IQ Scores by CategoryDate: 7/28/96 at 9:26:1 From: MIGUEL SPRUILL Subject: measures of dispersion, the normal distribution On standard IQ tests, the mean is 100, with standard deviation of 15. The results come very close to fitting a normal curve. Suppose an IQ test is given to a very large group of people. Find the percent of people whose IQ scores fall into the following categories: 1) greater than 115. 2) greater than 145. If you find both kinds of standard deviation, the sample standard deviation and the population standard deviation, which of the two will be a larger number for a given set of data? Find (a) the range, and (b )the standard deviation for each sample round fractional answers to the nearest hundredth. 1) 67, 83, 55, 68, 77, 63, 84, 72, 65 Date: 7/28/96 at 16:6:13 From: Doctor Anthony Subject: Re: measures of dispersion, the normal distribution To find the standardised values for use with normal tables you calculate z from z = (x-m)/s where m = mean and s = standard deviation. For greater than 115 we have z = (115.5 - 100)/15 = 15.5/15 = 1.0333. The tables give an area 0.8493, so tail area = 0.1507. So 15.07 percent of the population has an IQ above 115. For greater than 145 we have z = (145.5 - 100)/15 = 45.5/15 = 3.0333. The tables give an area .99878, so tail area = 0.00122. So 0.12 percent of the population has an IQ above 145. If you use a sample to estimate population mean and s.d. you lose one 'degree of freedom' in that with mean found, then you no longer have n independent values for finding the standard deviation. So in calculating variance from SIGMA(x-m)^2/n (sample variance), you would use SIGMA(x-m)^2/(n-1) if m and s are calculated from the sample as estimates for the population. So the standard deviation is greater if you use the sample to estimate population standard deviation. Find range and standard deviation of the sample 67, 83, 55, 68, 77, 63, 84, 72, 65. The range = 84 - 55 = 29. Variance of the sample is calculated from SIGMA[x^2/n] - mean^2 mean = 70.44 standard deviation = 8.995. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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