Arithmetic and numerical computation — AQA A-Level Psychology
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Arithmetic and numerical computation explained
Psychology papers print very small and very large numbers compactly, so a reaction time, a hormone concentration or a probability from a computer printout often arrives as a coefficient between one and ten multiplied by a power of ten.
Read the full explanation
To read it, move the decimal point once for every power, to the right for a positive power and to the left for a negative one, so 2.5 times ten to the power of minus three seconds is 0.0025 seconds. That skill is examined in two familiar places. A probability has to be compared against the five percent significance level before a result can be called significant, and a graph has to be drawn from plain frequencies: divide each category total by the grand total, multiply by 360, and label every sector with its category and its percentage.
Use ratios, fractions and percentages (to include: calculating the percentages of cases that fall into different categories in an observation study)
A structured observation produces tallies in behavioural categories, and raw tallies rarely compare fairly because the totals behind them differ. Converting a count into a proportion of its own total puts unequal samples on one scale: 18 aggressive acts out of 45 in one nursery and 24 out of 80 in another become 40 percent and 30 percent, which is the comparison the researcher actually wants. The same relationship can be written as parts, so 18 aggressive against 27 non aggressive acts is 2 to 3, or as a part of the whole, which is two fifths. Keep the raw counts alongside the converted figures, because an inferential test on nominal data works from the frequencies themselves and never from the percentages.
Estimate results (to include: commenting on the spread of scores for a set of data, which would require estimating the range)
Estimating is a checking skill rather than a shortcut. Round the numbers in a table to something convenient, do the arithmetic in your head, and use the approximate answer to judge whether a calculated figure is plausible, because a mean that falls outside the highest and lowest scores has to be wrong. Dispersion is where examiners ask for this most often: take the largest score and the smallest, subtract, and you have a quick measure of how variable a condition was, so times running from 7 up to 42 seconds vary by about 35 seconds. That measure is crude, since a single extreme score drives it and every middle score is ignored, which is why standard deviation, built from every score in the set, is the figure a researcher reports alongside the mean.
Your focus
- Calculate the decimal value of a number in standard form, moving the point one place per power, right for a positive power.
- Calculate each pie chart sector angle as the category frequency divided by the total number of observations, times 360 degrees.
- State whether a probability such as 1.2 times ten to the minus four is below 0.05 and so whether the result is significant.
Show all 12 objectives
- Explain why a pie chart suits one set of nominal data divided into categories while a scattergram displays a correlation.
- Calculate the percentage in an observational category as the count divided by the total behaviours recorded, times 100.
- Explain why converting tallies to percentages lets observations of unequal length or group size be compared fairly.
- Convert the same data between a ratio, a fraction and a percentage, and recover a raw frequency from a reported percentage.
- Calculate percentage agreement between two observers and state the 0.80 criterion used for inter observer reliability.
- Calculate the range as the highest score minus the lowest, written with its unit, and say how variable that condition was.
- Suggest whether a printed mean is plausible by estimating from rounded figures, since a mean outside the scores must be wrong.
- Explain why one extreme score makes the range crude and why standard deviation is the figure reported beside the mean.
- Explain what a small standard deviation shows about consistency and what a large one suggests about individual differences.
Arithmetic and numerical computation exam tips
Marking Points
- Reading the notation correctly: the coefficient is a number between one and ten, and the power of ten counts the places the decimal point moves, right for a positive power and left for a negative one, so 2.5 times ten to the power of minus three becomes 0.0025.
- Putting a whole results table into the same decimal form before any comparison, because 4.0 times ten to the power of minus two is larger than 8.0 times ten to the power of minus three even though the coefficient looks smaller.
- Turning frequencies into sector angles: the frequency in a category, divided by the total number of observations, multiplied by 360 degrees, with the sectors summing to 360 degrees and each one labelled.
- Judging a probability written this way against the accepted level of significance, so a value of 1.2 times ten to the power of minus four is smaller than 0.05, the null hypothesis is rejected and the difference is described as significant.
- Choosing the right display for the data: a pie chart shows how one whole set of nominal data divides into categories, a bar chart compares means or totals between conditions, and a scattergram displays a correlation between two co variables.
- The method stated plainly: the count in the category, divided by the total number of recorded behaviours, multiplied by 100, with the percent sign written in the answer.
- Saying why the conversion is made: percentages standardise, so behaviour tallied for different lengths of time, or in groups of different sizes, can be compared directly.
- Distinguishing the three forms: a ratio compares part with part in its simplest terms, such as 2 to 3, while a fraction and a percentage both express a part of the whole, such as two fifths or 40 percent.
- Converting between them accurately: divide the top of the fraction by the bottom to get 0.4, then multiply by 100 to reach 40 percent, and reverse the steps to recover a raw frequency from a reported percentage.
- Using percentages where the specification expects them elsewhere, for example percentage agreement between two observers, which is the number of agreements divided by the total number of observations multiplied by 100, alongside the usual correlational criterion of positive 0.80 or above for inter observer reliability.
- The range stated as a method: the highest score minus the lowest score, written with its unit, so 42 seconds minus 7 seconds is 35 seconds. Where a course text adds one to allow for the limits of each recorded score, show the subtraction so the method itself is credited.
- Standard deviation given in full: subtract the mean from each score to get the deviations, square each deviation, add the squared deviations together, divide by the number of scores minus one, then take the square root of that result.
- Interpreting the figures rather than only producing them: a small standard deviation means scores cluster close to the mean and the condition was consistent, while a large one means participants varied widely, which weakens the mean as a summary and points to individual differences or to an uncontrolled extraneous variable.
- Pairing the measure with the right average and the right data: the range with the median for ordinal or skewed data, and the standard deviation with the mean for interval data that is roughly normally distributed.
- Linking dispersion to the test that follows, since standard deviation is what a t test works from: interval data from a repeated measures design goes to the related t test, degrees of freedom are the number of pairs minus one, and the observed value of t must be equal to or greater than the critical value.
Examiner Tips
- 💡Expect standard-form values inside a Research Methods results table; convert them carefully before drawing or interpreting the requested display.
- 💡Show the working. A calculation mark is often available for the division and the multiplication even when the final figure is wrong, and an answer with no working scores nothing if it is wrong.
- 💡Give the answer to the accuracy the stem demands and keep the unit, so write 0.03 seconds when two decimal places are asked for, and do not round until the final step.
- 💡If the data are two co variables rather than categories, the display is a scattergram and the test is Spearman's rho, where the observed value of rho must be equal to or greater than the critical value at p less than or equal to 0.05 for that number of pairs.
- 💡When a table gives observation tallies, write the category count divided by the total count and multiplied by 100 before stating the percentage.
- 💡When asked to explain why the researcher reported percentages rather than the tallies, answer in terms of comparing groups with different totals, not in terms of it being easier to read.
- 💡If the follow up asks which test, say Chi Square and justify it as a test of difference or association with nominal data and independent groups. Degrees of freedom are the number of rows minus one, multiplied by the number of columns minus one, and the observed value of Chi Square must be equal to or greater than the critical value for the result to be significant.
- 💡Note the contrast in direction between tests, because examiners ask for it: Chi Square and the t tests need the observed value to be equal to or greater than the critical value, while Mann Whitney U and Wilcoxon need the observed value to be equal to or less than it.
- 💡The wording is usually briefly outline what the range tells the researcher about these data, or suggest why the psychologist reported the standard deviation as well as the mean. Answer about variability and consistency, never about the average.
- 💡Quote figures from the table when asked to comment on the results: name the condition with the wider spread, give both values, and tie the comparison back to the hypothesis.
- 💡Estimating from a graph counts too. Read the highest and lowest points off the vertical axis, subtract, and state the scale units in the answer.
- 💡Dispersion questions often lead into test choice, so be ready with the direction of the comparison: ordinal data from an independent groups design goes to Mann Whitney U and from a repeated measures design to Wilcoxon, and in both the observed value must be equal to or less than the critical value for significance.
Common Mistakes
- Moving the decimal point the wrong way for a negative power, so 3.0 times ten to the power of minus two is written as 300 instead of 0.03, which makes a fast reaction time look enormous.
- Reading a probability of 5.0 times ten to the power of minus four as bigger than 0.05 because the coefficient looks bigger, then wrongly retaining the null hypothesis when the result was in fact significant.
- Writing a coefficient that is not between one and ten, such as 25 times ten to the power of three in place of 2.5 times ten to the power of four.
- Drawing the pie chart from percentages that do not total one hundred, or leaving the sectors unlabelled, which loses the display marks however accurate the arithmetic was.
- Running Chi Square on percentages rather than on the raw frequencies. Chi Square compares observed counts with expected counts, so feeding it percentages changes every expected value and the whole calculation is void.
- Dividing by the wrong total, typically by the number of children observed rather than by the number of behaviours recorded, which produces a figure that cannot be a percentage of anything.
- Confusing a ratio with a fraction, so 2 to 3 is reported as two thirds of all behaviour when the aggressive acts were in fact two fifths of it.
- Dropping the percent sign or the required decimal places, so a correct calculation is written as 40 with no percent sign, or as 38 percent when one decimal place was specified.
- Quoting the two end scores, 7 and 42, as the answer instead of the single value 35, or calling the range a measure of central tendency when it measures dispersion.
- Treating two sets with the same range as equally variable, when one outlier has inflated a range that hides tightly clustered scores and the standard deviations are quite different.
- Mangling the standard deviation method, usually by taking the square root of each deviation instead of the whole total, or by forgetting to square the deviations at all so they cancel to zero.
- Estimating where the stem says calculate. An approximate figure earns nothing when an exact value is asked for, and a calculated value with no working earns nothing if it is wrong.