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The Normal Distribution: The Normal curve is a mathematical abstraction which conveniently describes ("models") many frequency distributions of scores in real-life.
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length of pickled gherkins: length of time before someone looks away in a staring contest:
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Francis Galton (1876) 'On the height and weight of boys aged 14, in town and country public schools.' Journal of the Anthropological Institute, 5, 174-180:
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An example of a normal distribution - the length of Sooty's magic wand... Length of wand Frequency of different wand lengths
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Properties of the Normal Distribution: 1. It is bell-shaped and asymptotic at the extremes.
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2. It's symmetrical around the mean.
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3. The mean, median and mode all have same value.
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4. It can be specified completely, once mean and s.d. are known.
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5. The area under the curve is directly proportional to the relative frequency of observations.
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e.g. here, 50% of scores fall below the mean, as does 50% of the area under the curve.
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e.g. here, 85% of scores fall below score X, corresponding to 85% of the area under the curve.
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Relationship between the normal curve and the standard deviation: All normal curves share this property: the s.d. cuts off a constant proportion of the distribution of scores:- -3 -2 -1 mean +1 +2 +3 Number of standard deviations either side of mean frequency 99.7%68%95%
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About 68% of scores will fall in the range of the mean plus and minus 1 s.d.; 95% in the range of the mean +/- 2 s.d.'s; 99.7% in the range of the mean +/- 3 s.d.'s. e.g.: I.Q. is normally distributed, with a mean of 100 and s.d. of 15. Therefore, 68% of people have I.Q's between 85 and 115 (100 +/- 15). 95% have I.Q.'s between 70 and 130 (100 +/- (2*15). 99.7% have I.Q's between 55 and 145 (100 +/- (3*15).
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85 (mean - 1 s.d.) 115 (mean + 1 s.d.) 68%
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Just by knowing the mean, s.d., and that scores are normally distributed, we can tell a lot about a population. If we encounter someone with a particular score, we can assess how they stand in relation to the rest of their group. e.g.: someone with an I.Q. of 145 is quite unusual: this is 3 s.d.'s above the mean. I.Q.'s of 3 s.d.'s or above occur in only 0.15% of the population [ (100-99.7) / 2 ].
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z-scores: z-scores are "standard scores". A z-score states the position of a raw score in relation to the mean of the distribution, using the standard deviation as the unit of measurement.
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1. Find the difference between a score and the mean of the set of scores. 2. Divide this difference by the s.d. (in order to assess how big it really is).
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Raw score distributions: A score, X, is expressed in the original units of measurement: z-score distribution: X is expressed in terms of its deviation from the mean (in s.d's) X = 65 X = 236 z = 1.5
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55 70 85 100 115 130 145 z-scores transform our original scores into scores with a mean of 0 and an s.d. of 1. Raw I.Q. scores (mean = 100, s.d. = 15):
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-3 -2 -1 0 +1 +2 +3 I.Q. as z-scores (mean = 0, s.d. = 1). z for 100 = (100-100) / 15 = 0, z for 115 = (115-100) / 15 = 1, z for 70 = (70-100) / -2, etc.
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Conclusions: Many psychological/biological properties are normally distributed. This is very important for statistical inference (extrapolating from samples to populations - more on this in later lectures...) z-scores provide a way of (a) comparing scores on different raw-score scales; (b) showing how a given score stands in relation to the overall set of scores.
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