What is Standard Deviation?
Standard deviation is used in probability theory and statistics. It shows how much variation exists in the data set from the average. The average can also be called as mean or expected value. If the variance of a data set is known, then the standard deviation can be obtained by finding the square root of it. The standard deviation can be calculated for a data set, random variable, probability distribution or statistical population. An important property of standard deviation that makes it the preferred method of statistical measure is that the standard deviation computed for a data set has the same unit as that of the data set.
Standard Deviation in Statistics
Statistics Standard Deviation is used to measure the dispersion of the set of data given. Standard Deviation Statistics shows how some data has high standard deviation and some have low standard deviation. Standard Deviation examples such as volatile stock and blue chip stock demonstrate this difference. The high standard deviation in volatile stock and low standard deviation in stable blue chip stock shows how standard deviation is used for statistical analysis of the given data sets. If the standard deviation is low, it indicates that the data values in the data set are very closely spread around the mean. If the standard deviation is high, it shows the wide dispersal of the data values over a larger range.
Steps to compute Standard Deviation
The first step in computing standard deviation is to find the mean of the data values of the given data set. The next step is to find out how far each data value is deviated from the mean. To compute this deviation, the mean is subtracted from each data value of the given data set and then the values are squared. Then find out the mean for the squared values. Then find the square root of this computed mean. Thus, by finding the square root of the mean of the squared values, you get the standard deviation.
Example of Standard Deviation
Let us consider an example for standard deviation: The students of a school have participated in Mathematics Olympiad exam. To know their level of knowledge in Mathematics, marks of few students is taken as a sample from the whole population of students who attended the exam and the standard deviation is determined for this sample. This is demonstrated below:
Here is the sample: the marks of 6 students are 91, 100, 100, 98, 96, and 100
Step 1: The mean of these marks will be (92+100+100+99+97+100)/6 which will be 98
Step 2: The next step is to find the squared differences from the mean and sum it: (92-98)^2 +(100-98)^2 +(100-98)^2 +(99-98)^2 +(97-98)^2 +(100-98)^2 = 50
Step 3: Find the square root of the above sum divided by one less than the number of students. Thus you will end up in Square root (50/5), which is equivalent to 3.1623. This resultant value 3.1623 is the standard deviation.
As this standard deviation is low, we can understand that the actual marks scored by the students are very near to their expected marks i.e. 100. As this is a sample data set, it implies that all the students who have attended the Mathematics Olympiad exam have scored good marks.
