Algebra — AQA A-Level Psychology
Test yourself on Algebra with AQA A-Level practice questions.
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Algebra explained
In a Psychology paper these signs are the shorthand a result is reported in, so each one has to be read exactly.
Read the full explanation
Each sign carries one meaning: = states exact equality, < states less than and > states greater than, while the doubled signs << and >> state much less than and much greater than. The sign ∝ marks a proportional relationship between two measures, and ~ marks an approximate or order of magnitude figure, as in a sample of ~ 30 people. The conventional way to report an inferential result is to set the probability against the accepted level, writing p < 0.05 for a significant result and p > 0.05 for one that is not. At the 0.05 level a psychologist accepts a five percent probability that the pattern arose by chance; the stricter 0.01 level cuts that to one percent and is chosen where a false claim would be costly, such as in drug trials.
Substitute numerical values into algebraic equations using appropriate units for physical quantities (to include: inserting the appropriate values from a given set of data into the formula for a statistical test, eg inserting the N value (for the number of scores) into the Chi Square formula)
A calculation question hands you a table of data and a printed formula, and the marks go to the candidate who puts the right number in the right place. Work through the formula symbol by symbol and write down what each one stands for before any arithmetic. For Chi Square the observed value of a cell is the frequency actually counted there, and the expected value is found by multiplying that cell’s row total by its column total and dividing by N, the total number of scores. For Spearman’s rho the figures to insert are the difference between each pair of ranks, squared and then summed, together with N as the number of pairs of scores. Keep the unit of the original measure, so seconds stay seconds and words recalled stay words recalled, and leave the substitution visible on the page.
Solve simple algebraic equations (to include: calculating the degrees of freedom for a Chi Square test)
Degrees of freedom decide which line of the Chi Square table you read; the wrong line gives the wrong critical value and decision. The rule is the number of rows minus one, multiplied by the number of columns minus one, written as (rows - 1)(columns - 1). Count only the cells holding observed frequencies, so a 2 by 2 contingency table has 1 degree of freedom and a 2 by 3 table has 2. Read the critical value at the chosen significance level. The AQA Chi Square table does not require choosing between one and two tails. Compare in the correct direction: Chi Square is significant when the calculated value is equal to or greater than the critical value. The same habit of rearranging a short equation turns a total and a number of participants into a mean.
Your focus
- State the meaning of each of =, <, <<, >>, >, proportional to and approximately equal to when used in a results section.
- Explain why p less than 0.05 reports a significant result while p greater than 0.05 reports one that is not.
- Outline an inferential outcome conventionally, quoting calculated and critical values, N or df, the tail and the level used.
Show all 12 objectives
- Justify reporting at the 0.01 level where a Type 1 error would be serious, and say what that does to Type 2 error risk.
- Calculate each expected frequency for Chi Square as its row total multiplied by its column total, divided by N.
- Identify what every letter in a printed formula stands for, so N is the number of scores or the number of pairs of scores.
- Calculate the sum of squared rank differences for Spearman's rho, squaring first so negatives do not cancel positives.
- Explain why the substituted line is written out in full, since the method mark survives an arithmetic slip.
- Calculate degrees of freedom for a Chi Square table as rows minus one times columns minus one, so 2 by 3 gives 2.
- State which rows and columns are counted, leaving out the totals row and totals column a printed table often carries.
- Explain what the degrees of freedom decide, namely which line of the critical values table is read and so the decision reached.
- Calculate a mean by rearranging the same kind of short equation, dividing the total of the scores by the number of scores.
Algebra exam tips
Marking Points
- Reading the sign in the direction the result needs: p < 0.05 says the probability of the finding arising by chance is less than five percent, so the result is significant, and p > 0.05 says it is greater than five percent, so it is not.
- Naming the level in full, as the 0.05 level or the five percent level, and then saying which hypothesis is rejected and which is retained, because a significant result rejects the null hypothesis and supports the alternative hypothesis.
- Writing the outcome in the conventional form, quoting the calculated value, the critical value, the number of participants or the degrees of freedom, whether the test was one tailed or two tailed, the level used, and closing with p < 0.05.
- Justifying a move to the 0.01 level where a Type 1 error would be serious, for instance in drug research, and noting that the stricter level raises the risk of a Type 2 error instead.
- Using the remaining signs accurately, so ∝ claims that one measure rises in direct proportion with another and ~ claims only an approximate or rounded size.
- Writing the formula out and saying what each letter stands for, so the marker can see N identified as the number of scores or the number of pairs of scores rather than the number of conditions or the number of groups.
- Showing the substituted line in full before the answer, because the method mark survives an arithmetic slip while a bare wrong number earns nothing at all.
- Calculating each expected frequency rather than counting it, using the row total multiplied by the column total divided by the overall total, and doing this for every cell of the contingency table.
- Giving the answer to the accuracy the question asks for, usually two decimal places, and carrying the unit of the original measurement wherever the figure has one.
- Squaring each rank difference before summing for Spearman’s rho, so that negative differences do not cancel positive ones.
- Quoting the rule as (rows - 1)(columns - 1) and showing the numbers put into it, so that a 2 by 3 table gives 1 multiplied by 2, which is 2 degrees of freedom.
- Counting rows and columns from the observed frequencies only, and leaving out the row of totals and the column of totals that a printed table often carries.
- Using the degrees of freedom together with the level of significance to read the critical value from the Chi Square table, noting that this specific table does not use one or two tails.
- Stating the comparison in the right direction, since the calculated Chi Square value must be equal to or greater than the critical value for the result to be significant.
- Rearranging other short equations correctly under exam pressure, such as making the mean the subject by dividing the total of the scores by the number of scores.
Examiner Tips
- 💡When explaining p < 0.05, include probability, the five-percent threshold and the decision about the null hypothesis, all tied to the variables in the study.
- 💡When asked to state whether a result is significant, write one sentence in the conventional form: calculated value, critical value, tails, N or degrees of freedom, level, decision, then p < 0.05 or p > 0.05.
- 💡Which way the comparison runs depends on the test. Chi Square, Spearman’s rho and both the related and unrelated t tests are significant when the calculated value is equal to or greater than the critical value; Mann Whitney U, Wilcoxon T and the sign test are significant when the calculated value is equal to or less than the critical value.
- 💡For a calculate command, show the values substituted into the equation with their units before simplifying to the final result.
- 💡The table in the stem is the source of every number you use, so label the totals you take from it and check that the row totals and the column totals reach the same grand total before substituting.
- 💡A calculation is often followed by a question asking you to explain why that test was chosen, so be ready to name the level of measurement, the experimental design, and whether a difference or a relationship is being tested.
- 💡Degrees of freedom appear as a calculate question attached to a contingency table, so count the table first and write the substituted rule rather than the bare answer.
- 💡Longer questions ask you to explain whether the result is significant, which needs the degrees of freedom, the level of significance, the calculated value, the critical value and the decision about the null hypothesis.
- 💡Where a table of critical values is printed, point to the exact figure you used, because the marks are for the comparison rather than for the arithmetic.
Common Mistakes
- Reversing the sign and writing p > 0.05 for a significant result. The probability of a chance finding has to be small, so the significant outcome is always the one where p falls below the stated level.
- Treating 0.05 as though it meant five, and writing the level as p < 5. The probability runs from zero to one, so five percent is written as 0.05 and one percent as 0.01.
- Claiming that p < 0.05 proves the hypothesis, or that it shows a ninety five percent chance the theory is true. It states only the probability that the results occurred by chance, and it proves nothing.
- Quoting a level of significance with no mention of the number of tails and no N or degrees of freedom, which leaves the marker unable to check that the right critical value was read.
- Putting the number of participants into a formula that asks for the number of scores. In a repeated measures design one participant supplies two scores, and for Wilcoxon the N read against the table is the number of pairs left after every difference of zero has been dropped.
- Substituting raw scores where the formula needs ranks, which happens whenever Spearman’s rho is worked straight from the original measurements without ranking each variable first.
- Mixing units inside one calculation, adding a time recorded in seconds to a time recorded in milliseconds without converting, so the total means nothing.
- Rounding at every intermediate step, which can push the final value across the critical value and reverse the decision about significance.
- Using the number of participants, or the grand total of the frequencies, as the degrees of freedom instead of the size of the table. A study of 60 people set out in a 2 by 2 table still has 1 degree of freedom.
- Counting the totals row and the totals column as data, so a 2 by 2 table is read as 3 by 3 and the critical value is taken from the wrong line of the table.
- Comparing in the wrong direction and calling a small calculated value significant because Mann Whitney U and Wilcoxon T work that way. Chi Square needs the calculated value to be equal to or greater than the critical value.
- Looking for one-tailed or two-tailed columns in the Chi Square critical values table, which are not present in the AQA specification table.