1. Statistics

2. Intro to statistics Presentations More on who to do qualitative analysis Tututorial time

3. Inferential statistics

8. Descriptive vs Inferential statistics Descriptive statistics like totals (how many people came?), percentages (what proportion of the total were adolescents?) and averages (how much did they enjoy it?) use numbers to describe things that happen. Descriptive data page Descriptive data page Inferential statistics infer or predict the differences and relationships between things. They also tell us how certain or confident we can be about the predictions.

9. Why statistics are important Statistics are concerned with difference – how much does one feature of an environment differ from another Suicide rates/100,000 people

10. Why statistics are important Relationships – how does much one feature of the environment change as another measure changes The response of the fear centre of white people to black faces depending on their exposure to diversity as adolescents

11. The two tasks of statistics Magnitude: What is the size of the difference or the strength of the relationship? Reliability. What is the degree to which the measures of the magnitude of variables can be replicated with other samples drawn from the same population.

12. Magnitude – what’s our measure? Suicide rates/100,000 people Raw number? Some aggregate of numbers? Mean, median, mode?

13. Arithmetic mean or average Mean (M or X), is the sum (  X) of all the sample values ((X 1 + X 2 + X 3.…… X 22 ) divided by the sample size (N). Mean/average =  X/N ABA*BCA*C Overall rating Gener al Unitec 2121 3002 43120 54 7 63 6 7 8 838 16 928 10 57___ 14___ N146 64

14. Compute the mean GeneralUnitec Total (  X) 1262493 N14664 mean8.647.70

15. The median median is the "middle" value of the sample. There are as many sample values above the sample median as below it. If the number (N) in the sample is odd, then the median = the value of that piece of data that is on the (N-1)/2+1 position of the sample ordered from smallest to largest value. E.g. If N=45, the median is the value of the data at the (45-1)/2+1=23 rd position If the sample size is even then the median is defined as the average of the value of N/2 position and N/2+1. If N=64, the median is the average of the 64/2 (32 nd ) and the 64/2+1(33 rd ) position

16. Other measures of central tendency The mode is the single most frequently occurring data value. If there are two or more values used equally frequently, then the data set is called bi- modal or tri-modal, etc The midrange is the midpoint of the sample - the average of the smallest and largest data values in the sample. (= (2+10)/2 =6 for both groups The geometric mean (log transformation) =8.46 (general) and 7.38 (Unitec) The harmonic mean (inverse transformation) =8.19 (general) and 6.94 (Unitec) Both these last measures give less weight to extreme scores

17. Compute the median and mode Overall ratingGeneralUnitec 211 302 430 547 636 7128 83816 92810 5714 N14664

18. Means, median, mode GeneralUnitec N14664 mean8.647.70 median98 mode108 geometric mean8.497.38 harmonic mean8.196.94

20. The underlying distribution of the data

21. Normal distribution

22. Data that looks like a normal distribution

23. Three things we must know before we can say events are different 1.the difference in mean scores of two or more events - the bigger the gap between means the greater the difference 2.the degree of variability in the data - the less variability the better, as it suggests that differences between are reliable

24. Variance and Standard Deviation These are estimates of the spread of data. They are calculated by measuring the distance between each data point and the mean variance (s 2 ) is the average of the squared deviations of each sample value from the mean = s 2 =  X-M) 2 /(N-1) The standard deviation (s) is the square root of the variance.

25. Calculating the Variance (s 2 ) and the Standard Deviation (s) for the Unitec sample Xn(X-Mu)(X-Mu) 2 *n Overall ratingUnitec 21-5.7032.5 32-4.7044.2 40-3.700.0 57-2.7051.1 66-1.7017.4 78-0.704.0 8160.301.4 9101.3016.8 10142.3073.9 N64 241.4 Mean Unitec (Mu)=7.70Variance=3.83 SD or s=1.96

26. All normal distributions have similar properties. The percentage of the scores that is between one standard deviation (s) below the mean and one standard deviation above is always 68.26% s

27. Is there a difference between Unitec and General overall OAP rating scores

28. Is there a significant difference between Unitec and General OAP rating scores ss

29. Three things we must know before we can say events are different 3.The extent to which the sample is representative of the population from which it is drawn - the bigger the sample the greater the likelihood that it represents the population from which it is drawn - small samples have unstable means. Big samples have stable means.

30. Estimating difference The measure of stability of the mean is the Standard Error of the Mean = standard deviation/the square root of the number in the sample. So stability of mean is determined by the variability in the sample (this can be affected by the consistency of measurement) and the size of the sample. The standard error of the mean (SEM) is the standard deviation of the normal distribution of the mean if we were to measure it again and again

31. Yes it’s significant. The mean of the smaller sample (Unitec) is not too variable. Its Standard Error of the Mean = 0.24. 1.96 *SE = 0.48 = the 95% confidence interval. The General mean falls outside this confidence interval ss

32. Is the difference between means significant? What is clear is that the mean of the General group is outside the area where there is a 95% chance that the mean for the Unitec Group will fall, so it is likely that the General mean comes from a different population as the Unitec mean. The convention is to say that if mean 2 falls outside of the area (the confidence interval) where 95% of mean 1 scores is estimated to be, then mean 2 is significantly different from mean 1. We say the probability of mean 1 and mean 2 being the same is less than 0.05 (p<0.05) and the difference is significant p

33. The significance of significance Not an opinion A sign that very specific criteria have been met A standardised way of saying that there is a There is a difference between two groups – p<0.05; There is no difference between two groups – p>0.05; There is a predictable relationship between two groups – p<0.05; or There is no predictable relationship between two groups - p>0.05. A way of getting around the problem of variability

34. If you argue for a one tailed test – saying the difference can only be in one direction, then you can add 2.5% error from side where no data is expected to the side where it is 2.5% of distri- bution 95% of distri- bution 2-tailed test 1-tailed test -1.96 +1.96 Standard deviations One and two tailed tests

35. T-test result t-Test: Two-Sample Assuming Unequal Variances General adultsUnitec adults Mean8.647.7 Variance2.343.83 Observations14664 t Stat for p<0.053.41 p one-tail0.00 t Critical one-tail1.66 p two-tail0.00 t Critical two-tail1.98

36. MasseyUnsworth Heights Mean9.238.33 Variance1.204.24 Observations5215 t Stat for p<0.051.62 p one-tail0.06 t Critical one-tail1.75 p two-tail0.12 t Critical two-tail2.12 malefemale Mean8.948.65 Variance1.552.28 Observations83125 t Stat for p<0.051.52 p one-tail0.07 t Critical one-tail1.65 p two-tail0.13 t Critical two-tail1.97

37. Correlations and Chi-square

38. The correlation with the glacier went unnoticed. The debate proceeded and receded with slow heated monotonous cold regularity although never reversing at the same point of disagreement. The correlation with the glacier went... The weight of paper and opinion now far-exceeding the frozen mountain, even at its zenith. But no amount of FSC vellum could paper over the crevasse cracked argument. The correlation with the glacier.... The blue-green water vein bled But no aerial artery replenished the source. The constant melt etching the message of increased bloodletting from the waning carcase

39. The correlation with the..... Lost in the science of the unknown. The pre-historic signpost, scarred by graffiti, slowly shrank and collapsed Its incremental deficit matched by political will. The correlation...... We are, we were, the new dinosaurs, like the sun-burnt beached berg doomed for demise in the new non-ice age. No-one will record its disappearance or ours. The correlation with humanity went unnoticed. http://allpoetry.com/poem/9257026- Correlation-by-JohnS http://allpoetry.com/poem/9257026- Correlation-by-JohnS Correlation by John S http://allpoetry.com/poem/9257026- Correlation-by-JohnShttp://allpoetry.com/poem/9257026- Correlation-by-JohnS

40. Yes it’s significant. The mean of the smaller sample (Unitec) is not too variable. Its Standard Error of the Mean = 0.24. 1.96 *SE = 0.48 = the 95% confidence interval. The General mean falls outside this confidence interval ss

41. Chi-square test - comparing OAP samples with the local populations MasseyOAP European56%49% Māori16%28% Pacific peoples18%13% Asian + MELAA16%8% Other ethnicity9%1% Total115%100% population49413300 The question: Is the Massey OAP sample representative of the cultural mix of the Massey population ?

42. What would we predict? Massey 2006 CensusOAP 2013 European146148 Māori4285 Pacific peoples4739 Asian + MELAA4225 Other ethnicity233 300 In red are the number of participants we would predict (we EXPECT) based on the percent in each category in the Massey population (2006). In blue is what we got (we OBSERVED). Is the match sufficiently close?

43. The chi-square ( χ2) test Culture OEO-E(0-E) 2 (0-E) 2 /E European1481461.913.660.03 Māori854243.261871.5044.84 Pacific peoples3947-7.9663.311.35 Asian + MELAA2542-16.74280.206.71 Other ethnicity323-20.48419.3617.86 N=300 chi-square= (the sum of (0-E) 2 /E) 70.79 Degrees of freedom = N-1 = 299 Value of chi-square (χ2) for p<0.05=335 Actual χ2 is less than 335, therefore there is no significant difference between the OAP sample and Massey population (O=Observed (OAP), E=Expected (2006 Census, Massey) Chi-square table Chi-square tableChi-square table Chi-square table click here to get the Chi-Square table

44. All the OAP sample show no significant difference (NS) compared with their local population chi- square (χ2)df=N-1 value to reach significance p<0.05outcome Total126.2410091075NS Massey70.79299335NS Glen Eden25.09238275NS Unsworth Heights62.5485102NS Avondale39.71120147NS Glendene67.53263300NS If the sample has the same cultural mix as the general population, that helps us in the claim that the outcomes of the research can be generally applied.

45. r=0.904 N=33 p<0.00

46. r =(  (X – M X )*((Y – M Y ))/(N*s X *s Y ) X = GDP purchasing power in $'000s Y= Better Life Index (0-10) M X =Mean of X = 25,200 M Y =Mean of Y= 6.34 s X =Standard deviation of X=7.02 s Y =Standard deviation of Y=1.44 r =correlation coefficient = +0.90 Is it significant? That depends on how big the sample is. For N=33, it is highly significant. Correlations are calculated using means and standard deviations and big samples are more reliable than small ones

47. Correlations vary from -1 to +1

48. To what extent has today's experience. 1=Hugely; 2=a good amount; 3=some-what; 4=a little bit; 5=not at all made your team more aware of what available in this community? made your team feel more a part of this community? encouraged team members to use a services/ resources they have come in contact with? put team members in closer touch with neighbours or friends helped team members make some new friends? given team members some ideas about changes they would like to make in their lives? made team members feel safer in this community? Overall rating: 10 = a wonderful day, 7-8 mostly fun, 5-6=good in parts, 3=mostly boring, 1 = no fun at all, where would you all rate the day?

49. 1-tail: p<0.050.0250.010.005 2-tail: p<0.10.050.020.01 DF=267 (=N-1)0.1020.1210.1440.159 N=268 r=-0.52 p<0.005

50. One or two tails? Have we made a prior prediction? Yes, that high engagement will create high satisfaction = 1 tailed test What degrees of freedom? df=N-1= 268-1 = 267 What level of significance should be chosen? It depends on the number of correlations. p<0.05 – there is only one correlation. Often there are 100’s – in which case a tougher criterion should be chosen, p<0.01. Where can we find the critical values of r? HERE HERE

51. Correlation and regression Correlation quantifies the degree to which two random variables are related. Correlation does not fit a line through the data points. You simply are computing a correlation coefficient (r) that tells you how much one variable tends to change when the other one does. Linear regression finds the best line that predicts the size of one variable when given another variable which is fixed. The regression co- efficient (r 2 ) tells how much of the variability of our fixed (dependent) variable is accounted for by the independent variable

52. A perfect relationship, but not a linear correlation

53. A powerful relationship, but not a correlation – what’s happening here?

54. Normality of the data and Homoscedasticity

55. r=0.904 N=33 p<0.00

57. How correlation is used and misused A - The Church Unlimited B - causes people to want freebies B - The Church Unlimited A - Misery C - Desire for Freebies There are so many ways that events can influence each other, that we have to take great about claiming causal relationships between events.