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28 | A2 PSYCHOLOGY: THE STUDENT’S TEXTBOOKSpearman’s RhoThis test is used in correlation research. If you arelooking at the relationship between two variablesand have drawn a scattergraph, then Spearman’sRho is the test to use. We’ll start by summarisingsome research involving correlation and using thereal data in our calculation.Aim:To investigate the relationship between ability atmathematics and ability at playing chess.Hypothesis:The better you are at maths, the better you will beat chess.Null hypothesis:Being good at maths does not mean that you willbe good at chess.Operationalising the variables:Maths ability was operationalised as ‘score out of100 on a maths test’.Chess ability was operationalised as ‘number ofchess problems out of 100 successfully solved’.The raw data:12345678910Maths Score58347759469022788967Chess Score67437261509920739070The graph:This is correlational research investigating therelationship between one variable and another. Theappropriate graph to draw is a scattergram.The relationship between maths and chess ability1008060402000 20 40 60 80 100Chess scoreMaths scoreThe descriptive statistics:MeanStandarddeviationMaths Score6222.81Chess Score64.522.78It seems from these data that there is indeed astrong positive correlation between maths abilityand chess ability. This is shown most clearly inthe graph. As maths score increases so too doeschess score, typical of a positive correlation. To bemore certain, and to see whether the correlationis significant, we need to carry out the Spearman’sRho statistical test.Doing Spearman’s RhoStep 1: Rank the scores for each variableRanking will become extremely straightforwardas you have to do it for three of the tests you arerequired to understand. It is simple enough, buttake care to get it right. Take your original tableand add two more columns. Ranking meansputting the scores in order, smallest to largest, andgiving the smallest the rank of 1, the next smallestthe rank of 2 and so on. You can see how this isdone by looking at the ranks for maths scores inthe table below. Notice, the smallest maths scorewas 22. This gets the rank of 1. The next smallestwas 34. This gets the rank of 2 and so on, right upto the highest score of 90, which gets the rank of10.12345678910MathsScore58347759469022788967ChessScore67437261509920739070MathsRank42753101896ChessRank52743101896The ranking of our data is uncomplicated. In othercircumstances, though, you may find that twoparticipants score the same on the same task. Thiscommon problem makes the ranking slightly moredifficult. Take a look at the section on tied ranksbelow for more on this.Extract from A2 Psychology: The Student’s Textbook © Nigel Holt and Rob Lewis ISBN: 9781845901004 www.crownhouse.co.uk
DATA ANALYSIS AND REPORTING INVESTIGATIONS | 29Step 2: Work out differences and square themIn our case we need to work out the differencebetween maths score and chess score for each ofour participants. To start you off:Participant number 1 was ranked at 4 for mathsand 5 for chess.4–5 = –1–12 = 1For each participant, the ‘difference’ number, andthen the figure which is the result of squaring thatnumber, are recorded in two new columns, to theright of those shown in the table above. Whenyou’ve done this for all your participants youshould end up with a table like this:12345678910MathsScore58347759469022788967ChessScore67437261509920739070MathsRankChessRankd(difference)d 2(d squared)4 5 -1 12 2 0 07 7 0 05 4 1 13 3 0 010 10 0 01 1 0 08 8 0 09 9 0 06 6 0 0Step 3: Add up the numbers in the d 2 column(∑d 2 – ‘sigma d 2 ’ in English)You need this number for working out the equation.In our case all we do here is add up thenumbers in the far right column:1+0+0+1+0+0+0+0+0+0 = 2Step 4: Identify N and N 2N refers to the number of pairs you have in yourdata. In this case we have 10. N 2 is simply thatnumber squared: in your case that’s (10 x 10)which is 100.Step 5: Get organisedGet all the numbers you need for your equation.r6(∑d 2 )s=1–N(N 2 –1)r s– this is the statistic we are after – Spearman’sRho.N – the number of people who provided data inyour correlation research.d – The difference between each rank of onevariable and the corresponding rank of the othervariable.Your equation requires ∑d 2 , N 2 and N. Make sureyou know what these are and write them downon the top of the paper on which you will do yourcalculations.∑d 2 = 2N = 10N 2 = 100Step 6: Insert your numbers into the equationWith your numbers it looks like this:6(2)r s=1–10(100–1)This looks a little less frightening and gets evensimpler. The first rule is to work out everythingwithin the brackets first. When you do that youget the following:6(2)r s=1–10(99)You’ll know this from maths, but it won’t hurt toremind you that you need to multiply the numberinside the brackets by the number on the outside:i.e. you need to calculate 6 x 2 and 10 x 99. Whenyou do that, your equation looks like this:12r s=1–990Next, work out the fraction. Remember, youdivide the top number by the bottom number,(in other words, 12÷990) to give you the statisticwe need which will indicate the strength of yourcorrelation.r s=1– 0.012= 0.988This is your correlation coefficient. You will recallthat anything positive means that as one variableincreases so too does the other (the absence of asign in front of the number means that it is positive).You will also know that the nearer to 1 yourcoefficient is, the stronger the correlation. It isclear that we have an extremely strong correlationhere, but is it significant? Might it have occurredby chance?Step 7: Finding the critical valueTo see whether the calculated r sis large enoughfor you to say that your results did not occur bychance you need to look in statistical tables. At theend of this book you will find an appendix withthe table you need for this test. To use it, you needto know:N The number of participants in your task: 10Extract from A2 Psychology: The Student’s Textbook © Nigel Holt and Rob Lewis ISBN: 9781845901004 www.crownhouse.co.uk
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