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- 1. BIOLOGY Spacebar to continue
- 2. Introduction • Biological studies deal with organisms which show variety • We cannot rely on a single measurement and so we must take a sample • This sample of data must be summarised and analyzed to find out if it is reliable Spacebar to continue
- 3. Summarising data • MEAN Sum of samples ÷ sample size x ÷ n • MEDIAN Middle number in a list when arranged in rank order: 2, 5, 7, 7, 8, 23, 31 • MODE The measurement which occurs most frequently ; 2, 5, 7, 7, 8, 23, 31 Spacebar to continue
- 4. Distribution Curves • A visual summary of data • They can be produced by; 1. Collect data 2. Split results into equal size classes 3. Make a tally chart 4. Plot a histogram of frequency against size class • Data can show normal distribution or skewed distribution Spacebar to continue
- 5. Distribution curves • Normal distribution • Symmetrical bell shaped curve around the mean • Use parametric tests to analyse data 16 14 12 10 8 6 4 2 0 Spacebar to continue
- 6. Distribution curves • Skewed data • Asymmetrical curve around the mode • Use non-parametric tests to analyse data 18 16 14 12 10 8 6 4 2 0 Spacebar to continue
- 7. Standard Deviation • Standard deviation (SD) is a measure of the spread of the data Large SD Small SD
- 8. Standard deviation • A high SD indicates data which shows great variation from the mean • A low SD indicates data which shows little variation from the mean value • By definition, 68% of all data values lie within the range MEAN 1SD • 95% of all values lie within 2SD Spacebar to continue
- 9. SD and confidence limits 14 • 12 10 8 6 4 2 0 68% 95%
- 10. Calculating SD • Can only be used for normally distributed data • Calculate as follows; – – – – – Sum the values for x2 ie ( x2) Sum the values for x, then square it ie ( x)2 Divide ( x)2 by n Take one from the other and divide by n Take the square root of this. (see hand-out) Spacebar to continue
- 11. Calculating SD S= x2 - (( x)2/n) n Spacebar to continue
- 12. Confidence limits • 95% of all values lie within 2SD of the mean • Any value which lies outside this range is said to be significantly different from the others • We say that we are working to 95% confidence limits or to a 5% significance level. Spacebar to continue
- 13. Comparison tests • To compare two samples of data we look at the overlap between the two distribution curves. • This depends on; – The distance between the two mean values – The spread of each sample (standard deviation) • The greater the overlap, the more similar the two samples are. Spacebar to continue
- 14. Comparison tests Mean Mean Sample 2 Overlap Sample 1 Spacebar to continue
- 15. Comparison tests When the SD is small, the overlap is less; Sample 2 Overlap Sample 1 Spacebar to continue
- 16. The null hypothesis • In order to compare two sets of data we must first assume that there is no difference between them. • This is called the null hypothesis • We must also produce an alternative hypothesis which states that there is a difference. Spacebar to continue
- 17. The t-test • Used to compare the overlap of two sets of data • Samples must show normal distribution • Sample size (n) should be greater than 30 • This tests for differences between two sets of data Spacebar to continue
- 18. The t-test • To calculate t; – Check data is normally distributed by drawing a tally chart – Work out difference in means |x1 – x2| – Calculate variance for each set of data (this is s2 n) – Put these into the equation for t: Spacebar to continue
- 19. The t-test |x1 – x2| t= s12 n1 s22 n2 Spacebar to continue
- 20. The t-test • Compare the value of t with the critical value at n1 + n2 – 2 degrees of freedom • Use a probability value of 5% • If t is greater than the critical value we can reject the null hypothesis… • … there is a significant difference between the two sets of data • … there is only a 5% chance that any similarity is due to chance
- 21. Mann-Whitney u-test • Compares two sets of data • Data can be skewed • Sample size can be small; 5<n<30 • For details refer to stats book Spacebar to continue
- 22. Chi squared • Some data is categoric • This means that it belongs to one or more categories • Examples include – eye colour – presence or absence data – texture of seeds • For these we use a chi squared test 2 • This tests for an association between two or more variables
- 23. Chi squared • Draw a contingency table • These are the observed values Blue eyes Green eyes Row totals Fair hair a b a+b Ginger hair c d c+d Column totals a+c b+d a+b+c+d
- 24. Chi squared • Now work out the expected values: • Where, (Row total) x (Column total) E= (Grand total)
- 25. Chi squared Blue eyes Fair hair Ginger hair Column totals Green eyes (a+b)(a+c) (a+b+c+d) (c+d)(a+c) (a+b+c+d) (a+b)(b+d) (a+b+c+d) (c+d)(b+d) (a+b+c+d) a+c b+d Row totals a+b c+d a+b+c+d
- 26. Chi squared • For each box work out (O-E)2 E • Find the sum of these to get 2 (O-E)2 2 = E
- 27. Chi squared • Compare 2 with the critical value at 5% confidence limits • There will be (no. rows – 1) x (no. columns – 1) degrees of freedom • If 2 is greater than the critical value we can say that the variables are associated with one another in some way • We reject the null hypothesis
- 28. Spearman Rank • Two sets of data may show a correlation • The data can be plotted on a scatter graph: Negative correlation Positive correlation No correlation
- 29. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank Data 2 Rank 12 24 14 29 18 29 18 38
- 30. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank 12 14 1 Data 2 Rank This is the Lowest value – So we call it rank 1 24 29 18 29 18 38
- 31. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank 12 14 18 18 Data 2 Rank 1 2 24 This is the 2nd lowest value – so we call it rank 2 29 29 38
- 32. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank 12 1 14 2 18 ? 18 ? Data 2 Rank These should be rank 3 & 4 – but they are the same. We find the average of 3 + 4 and give them this rank 24 29 29 38
- 33. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank Data 2 Rank 12 1 24 14 2 29 18 3.5 29 18 3.5 (3+4)/2 = 3.5 38
- 34. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank Data 2 Rank 12 1 14 2 18 3.5 29 18 3.5 38 Similarly on this side 24 29
- 35. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank Data 2 Rank 12 1 24 14 2 29 18 3.5 29 18 3.5 38 1
- 36. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank Data 2 Rank 12 1 24 1 14 2 29 2.5 18 3.5 29 2.5 18 3.5 The average of 2 & 3 38
- 37. Spearman Rank • We calculate the correlation by assigning a rank to the values: Data 1 Rank Data 2 Rank 12 1 24 1 14 2 29 2.5 18 3.5 29 2.5 18 3.5 38 4
- 38. Spearman Rank • • • • Find the difference D between each rank Square this difference Sum the D2 values Calculate the Spearman Rank Correlation Coefficient rs 6 D2 rs = 1 - n(n2-1)
- 39. Spearman Rank • Compare rs with the critical value at the 5% level • If it is greater than the critical value (ignoring the sign) then we reject the null hypothesis • … there is a significant correlation between the two sets of data • If the value is positive there is a positive correlation • If it is negative then there is a negative correlation
- 40. Quick guide Is your data interval data or is it categoric data (it can only be placed in a number of categories) Interval Categoric
- 41. Quick guide Are you looking for a correlation between two sets of data – eg the rate of photosynthesis and light intensity Yes No
- 42. Quick guide Use the Chi squared test Back End Chi squared
- 43. Quick guide Use the Spearman Rank test Back End Chi squared
- 44. Quick guide Are you comparing data from two populations? Yes No
- 45. Quick guide Is your data normally distributed? 16 14 12 10 8 6 4 2 0 Yes No
- 46. Quick guide Use a t-test t-test Back
- 47. Quick guide Use a Mann-Whitney U test Back Exit

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