calculating std dev

Calculating a Standard Deviation

According to Drummond & Jones (2006), a standard deviation "is the numerical value that describes the spread of scores away from the mean and is expressed in the same units as the original scores. The wider the spread of scores, the larger the standard deviation."

A standard deviation is calculated by subtracting the mean of a distribution from the value of each individual variable in the distribution, squaring each resulting difference, summing these squared differences, then dividing this sum by the number of variables, and finally taking the square root of this quotient. The formula for this process is often represented as follows:



Alternately, Drummond & Jones (2006) use s in place of s . They also suggest the following formula as a more convenient means of calculating a standard deviation:



Now, apply this formula to calculate a standard deviation for the following distribution:

X1 = 79

X2 = 85

X3 = 92

X4 = 87

X5 = 93

X6 = 99

There are several intermediate calculations which must be performed before proceeding to the standard deviation calculation. Let's calculate the mean first. Remember that the mean is calculated by summing the variables then dividing by the number (count) of variables included in the distribution. In this instance the sum of the variables (79+85+92+87+93+99) equals 535. The count equals 6. When we divide 535 by 6, we get a quotient of 89.17. This is the mean for this distribution. Let's go ahead and square the mean, getting a value of 7951.29. Next, let's square each variable, then sum them [(79 x 79)+(85 x 85)+(92 x 92)+(87 x 87)+(93 x 93)+(99 x 99)]. Performing this calculation yields a value of 47,949. Then, we divide this value by 6, giving us 7991.5. Now, we can subtract the squared mean (7951.29) from 7991.5. This gives us a value of 40.21. Finally, we must take the square root of this value, arriving at our standard deviation value of 6.34.

Following the formula mathematically looks like this:
















When we have a small sample (typically 20 or fewer variables) it is generally recommended that we substitute n-1 for n so that the standard deviation is not underestimated. Let's calculate a standard deviation using the original formula shown above, but substituting n-1 in place of n. We'll use the same variables as before. Remember, the mean equals 89.17. First, let's subtract the mean from each variable value: 79 - 89.17 = -10.17, 85 - 89.17 = -4.17, 92 - 89.17 = 2.83, 87 - 89.17 = -2.17, 93 - 89.17 = 3.83, and 99 - 89.17 = 9.83. Of course, 6-1 (n-1) equals 5. Next, we'll square these differences with the following respective results: 103.43, 17.39, 8.01, 4.71, 14.67, and 96.63. Then, let's sum those squares obtaining 244.84, divide by 5 (n-1 which is 6-1), obtaining 48.97 and take the square root of that quotient, giving us a standard deviation of 6.998, or 7.

Here it is presented mathematically:







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