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Statistics

Statistics. A Word on Statistics - Wislawa Szymborska. Led to error by youth (which passes): sixty, plus or minus. Those not to be messed with: four-and-forty. Living in constant fear of someone or something: seventy-seven. Capable of happiness: twenty-some-odd at most. Harmless alone,

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Statistics

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  1. Statistics

  2. A Word on Statistics - Wislawa Szymborska Led to error by youth (which passes): sixty, plus or minus. Those not to be messed with: four-and-forty. Living in constant fear of someone or something: seventy-seven. Capable of happiness: twenty-some-odd at most. Harmless alone, turning savage in crowds: more than half, for sure. Out of every hundred people, those who always know better: fifty-two. Unsure of every step: almost all the rest. Ready to help, if it doesn't take long: forty-nine. Always good, because they cannot be otherwise: four -- well, maybe five. Able to admire without envy: eighteen.

  3. Those who are just: quite a few, thirty-five. But if it takes effort to understand: three. Worthy of empathy: ninety-nine. Mortal: one hundred out of one hundred -- a figure that has never varied yet. Cruel when forced by circumstances: it's better not to know, not even approximately. Wise in hindsight: not many more than wise in foresight. Getting nothing out of life except things: thirty (though I would like to be wrong). Balled up in pain and without a flashlight in the dark: eighty-three, sooner or later.

  4. Today • Introduction to statistics • Looking at our qualitative data in a quantitative way • Presentations • More exploration of the data • Tutorials

  5. 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

  6. 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

  7. 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.

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

  9. How safe do MPHS people feel? • Feeling safe in their own home: yes=1, no=0 • Feeling safe in their local part of MPHS: yes =1, no=0 • Feeling safe in MPHS generally: yes=1, no=0 Total safety score = add 1-3. range=0 to 3. If people don’t refer to 1. above, score it as =1, If people score 0 on 2, they must be 0 on 3.

  10. Arithmetic mean or average Mean (M or X), is the sum (SX) of all the sample values ((X1 + X2 +X3.…… X22) divided by the sample size (N). Mean/average = SX/N - Carbon footprint scores

  11. Compute the mean

  12. 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=23rd 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=32, the median is the average of the 32/2 (16th) and the 32/2+1(17th) position. Why use the median?

  13. 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. • The geometric mean (log transformation) and the harmonic mean (inverse transformation) – both used where data is skewed with the aim of creating a more even distribution

  14. Compute the median and mode

  15. Mean, median, mode, mid-range

  16. The underlying distribution of the data

  17. Normal distribution

  18. Three things we must know before we can say events are different • the difference in mean scores of two or more events - the bigger the gap between means the greater the difference • the degree of variability in the data - the less variability the better

  19. 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 (s2) is the average of the squared deviations of each sample value from the mean = s2 = S(X-M)2/(N-1) The standard deviation (s) is the square root of the variance.

  20. Calculating the Variance and the standard deviation for the Polynesian sample

  21. 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%

  22. Is there a difference between Polynesian and “other” scores

  23. Is there a significant difference between Polynesian and “other” scores

  24. Three things we must know before we can say events are different • 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.

  25. 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

  26. Yes it’s significant. The mean of the smaller sample is not too variable. Its Standard Error of the Mean = 6.5/√13 = 1.80. The 95% confidence interval =1.96 SDs = 3.52. This gives a range from 71.2 to 78.2. The “Other” mean falls just outside this confidence interval Polynesian Mean =74.7 SD=6.5 N= 13 Distribution of Standard error of the mean

  27. Is the difference between means significant? What is clear is that the mean of the Other group is just outside the area where there is a 95% chance that the mean for the Polynesian Group will fall, so it is likely that the Other mean comes from a different population as the Polynesian mean. The convention is to say that if mean 2 falls outside of the area (the confidence interval) where 95% of mean 1 scores are 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

  28. 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

  29. 2.5% of M1 distri-bution 95% of M1 distri-bution 2.5% of M1 distri-bution One and two tailed tests 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 the side where no data is expected to the side where it is 1-tailed test 2-tailed test -1.96 +1.96 Standard deviations

  30. If we were to argue for a one tailed test – that Polynesian people were more eco-sustaintable, than the Others – the 95% confidence interval can all be to the left of the of the SEM distribution rather than equally distributed on either side. This means that instead of going to 47.5% line on the right we go to the 45% line = 1.65 SDs or 3.0 units Normal distribution Polynesian Mean =74.7 SD=6.5 N= 13 Distribution of Standard error of the mean

  31. T-tests t = (Mx-My)/Sx2/Nx + Sy2/Ny; where t is value generated and : Mx= the mean carbon footprint of participants with higher incomes My= the mean carbon footprint of participants with moderate to low incomes Sx2=the variance of the carbon footprint of participants with higher incomes Sy2= the variance of the carbon footprint of participants with moderate to low incomes Nx=the number of participants with higher incomes Ny=the number of participants with moderate to low incomes

  32. T-test result. This does exactly what we have done except it argues that in every sample the first data point is fixed and that other data points are free to vary in relation to it. Consequently, when estimating variance we should divide by (N-1) not N. That makes this test more conservative.

  33. Impact of gender on safety

  34. Impact of religion on safety

  35. Impact of work on safety

  36. Correlations and Chi-square

  37. The correlation with the glacier went unnoticed.The debate proceeded and receded with slow heated monotonous cold regularityalthough never reversing at the same point of disagreement.The correlation with the glacier went. . . The weight of paper and opinionnow 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 messageof increased bloodletting from the waning carcase

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

  39. Chi-square test - comparing MPHS samples with the local populations • Looks at the magnitude or size of the difference between observed and expected values (O-E) and then squares those differences to they are all positive - (O-E)2, • Adjust those differences so they are relative to the size of the expected values - (O-E)2/E. This is a variance measure and takes care of effects that are due to the size of the expected value, which in turn is related to the sample size. • Calculates a chi-square value which is the sum of the adjusted differences ( S(O-E)2/E)=14.03). This is compared with the value that chi-squared would have to reach to be significant for the number of categories used (n). • The question: Is the MPHS sample representative of the cultural mix of the MPHS population?

  40. What would we predict? In red are the number of participants we would predict (we EXPECT) based on the percent in each category in the MPHS population (2006). In blue is what we got (we OBSERVED). Is the match sufficiently close?

  41. Does the MPHS sample match the population age distribution?

  42. Does the MPHS sample match the population age distribution?

  43. r=0.904 N=33 p<0.00

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