D.1 – handling data — OCR A-Level Psychology

Significant figures are counted from the first non-zero digit. In a correlation coefficient of 0.06428, leading zeros are placeholders; the first significant figure is six. This rounds to 0.0643 (three significant figures) and 0.064 (two significant figures). Rounding examines the next digit, rounding up from five. Coefficients are expressed to two or three significant figures because excessive decimal places imply false precision. Crucially, compare observed and critical values before rounding. An observed Spearman's rho of 0.5638 against a critical value of 0.564 fails to reach significance, but rounding first to 0.564 would falsely suggest significance.

The arithmetic mean is the sum of all scores divided by the number of scores. In a two-condition class experiment, means are calculated separately for each condition using raw data to allow comparison. The mean is suitable for interval data because it uses every value. However, this makes it sensitive to extreme scores; one participant scoring unusually low will drag the mean down, making the median a safer summary if outliers exist. For example, Loftus and Palmer (1974) reported mean speed estimates of 40.8 mph for the verb 'smashed' and 31.8 mph for 'contacted'. Report a mean to one more decimal place than the raw scores and pair it with a measure of dispersion.

A frequency table organises raw data into counts per category or class interval, dictating the appropriate graph. Categorical data (e.g., number of participants conforming) requires a bar chart with gaps between bars, representing discrete categories. Continuous data grouped into equal class intervals requires a histogram, featuring touching bars and a continuous numerical scale on the x-axis, where area represents frequency. A correlation requires a scattergram. When sketching any display, both axes must have labels and units, and the title should clearly name the variables. Interpreting these diagrams involves reading values accurately and describing the distribution's shape.

A probability value (p) represents the chance of obtaining a result at least as extreme as the one found, assuming the null hypothesis is true. The significance level (alpha) is the preset threshold used to decide if this p-value is small enough to reject the null hypothesis. Psychology conventionally accepts a result as significant when the p-value is less than or equal to the 0.05 (5%) significance level. A stricter level of 0.01 (1%) is chosen when a false claim would be costly, such as in drug trials. Lowering the significance level reduces the risk of a Type I error (rejecting a true null hypothesis) but increases the risk of a Type II error (retaining a false null hypothesis). The 0.05 level serves as a conventional compromise between these two errors.

A sample is drawn from a target population, and how it is drawn decides how far the findings can travel. A random sample gives every member an equal chance of selection, which in practice means a complete list of the population and then a lottery or a random number generator; it removes researcher bias but needs that list and can still throw up an unrepresentative set by chance. A stratified sample identifies the subgroups that matter, works out the proportion each holds in the population, and then selects randomly inside each stratum so those proportions are reproduced. Milgram recruited by newspaper advertisement, so his obedience figures rest on self selected men, and the broader problem is that published samples remain heavily Western and student based.

A measure of central tendency is a single figure standing in for a whole set of scores, and the examinable skill is choosing between three of them rather than reciting how each is worked out. The arithmetic average uses every score, which makes it sensitive and also makes one extreme value drag it, so it suits interval data that are roughly symmetrical, such as the speed estimates in miles per hour collected by Loftus and Palmer (1974). The middle value of an ordered set ignores how far out the outliers lie, which is why it is the safer summary of reaction times or a lopsided set of aggression scores. The most frequent value is the only option for named categories, for instance the commonest coded behaviour in an observation. Examiners reward a justification that names the level of measurement and the shape of the data together.

Standard deviation is a measure of dispersion representing the square root of the mean squared deviation from the mean. It answers whether a condition produced a consistent effect or if one summary figure hides widely varying scores. Calculate it in fixed steps: find the mean, subtract it from each score to find the deviations, square each deviation so negatives stop cancelling, add the squared deviations, divide by n (or n-1 for a sample) to find the variance, and finally take the square root so the answer returns to the original units. The figure is highly useful comparatively; two experimental conditions might share a mean, but one group is tightly clustered (small standard deviation) while the other is spread out.

This is the picture a correlational design makes: one co-variable on each axis, one point for every participant placed where their two scores meet, and no line joining the points up. The pattern is read from the slope, so points running from bottom left to top right show that high scores on one measure go with high scores on the other, points running from top left to bottom right show that high on one goes with low on the other, and a shapeless cloud shows no relationship at all. Maguire and colleagues (2000) found that the volume of posterior hippocampal grey matter in London taxi drivers rose with the years they had spent driving a cab while anterior volume fell, which is one relationship of each direction in the same brain. Direction is not strength, and neither of them is cause.

To calculate a non-parametric test of differences using data from a given experiment, specific steps apply. For the binomial sign test, assign a plus or minus to each change. For example, if memory scores change by +2, 0, -4, and +2, discard the zero, leaving two pluses and one minus. The less frequent sign is minus (count = 1), so the calculated value is 1 (N=3). For the Wilcoxon signed ranks test, calculate each pair's difference, discard zeros, rank absolute differences, sum positive and negative ranks separately, and take the smaller sum. For the Mann-Whitney test, rank all scores together, sum ranks per group ($R_a$, $R_b$), and use $U_a = n_a n_b + \frac{n_a(n_a+1)}{2} - R_a$ to find $U_a$. Calculate this for both groups; the smaller result is the calculated $U$.

This is deliberate approximation: round the awkward numbers to something you can hold in your head, do the arithmetic, then check the answer sits in a sensible band rather than being ten times too large or too small. The examinable version is recovering an average from a total, since dividing an overall score by the number of participants returns the mean without anyone ever seeing an individual score, and the reverse step multiplies a reported mean by the sample size to recover the total. It also polices other people figures, so a study claiming an average of forty words recalled from a list of twenty has gone wrong somewhere, and a count of helping responses larger than the number of trials run cannot stand. Marks come from working that shows which numbers were rounded and in which direction the estimate will therefore lean.

The level of a measure describes what its numbers represent, driving decisions about summarising and testing. Counts of cases in named categories are nominal, so the numbers label rather than measure and only frequencies can be compared, as when Simons and Chabris recorded how many observers noticed the unexpected event. Scores that can be placed in order but whose gaps are not known to be equal are ordinal, covering rankings and rating scales. Scores on a scale with genuinely equal units, such as time in seconds, are interval. Work out how the researcher recorded the behaviour before you state the level of measurement used in a study to distinguish between them accurately.

A distribution is the shape a set of scores makes once frequency is plotted against score. The normal curve is symmetrical and bell shaped, most scores cluster near the centre and the tails fall away evenly, and because of that symmetry the mean, the median and the mode all sit together at the peak. Skew is what happens when one tail is stretched. Where a few unusually high scores drag the tail out to the right the distribution is positively skewed and the mean is pulled furthest in that direction, leaving the order mode, median, mean reading from left to right; where a few unusually low scores stretch the tail to the left that order reverses. Floor and ceiling effects are the usual cause in psychology, so a task almost everyone completes gives a negative skew and one that defeats most participants gives a positive skew.

Selecting a suitable inferential test for a given practical investigation requires evaluating three criteria. First, does the hypothesis predict a difference or a correlation? Second, what is the experimental design: independent or repeated/matched? Third, what is the level of measurement? For nominal data, use Chi-square (independent) or the Binomial Sign test (repeated). For ordinal data, use Mann-Whitney U (independent), Wilcoxon (repeated), or Spearman's Rho (correlation). For interval data meeting parametric criteria, use an unrelated t-test, related t-test, or Pearson's r. Explaining why the chosen test is appropriate involves explicitly linking these three criteria to the investigation's details.

A critical value is the figure a calculated result must beat to be considered significant. You use statistical tables to determine significance by entering with specific information: the level of significance (e.g. the 0.05 level of significance), whether the hypothesis is for a one-tailed test or two-tailed test, and the sample size (N). Chi-square uses degrees of freedom (df). Compare the critical value with your observed value. For Wilcoxon, Mann-Whitney and the Sign test, the observed value must be $\le$ the critical value; for Chi-square and Spearman's rho, it must be $\ge$. For example, using an extract from statistical tables, if a given observed value for a Wilcoxon test is 4 (N=8), the critical value at the 0.05 level for a one-tailed test is 5. Since $4 \le 5$, the result is significant.

Measures of dispersion describe the spread of a dataset. The range is calculated by subtracting the lowest score from the highest (e.g. $15 - 3 = 12$). While simple, it relies solely on two extreme values, making it highly sensitive to outliers. The standard deviation (SD) represents the average amount that scores differ from the mean. Because SD incorporates every score in the dataset, it is often a more useful measure of dispersion than the range when an outlying score is present. If one participant scores 50 while others score 10 to 15, the range is heavily distorted by the 50. The SD dilutes this extreme value across the sample, providing a more representative summary of the overall spread. Students must understand these concepts but are not required to calculate the standard deviation in the exam.

Numbers and words answer different questions, and the examinable skill is moving between them. Counted or measured scores can be averaged, graphed and tested for significance, which is how Baron-Cohen et al. (1997) compared eyes task scores between adults with autism and controls. Words, images and transcripts carry meaning and reasons that a score cannot, which makes Freud (1909) on Little Hans rich and unrepeatable. Content analysis is the bridge: the researcher fixes coding categories, breaks the material into units, decides which category each unit falls into and tallies the totals, much as Hancock et al. (2011) did in counting language features in the crime narratives of psychopathic and non-psychopathic murderers.

The distinction turns on who gathered the information and what for, not on whether it is words or numbers. Anything a researcher collects at first hand for the question in front of them counts as original data, whether that is timing participants or writing down interview answers. Anything that already existed, gathered by someone else or for another purpose, is second hand: official crime figures, medical records, or the wartime Army intelligence test scores that Gould (1982) re-examined. Casey et al. (2011) carries both at once, since the childhood delay of gratification measures came from an existing longitudinal sample while the go/no-go performance and the brain scans were gathered fresh.