4.5:
Calculating Standard Deviation
Consider the dataset of the summertime temperatures in different American states. By calculating the standard deviation of the dataset, one can estimate the spread or deviation of each of these values from the mean.
Since this dataset represents a sample drawn from a larger population, the formula for the sample standard deviation is used.
Begin by computing the mean of the data values, denoted as x bar. Then, subtract the mean from each sample data value, x. These resulting values are known as deviations.
Square each of these deviations and add them. Next, divide this sum of the squares by the sample size, n minus one. In this case, since the sample size is 5, the denominator is five minus one, four.
Lastly, find the square root of this value, which can be rounded off to 4.0 for convenience. Thus, the calculated standard deviation of the summertime temperatures is four degrees Celsius.
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