A standard deviation tells you how the numbers are spread out. It is the most common and popular way to quantify the data set. Though you can use your calculator which has a built-in button to calculate the standard deviation, it is good to know how it is done by hand. Rather than memorize the formula, it is easy to remember how it is calculated step-by-step. The steps below help you to understand better and easier to remember.
Calculate the Mean: It is essential to first look at the data set in hand before you make any calculation. Find out how many numbers are there in the sample, then look at the range these numbers vary, is the difference small having a few decimal places or large. Next, determine what these numbers represent, is it a test score, height, weight.
Ex: It can be a test score of 4,8,8,10,8 and 10.
The average of all the data points is the mean. It is calculated by adding all the numbers and dividing it by how many numbers are available in the sample. In the above example, the number of samples is 6.
The mean of the number is:
4+8+8+10+8+10 = 48
The number of test scores in the sample is 6.
Mean = 48/6 = 8
Calculate the Variance: The variance shows how far the data in the data set is from the mean and helps to compare the distribution of data sets. It gives an idea of the data spread, data that have low variance is close to the mean, and those that have high variance is away from the mean.
Find the difference between the mean and every number in the sample and that gives how much the data differs from a mean. In the above example test score, 4 – 8= -4, 8 – 8=0, 8 – 8=0, 10-8=2, 8 – 8=0 and 10 – 8= 2
Find the square of the difference between the mean and the numbers in the sample.
Square of 4 = 16, square of 0 = 0, square of 2 = 4.
Adding all of them together results in
16+0+0+4+0+4 = 24
Divide the sum of squares by number of samples -1: Using the above example, 24/(6-1) = 24/5 = 4.8
To calculate the Standard Deviation using the following formula:
Standard deviation = square root of the variance.
Using the above example, the standard deviation is 2.19
Advantages of Standard Deviation:
- The use of standard deviation is the best way to calculate variance as it is based on every sample of the data set. The algebraic operations are least affected by sampling fluctuations than most of the other dispersion methods.
- You can also calculate the standard deviation of more than two groups by combining them, and this is not possible with other measurement methods.
- It is the most appropriate way of comparing the variables in distribution and coefficient variation and is based on mean standard deviation.
- It is used as a measurement unit for normal distribution and also in sampling.
The calculation can be complicated and it does not give a full data range. A normal pattern of distribution is assumed and it can be tough to calculate.
Standard Deviation is used in statistical analysis for example in correlation, skew computation, and much more.