Correlation
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Correlation

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Correlation Correlation Presentation Transcript

  • Correlation.
  • What is it?
    • Two things correlate when they vary together.
    • E.G such as temperature decreasing with altitude or land values falling with distance from the city centre.
    • If, as one variable increases in value so does the other this is positive correlation.
    • If one goes up as the other goes down this is a negative correlation.
  • Positive. Negative.
  • Correlation.
    • Correlation is useful for three reasons.
    • It is more precise than a graph. While two graphs showing correlations may look similar, the correlation coefficients for the sets of data may well be slightly different.
    • If we wanted to compare several pairs of data, such as the relationship between temperature and altitude on twenty slopes, it would be far easier to compare twenty numbers than twenty graphs.
    • It is possible to test the correlation to see if it is really significant or whether it could have occurred by chance.
  • WARNING!!!!!!.
    • The fact that two things correlate proves nothing. We can never conclude from statistical evidence alone that, because two things correlate, one must be affecting the other.
    • All statistical tests must be supplemented with research regardless of the result.
  • Spearmans Rank.
    • This technique is among the most reliable methods of calculating a correlation coefficient.
    • This is a number which will summarise the strength and direction of any correlation between two variables.
  • Method.
    • Stage 1- Tabulate the data- I will show you how to do this with an example.
    • Stage 2 Find the difference between the ranks of each of the paired variables (d). Square these differences (d ²) and sum them ( Σ d ²).
    • Stage 3 Calculate the coefficient from (r s ) from the formula…
    • R s = 1- 6 Σ d ²
    • n ³-n
    • Where d= The difference in rank of the values of each matched pair.
    • N= the number of pairs.
    • The result can be interpreted from the scale.
    • +1.0 0 -1.0
    • Perfect no Perfect
    • postive Correlation negative
    • correlation correlation
  • Next.
    • Now you determine whether the correlation you have calculated is really significant, or whether it could have occurred by chance.
    • Stage 4 Decide on the rejection level ( ).
    • This is simply how certain you wish to be that the correlation you have calculated could not just have occurred by chance. Thus, if you wish to be 95 % certain your rejection level is calculated as follows…
    • = 100-95
    • 100
    • =0.05.
  • Stage 5.
    • Calculate the formula for T.
    • T= R s n-2
    • 1- R s ²
    • Where R s = spearmans rank correlation coefficient.
    • N= number of pairs.
    • Calculate the degrees of freedom.
    • Df = n-2.
    • Where n = the number of pairs.
  • Stage 7
    • Look up the critical value in the t- table using the degrees of freedom and the rejection level.
    • If the critical value is less than your t-value then the correlation is significant at the level chosen (95 %).
  • But what if my critical value is higher than my t value???
    • This means that you cannot be certain that the correlation could not have occurred by chance. This may mean one of two things.
    • A- The relationship is not a good one and it is thus not really worth pursuing it any further.
    • B- The size of the sample you are using is too small to permit you to prove correlation.
  • Example.
    • Population size and number of services in each of 12 settlements.
    • Draw a graph for the following data set…
  • 19 2 362 12 11 1 016 11 35 4 981 10 73 6 781 9 81 9 982 8 72 8 763 7 114 15 739 6 4 220 5 87 10 714 4 43 6 793 3 41 5 632 2 3 350 1 No. of Services. Population Settlement
  • Stages 1-2 0 0 1 114 1 15 739 0 0 2 87 2 10 714 4 2 5 72 3 8 763 1 1 3 81 4 7 982 1 1 6 43 5 6 793 4 2 4 73 6 6 781 0 0 7 41 7 5 632 0 0 8 35 8 4 981 0 0 9 19 9 2 362 0 0 10 11 10 1 016 1 1 12 3 11 350 1 1 11 4 12 220 d ² Difference between ranks (d) Rank No. of services. Rank Settlement Population
  • You Complete stages 3-7.
    • R s = 1- 6 Σ d ² n ³-n
    • = 1- 6x12.
    • 12³-12
    • =+0.96 (a strong positive correlation)
  • Stage 4 and 5.
    • Stage 4 Rejection level = 95%
    • = 0.05.
    • Stage 5. T= R s n-2
    • 1- R s ²
    • = 0.96 12-2
    • 1-0.96²
    • = 10.73
  • Stage 6 and 7
    • Stage 6
    • Df= (n-2) = (12-2) = 10.
    • Stage 7 df = 10
    • Rejection value = 0.05
    • Therefore critical value of t =2.23.
    • The critical value is less than our t- value (10.73). We can therefore conclude that there is a significant correlation between settlement size and the number of services offered in each.