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To calculate this by hand, 1 Calculate the mean: ( x ) 2 Subtract each value in the set from the mean (thus creating another set): ( x x i ) 3 Square each of the values: ( x x i )2
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n 4 Add together all the values in this set (that is, sum the squares): i =1( x x i )2 n 5 Divide the sum by the number of values in the set: i =1 ( x x i )2 /n n ( x x )2/ n 6 Take the square root of this value: i =1 i
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The other type of standard deviation question you might encounter on the GMAT provides you with the standard deviation for a set and tests whether you know what that means A question could appear like the following: A set has an average (arithmetic mean) of 42 and a standard deviation of 18 Which of the following values is two standard deviations away from the mean A 36 B 384 C 402 D 438 E 756 To find the numbers that are one standard deviation away from the mean, you add the standard deviation to, and subtract it from, the mean To find the numbers that are two standard deviations from the mean, you add the standard deviation to the mean twice or subtract it twice from the mean In this question, this means 42 + (18) + (18) = 456 or 42 (18) (18) = 384 Thus, answer B is the correct answer If you wanted a number that was three standard deviations from the mean, you would add the standard deviations to the mean three times or subtract it three times from the mean, and so on You should have enough information now to answer the question posed at the beginning of the chapter: If the set S is composed of the following numbers {99, 100, 100, 105, 106, 116, 123}, which of the following is largest A The average (arithmetic mean) of set S B The median of set S C The mode of set S D The range of set S E The standard deviation of set S Let s start with choice A We ve calculated the mean, and we know that it is 107 This is larger than the median, which is 105; even if we had not calculated the mean exactly, we could estimate that the mean is greater than the median because the three numbers greater than the median are relatively farther away from 105 than the three numbers less than the median We know that the mode is 100, which is less than 107 We know that the range is 123 99 = 24, which is less than 107 And while we don t want to waste time calculating the standard deviation of this set, we can see that the numbers
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aren t too far apart, so we can assume that the standard deviation is a relatively small number (it s actually around 9) Therefore, the answer has to be A, the average of set S
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The mean The median is the average of a set, or the sum of its values divided by the number of elements is the middle value of a set Note that if there is an odd number of elements in the set, the median will be in the set, but if there is an even number of elements, the median will be the value halfway between the two middle-most elements The mode The range The standard deviation is the most common element in the set Note that there may be more than one mode is the span of the set It is the smallest value subtracted from the largest is a measure of the degree to which the values in a set are spread out
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