Standard Deviation

Consider a situation in which you have a particular amount that you want to invest. So, what are the things you consider before investing your money? Typically, you consider the risk of making that particular investment. And yes, you also take a good at the highest and lowest returns on the investment. So, when you are observing a range of returns, you’re mainly looking at the deviation of returns from its expected value shown in the advertisements. Standard deviation is a broad concept that encircles all such elements. In this article, you can learn about standard deviation statistics, its formula, and steps to solve it.


What are the Variance and Standard Deviation? 

Variance: It helps to measure how far the data has spread out. When all the data values are identical, then variance turns out to be zero. And all non-zero variances get considered as positive. A lower variance denotes that the data values are closer to each other and the mean. When the data points are far away from one another and the mean, it means the variance is high. In simple words, you can define variance as the average of squared distances from every single point to the mean.       

Standard deviation: It helps to determine the dispersion from the mean. Dispersion refers to a value by which an object differs from another object; in this case, it’s an arithmetic mean. Standard deviation denotes the typical deviation from the mean. It’s the most favourite option of gauging deviation because it returns to the primary units of measurement of the data set. 

Also, it’s similar to variance; when the data points are closer to mean, the variation is minimal. And when data points are far away from the mean, then there’s high variance. 


Standard Deviation v/s Variance

You can obtain the variance by taking the mean of the data points and subtracting mean from each data point separately. After that, you have to square the results and take another mean of those squared values. Standard deviation is nothing but the square root of the variance value.  

Variance and Standard Deviation Formula 

The following are the formulas for variance and standard deviation. Typically, it gets denoted by sigma (σ).

Standard Deviation Formula:


 \[\sigma\] = \[\sqrt{\frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2} }{n-1}}\] 


And the Formula for Variance: 


 \[\sigma ^{2}\]   = \[\sqrt{\frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2} }{n}}\] 


Here, ‘σ’ stands for standard deviation. ‘X’ represents each value of the population. ‘x̅’ indicates the mean of all the values and ‘n’ shows the total number of values.


How to Calculate the Standard Deviation? 

In statistical analysis, the uses of standard deviation are highly influential. It helps you with measuring dispersion. The standard deviation has three distinct features. First is, it gets measured by arithmetic mean. You can say that deviation gets measured by taking the mean of the reference. Two, it deals with positive values. And finally, the standard deviation value is always positive because it’s a square root. Below you can learn the step-by-step way of measuring standard deviation. 


Steps to Calculate the Standard Deviation:

  • By adding all the data points and dividing by the number of data points, you can obtain the mean value.

  • By subtracting the value of data points from the mean, you can get the variance for each of the points. Next, you need to square the obtained values and sum the results. After that, you need to divide the results by the number of data points.

  • Now, you need to take the square root of the variance from the previous step. And the resulting value is your standard deviation.  

Solved Examples

Question: Find the standard deviation of the numbers given (3, 8, 6, 10, 12, 9, 11, 10, 12, and 7).


Answer: Step 1: First, we need to find the mean of those ten given values.

x̅ = (3 + 8 + 6 + 10 + 12 + 9 + 11 + 10 + 12 + 7)​/ 10 = 88 / 10 = 8.8

Step 2: You need to make a table as below with three columns. One column holds the values of x, the second column holds the deviations, and the third has squared deviations. (Note: You may not need a table when computing a lesser number of values.)    

 

Value (x)

X – x̅

( x – x̅ ) 2

3

-5.8

33.64

8

-0.8

0.64

6

-2.8

7.84

10

1.2

1.44

12

3.2

10.24

9

0.2

0.04

11

2.2

4.84

10

1.2

1.44

12

3.2

10.24

7

-1.8

3.24

Total

0

73.6


Step 3: Since the data is not in the form of sample data, you need to use the formula of the population variance. 

The standard deviation formula is, σ = √ ∑i=1n​ (xi ​– x̅)2​ / N. 

Now, you have ​ σ = √ 73.6 / 10 = √ 7.36.

Finally, the standard deviation you get is 2.71.  

FAQ (Frequently Asked Questions)

1. What is standard deviation?

The standard deviation refers to the statistic that calculates the dispersion of a particular dataset concerning its mean value. It also gets referred to as the square root of the variance. In statistics, it gets used to find the values of certain data that has dispersed. You can define the standard deviation as the deviation of data or values from their average mean. If the standard deviation is low, then the values are closer to their average. And higher deviation means that values are far from their mean value. Keep in mind that the standard deviation value is never negative. 


2. What is variance and how to calculate it? 

Variance is a crucial component of statistics and probability theory, under the branch of mathematics. It’s a way to calculate how far a set of numbers has spread out. You can use it to determine how much a random variable deviates from its expected value. Variance is significant to visualize and understand the data under consideration. You can define it as the average of the squared differences from the mean. To calculate the variance: first, you have to find out the mean (simple average). Then, you need to subtract the mean and square the obtained value for each number. In the end, figure out the average of those squared differences.