quick and easy to calculate | Uses only two values; distorted by a single outlier; ignores the middle of the data |
| Standard deviation (SD) | The average distance of each value from the mean | Uses all the data; far more informative; underpins the most powerful tests | Harder to calculate; still somewhat affected by outliers; needs interval data |
Key Definition: The standard deviation is a measure of dispersion indicating the average distance of each data point from the mean. A small SD means scores cluster tightly around the mean; a large SD means they are widely spread.
The standard deviation is calculated as
s=n−1∑(x−xˉ)2
where x is each score, xˉ is the mean, and n is the number of scores. In words: find each score's deviation from the mean (x−xˉ), square it (so negatives do not cancel), sum the squares, divide by n−1 (giving the variance), then take the square root to return to the original units.
Two design features of this formula reward a moment's thought. The deviations are squared for a precise reason: adding the raw deviations would always give zero, because positive and negative deviations cancel exactly around the mean; squaring removes the signs so that the spread does not vanish, with the side-effect that large deviations are weighted especially heavily. Taking the square root at the end then undoes the squaring, returning the answer to the original measurement units (so an SD of reaction times is in milliseconds, not "milliseconds squared").
Worked example. Consider the scores: 4, 6, 7, 8, 10.
Interpreting the SD. Because the SD is in the same units as the original data, it can be read concretely: a reaction-time mean of 320 ms with an SD of 15 ms tells us most participants responded within roughly 305–335 ms, whereas the same mean with an SD of 90 ms tells us responses were scattered far more widely. A small SD means participants responded similarly, so the mean represents them well; a large SD signals high variability, so the mean is a less trustworthy summary and there may be important individual differences at work.
The range in more detail. The range is the crudest measure of dispersion because it depends entirely on the two most extreme scores and ignores everything between them. Two datasets can share an identical range yet have utterly different spreads: for the sets {2, 2, 2, 2, 10} and {2, 4, 6, 8, 10} the range is 8 in both cases, yet the first is bunched at the low end with one outlier while the second is evenly spread. It remains useful as a quick, easily understood check before the SD is computed, particularly for small datasets.
Converting raw frequencies to percentages standardises data so that groups of different sizes can be compared:
percentage=wholepart×100
For example, if 18 of 24 participants recalled a word, that is 2418×100=75%. Percentages also make change easy to express: if a mean error rate falls from 40 to 30, that is a fall of 10 percentage points but a 4040−30×100=25% reduction relative to the starting figure — a distinction examiners sometimes probe when a question asks for the size of an effect.
Exam Tip: If a question asks you to compare two conditions, look at both the means and the SDs. Two groups can share an identical mean yet differ greatly in spread — the SD reveals a consistency (or inconsistency) that the mean alone conceals. Quoting the SD in the original units (e.g. "most scores fell within one SD, i.e. 305–335 ms") shows genuine understanding rather than rote recall.
The display chosen must match the data type and the purpose.
| Display | Used for | Key feature |
|---|---|---|
| Bar chart | Categorical/nominal data, or summary values such as condition means | Discrete bars with gaps; height = frequency or value |
| Histogram | Continuous (interval) data | Bars with no gaps; the x-axis shows continuous intervals (bins); area represents frequency |
| Scattergram | Correlational data (two co-variables) | One dot per pair of scores; the pattern shows direction and strength; a line of best fit can be added |
| Frequency table | Summarising raw data before graphing or calculation | Each value or class interval listed with its frequency |
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