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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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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.
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.
Today • Introduction to statistics • Looking at our qualitative data in a quantitative way • Presentations • More exploration of the data • Tutorials
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
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
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.
Magnitude – what’s our measure? • Raw number? Rate? • Some aggregate of numbers? Mean, median, mode? Suicide rates/100,000 people
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.
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
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?
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
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
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.
Calculating the Variance and the standard deviation for the Polynesian sample
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%
Is there a significant difference between Polynesian and “other” scores
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.
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
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
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
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
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
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
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.
Gender Identity Conflict – Finland 1,142 (698 women and 444 men). Controls vs Gender conflicted men and women. Gender Identity Scale for Males (Freund et al. 1977). Questions for men: Have you ever felt that you actually are a woman? Have you ever wished you had a woman’s body? Questions for women Have you ever felt that you actually are a man? Have you ever wished you had a man’s body? Participants who had answered yes to at least one of these questions were coded as having a conflicted gender identity. Explore the data for significance (sample size, differences between means, p value), difference between men and women, what’s it mean? Mistakes in the data?
r=0.904 N=33 p<0.00
Correlations r =(S(X – MX)*((Y – MY))/(N*SX*SY) X = GDP purchasing power in $'000s Y= Better Life Index (0-10) MX=Mean of X = 25,200 MY =Mean of Y= 6.34 SX=Standard deviation of X=7.02 SY=Standard deviation of Y=1.44 r =correlation coefficient = +0.90
One or two tails? Have we made a prior prediction? Yes, that life satisfaction will increase with wealth = 1 tailed test What degrees of freedom? df=N-1= 33-1 = 32 What level of significance should be chosen? It depends on the number of correlations. p<0.05 – there is only one correlation. Often there 100’s – in which case a tougher criterion should be chosen. Where can we find the critical values of r? HERE
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 (r2) tells how much of the variability of our fixed (dependent) variable is accounted for by the independent variable
Client factors associated with treatment completion in a substance abuse treatment facility. Addiction Research and Theory, 18(6): 667–680 Look at the correlations. • What is significantly correlated with treatment completion? • What’s the direction of the correlation? • What should be the significance leve1 (p value)? • What’s correlated with the things that are correlated with treatment completion? • What can you tell from table 3 – the final model? • What’s it all mean in practice?
Client factors associated with treatment completion in a substance abuse treatment facility. Addiction Research and Theory, 18(6): 667–680 Look at the correlations. • What is significantly correlated with treatment completion? • What’s the direction of the correlation? • What should be the p-value for significance • What’s correlated with the things that are correlated with treatment completion? • What’s it all mean?
