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 look 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.

Standard deviation is an important topic of statistics. It actually measures the amount of variation of a specific set of values. A low standard deviation means that the set of values are not much deviated from the mean values of the set and a high denotes a greater deviation from the mean of the set. It is thus able to determine the uncertainty in a set of values. 

In Physics, to get precision in the measurement standard deviation plays a role since it gives an idea of how much the value is deviated from the actual value. It is thus often used in day-to-day life to compare data from the test model or the actual expected value. It is often denoted by SD and represented by the symbol. The details of standard deviation can be learnt from this article.

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 favorite option for 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 the 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 the 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.)    

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.  

Understand the definition of standard deviation and its difference with variance. Focus on learning the derivation and utilization of the formula for better applications.