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Thread: Link between a questionnaires results, and individual scores

  1. #1 Link between a questionnaires results, and individual scores 
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    Hi,

    I need to know if there is a mathematical link between the scores obtained on a test, and the scores of individual users.

    For example,

    2 test takers

    Question 1 - Total correct score = 2
    Question 2 - Total correct score = 1

    Person 1 therefore must have got full marks (you can only take each question once) and person 2 got 50% right.

    I need to know what the formula is to work this out on a bigger scale (21 questions with 100 test takers) to find out each users individual score.

    Thanks


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  3. #2  
    Forum Masters Degree bit4bit's Avatar
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    I don't think you can do that, because as you start introducing more users, and more questions, there will be more than one combination of possible individual results.

    Also it's not an effective way to find which user got which score from this method. For your example, maybe person 2 got 100%, and person one got 50%. Theres two possible outcomes. As you have more users, aand more questions, there will be even more outcomes. You lost your test papers or something? :P


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  4. #3  
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    Haha yea I lost em and I only got the results of which questions were answered correctly. So there is no way that someone else can work out one from the other i.e. use this data to know that I didn't actually use the papers to come up with individual scores.

    Its not actually a test paper - so no one will be disadvantaged but is a questionnaire so it doesn't actually matter WHICH person got WHAT score but it would be good if the numbers tied up.

    Much appreciated
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  5. #4  
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    I've had a revelation.

    Is it true that if I work out the mean of the amount of answers returned correctly this should equal the mean of the individual scores?

    For example:

    There are 21 questions in my paper. 100 people took part.


    Question 1 was answered correctly 67 times
    Question 2 was answered correctly 43 times
    Question 3 was answered correctly 64 times
    Mean = 58

    So for the numbers to tally user 1 could have scored 50%
    user 2 could have scored 30%
    user 3 could have scored 94%

    which gives the same mean of 58

    The mean of the correctly returned scores for my actual paper is 60.4 (therefore the mean of incorrectly answered questions is 100-60.4 = 39.6).

    So the mean of the individuals persons scores must equal 60.4% of the paper or 21*0.604 = 12.69 or 13 questions correct.

    So for the numbers to tally up the mean of the correctly returned answers being 60.4 the mean of the individual users scores (for the whole paper) must be 13?
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  6. #5  
    Forum Professor serpicojr's Avatar
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    Yes. Let N be the number of individuals, let i range from 1 to N, i.e. each value of i represents a different individual, let q represent the number of questions, let c<sub>i</sub> represent the number of questions person i got right. Then the individual means are (for individual i):

    c<sub>i</sub>/q

    The total mean (i.e., the total percent of questions answered correctly) is:

    (c<sub>1</sub>+...+c<sub>N</sub>)/Nq

    Then the mean of the individual means is:

    ((c<sub>1</sub>/q)+...+(c<sub>N</sub>/q))/N = ((c<sub>1</sub>+...+c<sub>N</sub>)/q)/N = (c<sub>1</sub>+...+c<sub>N</sub>)/Nq

    One important piece of data you have lost is the standard deviation--you have no idea how spread out your data set may have been. Depending on what you want to do with your data, this could be fine.
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  7. #6  
    Forum Masters Degree bit4bit's Avatar
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    I'm not sure if you can actually say that the mean of the scores for each question (x') is equal to the mean of the scores for each user (y'). I can't prove whether it is true or not though. If you picture a matrix with the question numbers along the rows, and the user numbers down the columns, you have a 21x100 matrix, where each entry in the table will be a zero or one (right or wrong). Using s<sub>Q</sub> for the total per question, and s<sub>U</sub> for the total per user, I got this far:

    x'=y' (If what you say is valid)

    x' = (∑ s<sub>Q</sub>) / 21
    y' = [∑ (s<sub>U</sub> / 21)*100] / 100 = ∑ (s<sub>U</sub> / 21)

    (∑ s<sub>Q</sub>) / 21 = ∑ (s<sub>U</sub> / 21)

    I'm not sure how I can extend that any further to prove whether or not you can say that x' = y', but I have a feeling 1.) I'm thinking too hard about it, and there is a simpler answer, 2.) I'm nnot thinking hard enough about it, 3.) some sort of experiment with a real set of data is needed to show it....Anyway I'm having fun thinking about this, but can't give you an answer.

    One thing I am fairly certain of is that if you can say x'=y', then you would have to make up the 100 individual percentages, whose mean adds up to 60.4%, since you couldn't know the distribution of the results among the students.
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  8. #7  
    Forum Masters Degree bit4bit's Avatar
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    oops sorry Sepicojr, you beat me to it.
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  9. #8  
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    Thanks for all your help

    So just to clarify...as long as I make the two means match up...there's no way of anyone knowing that I calculated one from the other (i.e. I didn't actually go through the papers calculating each persons score).

    Much appreciated!
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