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Sampling Distributions Suppose I throw a dice 10000 times and count the number of times each face turns up: Each score has a similar frequency (uniform distribution)

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Sampling Distributions If instead you throw the dice 10 times (or throw ten dice) and take the average score each time, you get something like this: + ++ + + + + + + } { 10

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Sampling Distributions Compare averaging 10 vs 20 throws each go:

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10 x 20 x Note what happens to the spread and shape of the distribution of average scores 1 x

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1.Central Limit Theorem This is a theorem of statistics and probability that implies that the distribution of a sum (or average) of any set of scores approaches a Normal Distribution as the number of scores involved in the sum (or average) gets larger and larger. Single ThrowsAverage of N Throws Light Bulb Life Average Life

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2. Relation between the variation between individual scores and the variation between the averages of several scores. If the individual scores (values) in a population have a Variance of X then the variance of the averages of samples of size n has a variance of X/10. This is intuitive – think of individual heights:

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Population distribution of Individual Heights 5’5” 4’0” 7’0” (Population SD) approx = 7”

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Population distribution of raw scores 68% of scores lie within 1 standard deviation Of the mean 5’5” 4’0”7’0” 68% of people have a height between 4’10” and 6’0” 6’0”4’10”

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Suppose we take a random sample of 10 people and measure their heights: 5’5” 4’0” 7’0” The mean of the sample (x ) will tend to be quite close to the average height: x

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Keep taking samples of 10 people and measure average height: 5’5” 4’0” 7’0” x Back to17

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Keep taking samples of 50 people and measure average height: 5’5” 4’0” 7’0” x Back to17

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Distribution of Sample Means x cluster around the population mean more closely than the raw scores do: 5’5” 4’0” 7’0”

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The degree of spread (standard deviation of the sample means) around the population mean depends on the number (n) in each sample. 5’5” 4’0” 7’0” n=10 n=20 n=30

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Variance and SD As we observed before the Variance of sample means is the variance of the population of individual scores divided by the sample size. Because the Standard Deviation is the square root of the Variance, the Standard Deviation of the sample means is equal to the Standard Deviation of the individual scores divided by the square root of the sample size.

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The amount of variation (standard deviation of the sample means) around the population mean depends on the number (n) in each sample. The standard deviation of sample means of size n around the population mean is equal to the population standard deviation divided by √n and is called the standard error of the mean (se)

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5’5” 4’0” 7’0” Raw scores SD= 7” Samples of size 10 SD of the sample means = 7/sqrt(10) = 7/3.16 = 2.2 7” 2.2”

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Quick Summary We get an idea of the amount of variation in the population of individual scores from the variation within our sample (i.e. the data). Given that our sample average is from x number of scores we know how the sample averages would be expected to vary from one sample to the next.

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T-Test The T-Test works by assuming the data collected in two conditions is equivalent to collecting two samples from the same ‘parent’ population (this is the null hypothesis). The variation within the data is a good estimate of the variation in the parent population. This, together with the size of the samples, allows one to predict how much variation to expect in the means of one sample to the next. E.g.

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T Test If the two sample means obtained in the experiment conditions vary by more than we’d expect from this simple relation between the variation of individual scores and sample averages then it is unlikely that the data in the two conditions is equivalent to two samples from the same parent population. It is more likely they reflect two samples from different parent populations (i.e. one’s with different means)

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I.e. if the data does reflect samples from the same population we expect our samples, say of size 10, to cluster around the population mean quite closely: 5’5” 4’0” 7’0” Parent population of individual scores Expected variation of samples of size 10

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Not: 5’5” 4’0” 7’0” Expected variation of samples of size 10 Parent population of individual scores

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It is more likely that the real situation is that the two samples come from different parent populations: 5’5” 4’0” 7’0” 6’5”

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So an experiment selects 8 babies at random and feeds half Marmite and half Bovril. Heights measured at 20 years. Vs.

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6’5” It is more likely that the real situation is that the two samples come from different parent populations: 5’5” 4’0” 7’0”

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T-Test & ANOVA The T-Test works by computing the likelihood of getting a certain difference between two sample means. If you have experiments with more than 2 conditions there is no single distance between two means. Instead you can examine the ‘average’ distance or variation between them. The Variance of those condition means is just such a measure. ANOVA works out how likely it is to get the observed amount of variation (Variance) between several sample means if they really had been drawn from the same parent population.

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In a nutshell The data from the conditions of an experiment can be conceptualised as samples from a parent population. The null hypothesis assumes that these samples have been drawn from a single population. If the variation (or just difference in the case of a T-Test) between the means of these ‘samples’ is greater than we would expect given the samples size used, then we conclude that it is unlikely that they can be thought of as having been drawn from a single population but instead come from separate ones (i.e. ones that have different means).

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Some minor details: The T-test actually works out the sampling distribution of the difference between two means. When the probability of getting the observed difference is less than 5% H0 is rejected – i.e. the two populations from which the means were drawn are assumed not to be equal. ANOVA works out: 1. How the sample means vary and 2. How they should vary given their size and the individual variation If these two estimates differ widely then H0 is rejected.

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8 Sampling Distribution of the Mean Chapter8 p197 8.1 Sampling Distributions Population mean and standard deviation, and unknown Maximal Likelihood.

8 Sampling Distribution of the Mean Chapter8 p197 8.1 Sampling Distributions Population mean and standard deviation, and unknown Maximal Likelihood.

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