Standard Deviation Examples

Standard Deviation Examples

The following standard deviation example outlines the most common deviation scenarios. Standard deviation is the square root of the variance, calculated by determining the variation between the data points relative to their mean. Below is the standard deviation formula.

• The standard deviation example illustrates common scenarios of deviation in a dataset.
• Standard deviation is the square root of the variance and measures the variation between data points and their mean.
• It is a valuable tool for statisticians, scientists, financial analysts, and others to assess a dataset's volatility and performance trends.
• Standard deviation provides insights into the distance of data points from the mean and facilitates the comparison of different datasets with the same average.

Where,

• x_i = Value of the i^th point in the data set
• x = The mean value of the data set
• n = The number of data points in the data set​

It helps statisticians, scientists, financial analysts, etc., measure a data set's volatility and performance trends. But first, let’s understand the concept of standard deviation using some examples.

Example #1

In the financial sector, the standard deviation is a measure of ‘risk’ used to calculate the volatility between markets, financial securities, commodities, etc. A lower standard deviation means lower risk and vice versa. Also, the risk highly correlates with returns, i.e., with low risk comes lower returns.

E.g., a financial analyst analyzes the returns of Google stock and wants to measure the risks on returns if investments are in a particular stock. Therefore, he collects data on the historical returns of google for the last five years, which are as follows:

Year	2018	2017	2016	2015	2014
Returns (%) (xi)	27.70%	36.10%	10.50%	6.80%	-4.60%

Calculation:

Thus, Google's stock's standard deviation (or risk) is 16.41% for annual average returns of 16.5%.

Interpretation

#1 - Comparison Analysis

Let’s say Doodle Inc has similar annual average returns of 16.5% and SD ( σ ) of 8.5%. i.e., with Doodle, you can earn similar yearly returns as with Google but with lesser risks or volatility.

Again let’s say Doodle Inc has annual average returns of 18% and SD ( σ ) 25%. We can surely say that Google is the better investment compared to Doodle because the standard deviation of Doodle is very high compared to the returns it provides. In contrast, Google provides lower returns than Doodle but with very low-risk exposure.

#2 - The Empirical Rule

For normal distributions, nearly all (99.7%) of the data falls within three standard deviations of the mean, 95% falls within 2 SD, and 68% falls within 1 SD.

In other words, we can say that 68% returns of Google fall within + 1 time the SD of mean or ( x + 1 σ) = (16.5 + 1 * 16.41) = (0.09 to 32.91%). i.e., 68% returns of an investor of Google can go low to 0.09% and rise to 32.91%.

Example #2

John and his friend Paul argue about the heights of their dogs to properly categorize them as per the rules of a dog show where various dogs will compete with different heights based on categories. John and Paul decided to analyze their dogs' heights' variability using the concept of standard deviation.

They have five dogs with all types of heights, so they noted their heights as given below:

The heights of the dogs are 300mm, 430mm, 170mm, 470mm, and 600mm.

Calculation:

Step 1: Calculate the mean:

Mean ( x ) = 300 + 430 + 170 + 470 + 600 / 5 = 394

The red line in the graph shows the average height of the dogs.

Step 2: Calculate the variance:

Variance ( σ^2 ) = 8836 + 1296 + 50176 + 5776 + 42436 / 5 = 21704

Step 3: Calculate the standard deviation:

Standard Deviation (σ) = √ 21704 = 147

Now, using the empirical method, we can analyze which heights are within one standard deviation of the mean:

The empirical rule says that 68% of heights fall within + 1 time the SD of mean or ( x + 1 σ ) = (394 + 1 * 147) = (247, 541). i.e. 68% of heights fluctuate between 247 and 541.

The theory of the Empirical Method applies only to data-sets that are normally distributed and whose shape appears like a bell curve.

Example #3

Outliers can artificially inflate standard deviation, so identify them and remove them from the better analysis.

For example, 20 students in a mathematical class grade an average of 60% marks on a practice test. The teacher seems concerned with the poor results, so she calculates the standard deviation of marks to check whether students score far or close to mean marks.

As per the standard deviation calculation, it is 22.26%, which he thinks is very high. So let’s examine the teachers’ concerns.

Roll no.	1001	1002	1003	1004	1005	1006	1007	1008	1009	1010
Marks (%)	72	45	58	84	60	10	91	65	55	60

Calculation:

• Using an Empirical concept, he finds 95% of students’ marks fluctuate between ( x + 2 σ ) e.15.5% and 100%. i.e., few students fail in the subject if passing marks are 30%.
• On closely analyzing the marks, he found a very low-scoring student, roll no. 6, who scored only 10%.
• Roll no. 6 is an outlier that disturbs the analysis by artificially inflating the standard deviation and decreasing the overall mean.
• The teacher decides to remove roll no. 6 to re-analyze the performance of the class and found the following result.

Calculation:

• Again, using an empirical concept, he found that 95% of students’ marks fluctuate between 36.50% and 80%. i.e., neither student is failing in the subject.
• However, the teacher has to put extra effort into improving the ‘outlier’ Roll no. 6 because, in real life, a student cannot be removed where a teacher finds hope for improvements.

Conclusion

Statistics informs how tightly various data points clustered around the mean in a normally distributed data set. For example, if the data points bunches closely near the mean, the standard deviation will be a small figure. As a result, the bell curve will be steeply shaped and vise-versa.

The more popular statistical measures like mean (average) or median may mislead the user due to extreme data points. Still, standard deviation educates the user about how far the data point lies from the mean. Also, it helps compare two different data sets if the averages are the same for both data sets.

Hence, they present a complete picture where basic means can be misleading.

Frequently Asked Questions (FAQs)

This article is a guide to the Standard Deviation Examples. Here, we discuss its examples along with a step-by-step explanation. You can learn more about accounting from the following articles: -

• Formula of Sample Standard Deviation
• Formula of Relative Standard Deviation
• Standard Deviation Excel Graph
• Portfolio Standard Deviation