Transforms What does the word transform mean?

Transforms What does the word transform mean? –Changing something into another thing

Transforms What does the word transform mean? –Changing something into another thing In statistics it refers to changing a distribution into a different distribution

Transforms What does the word transform mean? –Changing something into another thing In statistics it refers to changing a distribution into a different distribution –How can you change a distribution?

Linear Transforms A linear transform happens when you add or multiply a constant with EACH number in a distribution –Usually has the form z i = a(x i ) + b Where y is the new “transformed” number, x is the old “untransformed number” and a and b are any constants (including zero!)

Linear Transforms What would happen to the mean, the variance, the standard deviation if you applied a linear transform?

Linear Transforms adding or multiplying by a constant affects the mean multiplying by a constant affects the variance and standard deviation but adding a constant does not!

Linear Transforms More formally, when adding a constant: the mean of a new distribution z is the mean of the old distribution x plus the constant c

Linear Transforms More formally, when adding a constant: the variance of the distribution z is the same as the variance of the distribution x

Linear Transforms More formally, when multiplying by a constant: the mean of the distribution z is the mean of x multiplied by the constant c

Linear Transforms More formally, when multiplying by a constant: the variance of the distribution z is the variance of x multiplied by the square of the constant c

Linear Transforms More formally, when multiplying by a constant: And the standard deviation

Linear Transforms If these features of transforms aren’t intuitive for you, work through pages 34 and 35!

Linear Transforms Notice that you can work backward from what you want the mean or standard deviation to be because: –Add or subtract a constant to change the mean –Multiply or divide a constant to change the standard deviation

The Z Transform

What if you wanted a very specific distribution - one with a mean of zero and a standard deviation of one Why on earth would you want THAT? The Z Transform

Often we can assume that a set of numbers are normally distributed Normally distributed numbers have interesting characteristics Normal Distributions

Importantly, the probability of any number being of a particular value can be computed using the gaussian function: Which you will almost certainly never need to compute yourself! Normal Distributions

Normal Distribution Probability of a score is the height on the curve mean standard deviation

The Normal Distribution 34% of scores fall between the mean and 1 standard deviation above the mean Standard Deviations 34%

The Normal Distribution 34% of scores fall between the mean and 1 standard deviation below the mean Standard Deviations 34%

The Normal Distribution 68% of scores fall between the 1 standard deviation below and 1 standard deviation above the mean Standard Deviations 34%

The Normal Distribution 96% of scores fall between the 2 standard deviations below and 2 standard deviations above the mean Standard Deviations 48%

The Normal Distribution 95% of scores fall between 1.96 standard deviations below and 1.96 standard deviations above the mean Standard Deviations 48%

The Normal Distribution The Normal distribution reveals the proportions (i.e. probabilities) of scores that fall within certain ranges when the ranges are expressed in terms of standard deviations

The Normal Distribution The Normal distribution reveals the proportions (i.e. probabilities) of scores that fall within certain ranges when the ranges are expressed in terms of standard deviations If only there was some way to transform scores into units of standard deviation…

The Z Transform

Break that down: –Remember that –And that The Z Transform

Break that down: –If we used this to make the new mean zero by plugging in the negative of the old mean for c The Z Transform

Break that down: –If we used this to make the new mean zero by plugging in the negative of the old mean for c –And we use this to make the standard deviation equal 1 by plugging in 1 / the old standard deviation The Z Transform

-subtract the mean from each score -Divide each score by the standard deviation

–Then any score that was exactly the mean would be zero standard deviations from the mean (Z = 0.0) –A score that was 1 standard deviation from the mean would now be Z = 1.0 –2 standard deviations from the mean would be Z = 2.0 –Half way between 1 and 2 std. dev. from the mean would be 1.5, etc. The Z Transform

Z scores are standardized Z scores are in units of standard deviation One can think of Z scores as the ratio of a score’s difference from the mean to the average difference from the mean Or one can think of Z score as “what percentage of one standard deviation from the mean is this score’s distance from the mean?” The Z Transform

Uses of Z scores: –allows comparison across different samples (e.g. 25 degrees in Vancouver vs. 25 degrees in Lethbridge) –If one assumes that scores are normally distributed, the Z score reveals the probability of that particular score occurring by chance The Z Transform

The Standard Normal Distribution or Z Distribution For any distribution: –probability for each number (on x axis) is given by height of curve –probability for getting one out of a range of numbers is given by the area under the curve

Standard Normal Distribution For the Standard Normal (a.k.a. Z distribution), the area under the curve for a given range is found in a Z table e.g. pg 111

Standard Normal Distribution Table shows areas between  and any z score you wish What’s  !?  is the mean of the population of possible scores (more on that later)

Standard Normal Distribution Using the Z table –note that negative z scores yield the same probabilities because the curve is symmetric –total area under the curve = 1.0 (probability that something will happen is 1 ! ) –Examples probability of getting a z score between 0 and 1 is.3413 probability of getting a z score within 1 std. dev. of the mean is =.6826 or ~ 68%

Standard Normal Distribution What range above and below the mean contains 95% of all the z scores?

Standard Normal Distribution What range above and below the mean contains 95% of all the z scores? –Z table tells you the positive half of the curve

Standard Normal Distribution What range above and below the mean contains 95% of all the z scores? –Z table tells you the positive half of the curve –1/2 of 95% = 47.5% or.475 is on each side of the mean

Standard Normal Distribution What range above and below the mean contains 95% of all the z scores? –Z table tells you the positive half of the curve –1/2 of 95% = 47.5% or.475 is on each side of the mean –.475 corresponds to a z score of 1.96 (or negative 1.96! )

Standard Normal Distribution What range above and below the mean contains 95% of all the z scores? –Z table tells you the positive half of the curve –1/2 of 95% = 47.5% or.475 is on each side of the mean –.475 corresponds to a z score of 1.96 (or negative 1.96! ) –Thus or 95% of z scores fall between + / standard deviations of the mean

Standard Normal Distribution What do we do with this knowledge?

Standard Normal Distribution What do we do with this knowledge? Knowing the probability of getting particular z scores helps us to know what population a given sample came from

Standard Normal Distribution For example: Blood Doping in Cross-Country Skiing

Standard Normal Distribution For example: Blood Doping in Cross-Country Skiing At the World Championships in Lahti Finland, 13% of the athletes were found to have a red blood cell count between 3.5 and 5.5 standard deviations from the presumed population mean for all athletes (that was measured in previous IOC study)

Standard Normal Distribution What percentage of athletes would you expect to be greater than 3.5 standard deviations from the mean?

Standard Normal Distribution What percentage of athletes would you expect to be greater than 3.5 standard deviations from the mean? –look up z = 3.5 –the associated probability is.4998 so 2x.4998 =.9996 or 99.96% should fall within +/- 3.5 ! –less than =.0004 or.04% should have red blood cell z scores greater than +/-3.5! –only half (.02%) of that should be above +3.5 !

Standard Normal Distribution.02% is what you’d expect - 13% is what they observed! What this tells us is that the sample of athletes at the Lahti World Championships was almost certainly not taken from the same population as the “normal” athletes in the IOC study At least some of the athletes sampled in Lahti had done something to artificially elevate their red blood cell count

Problem: What good is it to know about normal distributions if there’s no guarantee that your scores will be normally distributed?