4.5: Calculating Standard Deviation
The standard deviation is the most common measure of variation. It is a value that tells us how far a data value is from the mean value in a dataset. Further, the standard deviation is always a positive value or zero.
The standard deviation value is small when all the data is concentrated close to the mean. Here the data exhibits low variation. The standard deviation value is larger when the data values are more spread out from the mean. Here, the data displays high variation.
Let us consider a dataset with test scores (out of 100) of 5 students: 91, 89, 70, 76, and 80.
Initially, we determine the sample mean denoted as x bar. The mean score of the above sample test scores is 81.2. Next, we find the difference between each data value, x, and the mean. This is known as a deviation.
For the above example, the deviations calculated as" x-mean" are 9.8, 7.8, -11.2, -5.2, and -1,2, respectively. Now, we square all the deviations and add them up. After calculating the sum of the square of the deviations, we get a value of 310.8.
Next, we divide this value by sample size, n minus 1. For the above example, the sample size is 5, so n-1 equals 4. Upon dividing the value of 310.8 by 4, we get a value of 77.7. Finally, we find out the square root of this value to get the sample standard deviation,s. The sample standard deviation,s for this above example is 8.8.
This text is adapted from Openstax, Introductory Statistics, Section 2.7 Measures of the Spread of the Data