Calculating the Standard Deviation

							Mulhorn, ANE 560 Calculating the Standard Deviation

Calculating the Standard Deviation Two methods of calculating the standard deviation are described in order to elucidate the mathematics involved. Basically there are two different standard deviations: one has DIVISOR n, the other has DIVISOR n-1

NB we have omitted subscripts for clarity
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The "divisor n" formula is used if the data set is known to be the whole population, i.e. the complete set of data you are interested in. (This formula is used at GCSE and in introductory courses.) The "divisor n-1" formula is used when you are working with a sample data set, and wish to estimate the standard deviation of the parent population. (This is a more sophisticated concept, met when studying statistical inference.)

The deviation method is considered first, since it closely parallels the concept of standard deviation. The raw score method is presented as a convenient computation alternative. Deviation method for calculating standard deviation Consider the observations 8,25,7,5,8,3,10,12,9. 1. First, calculate the mean and determine N. 2. Remember, the mean is the sum of scores divided by N where N is the number of scores. 3. Therefore, the mean = (8+25+7+5+8+3+10+12+9) / 9 or 9.67 4. Then, calculate the standard deviation as illustrated below.

Mulhorn, ANE 560
squared deviation 2.79 235.01 7.13 21.81 2.79 44.49 .11 5.43 .45 320.01

score 8 25 7 5 8 3 10 12 9

mean deviation* 9.67 - 1.67 9.67 +15.33 9.67 - 2.67 9.67 - 4.67 9.67 - 1.67 9.67 - 6.67 9.67 + .33 9.67 + 2.33 9.67 - .67 sum of squared dev=

*deviation from mean=score-mean

Standard Deviation = = = =

Square root(sum of squared deviations / (N-1) Square root(320.01/(9-1)) Square root(40) 6.32

Raw score method for calculating standard deviation Again, consider the observations 8,25,7,5,8,3,10,12,9. 1. 2. 3. 4. First, square each of the scores. Determine N, which is the number of scores. Compute the sum of X and the sum of Xsquared. Then, calculate the standard deviation as illustrated below.

score(X) 8 25 7 5 8 3 10 12 9 --87

Xsquared 64 625 49 25 64 9 100 144 81 ---1161 N = 9 sum of X = 87 sum of Xsquared = 1161

Standard Deviation = of X)/N))/ (N-1)) ] = = = = = =

square root of[ (sum of Xsquared -((sum of X)*(sum square square square square square 6.32 root[(1161)-(87*87)/9)/(9-1)] root[(1161-(7569/9)/8)] root[(1161-841)/8] root[320/8] root[40]

Mulhorn, ANE 560 Well, we are all glad that we have computer software like JMP IN to make statistical calculations. Even simple statistics, such as the standard deviation, are tedious to calculate "by hand".