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THE NORMAL DISTRIBUTION CURVE AND Z-SCORES LEARNING OBJECTIVES At the end of lecture students must be able to know, Properties of the normal curve. Mean and standard deviation of the normal curve. Area under the curve. Calculating z-scores. Probability. CARL GAUSS The normal curve is often called the Gaussian distribution, after Carl Friedrich Gauss, who discovered many of its properties. Gauss, commonly viewed as one of the greatest mathematicians of all time (if not the greatest), is honoured by Germany on their 10 Deutschmark bill. PROPERTIES OF THE NORMAL DISTRIBUTION: 1. IT IS BELL-SHAPED AND ASYMPTOTIC AT THE EXTREMES. 2. IT'S SYMMETRICAL AROUND THE MEAN. 3. THE MEAN, MEDIAN AND MODE ALL HAVE SAME VALUE. 4. IT CAN BE SPECIFIED COMPLETELY, ONCE MEAN AND SD ARE KNOWN. 5. THE AREA UNDER THE CURVE IS DIRECTLY PROPORTIONAL TO THE RELATIVE FREQUENCY OF OBSERVATIONS. e.g. here, 50% of scores fall below the mean, as does 50% of the area under the curve. RELATIONSHIP BETWEEN THE NORMAL CURVE AND THE STANDARD DEVIATION: frequency All normal curves share this property: the SD cuts off a constant proportion of the distribution of scores:- 68% 95% 99.7% -3 -2 -1 mean +1 +2 +3 Number of standard deviations either side of mean About 68% of scores fall in the range of the mean plus and minus 1 SD; 95% in the range of the mean +/- 2 SDs; 99.7% in the range of the mean +/- 3 SDs. e.g. IQ is normally distributed (mean = 100, SD = 15). 68% of people have IQs between 85 and 115 (100 +/- 15). 95% have IQs between 70 and 130 (100 +/- (2*15). 99.7% have IQs between 55 and 145 (100 +/- (3*15). 68% 85 (mean - 1 SD) 115 (mean + 1 SD) We can tell a lot about a population just from knowing the mean; SD; and that scores are normally distributed. 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 IQ of 145 is quite unusual (3 SDs above the mean). IQs of 3 SDs or above occur in only 0.15% of the population [ (10099.7) / 2 ]. 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. z raw score mean standard deviation for a population : z X μ σ for a sample : z X - X s RAW SCORE DISTRIBUTIONS: A score, X, is expressed in the original units of measurement: X = 236 X = 65 X 50 s 10 X 200 s 24 z = 1.5 X0 s 1 z-score distribution: X is expressed in terms of its deviation from the mean (in SDs). Z-SCORES TRANSFORM OUR ORIGINAL SCORES INTO SCORES WITH A MEAN OF 0 AND AN SD OF 1. Raw IQ scores (mean = 100, SD = 15): 55 70 85 100 115 130 145 +1 +2 +3 IQ as z-scores (mean = 0, SD = 1). z for 100 = (100-100) / 15 = 0, z for 115 = (115-100) / 15 = 1, z for 70 = (70-100) / -2, etc. -3 THANK YOU -2 -1 0