800 CHAPTER 12 Statistics Now we will develop a formula for determining the standard deviation of a set of data. If we call the individual data x and the mean x, we could write the three column heads Data, − Data Mean, and − (Data Mean)2 in Table 12.5 as − − x x x x x ( )2 Let’s follow the procedure we used to obtain the standard deviation in Example 2. We determined the sum of the − (Data Mean)2 column, which is the same as the sum of the − x x ( )2 column. We can represent the sum of the − x x ( )2 column by using the summation notation, Σ − x x ( ) .2 Thus, in Table 12.6, Σ − = x x ( ) 80. 2 We then The sum of the values in the − Data Mean column should always be zero; if not, you have made an error. (If a rounded value of x is used, the sum of the values in the − Data Mean column will not always be exactly zero; however, the sum will be very close to zero.) Table 12.5 Data − Data Mean − Data Mean ( )2 7 − = − 7 12 5 9 − = − 9 12 3 11 − = − 11 12 1 12 − = 12 12 0 15 − = 15 12 3 18 − = 18 12 6 0 Next square the values in the second column and place the squares in the third column (Table 12.6). Table 12.6 Data − Data Mean − Data Mean ( )2 7 −5 − = − − = ( 5) ( 5)( 5) 25 2 − = − − = ( 3) ( 3)( 3) 9 2 − = − − = ( 1) ( 1)( 1) 1 2 = = (0) (0)(0) 0 2 = = (3) (3)(3) 9 2 = = (6) (6)(6) 36 80 2 9 −3 11 −1 12 0 15 3 18 6 0 Add the squares in the third column. In this case, the sum is 80. Divide this sum by one less than the number of pieces of data − n( 1). In this case, the number of pieces of data is 6. Therefore, we divide by 5 and get = 80 5 16 Finally, take the square root of this number. Since = 16 4, the standard deviation, symbolized s, is 4. 7 Now try Exercise 13 MATHEMATICS TODAY Statistics and Opera Houses Architects have developed a mathematical rule based on statistics to help them construct opera houses with exceptional acoustics. The rule was first developed by having conductors rate the overall sound quality in 23 opera houses. Then acoustical engineers measured several acoustical properties in those 23 buildings. By using statistical analysis, the engineers were able to determine which combination of properties produced exceptional sound and which of the acoustic characteristics were most important. This mathematical rule is now used in the development of new opera houses. Why This Is Important Statistics is used is many different professions, including architecture and engineering. m Sydney Opera House in Sydney, Australia Bella Falk/Alamy Stock Photo
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