Standard Deviation Examples With Step By Step Explanation

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 formulaStandard Deviation FormulaStandard deviation (SD) is a popular statistical tool represented by the Greek letter ‘σ’ to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data’s reliability.read more.

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Where,

xi = Value of the ith point in the data setx = The mean value of the data setn = 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 volatilityCalculate The VolatilityVolatility is the rate of change of price of a security. It is measured by calculating the standard deviation of annual returns and giving out minimum and maximum price. read more 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.

Note:

Remember, there are no good or bad standard deviations; It is just a way to represent data. But generally, a comparison of SD with a similar data set is being made for better interpretation.

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:

Calculation:

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

#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 distributionsNormal DistributionsNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more, 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.

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 ruleThe Empirical RuleEmpirical Rule in Statistics states that almost all (95%) of the observations in a normal distribution lie within 3 Standard Deviations from the Mean.read more 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.

Example #3

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

The theory of the Empirical Method applies only to data-sets that are normally distributed and whose shape appears like a bell curveBell CurveBell Curve graph portrays a normal distribution which is a type of continuous probability. It gets its name from the shape of the graph which resembles to a bell. read more.

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.

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.
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

statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more 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 meanMeanMean refers to the mathematical average calculated for two or more values. There are primarily two ways: arithmetic mean, where all the numbers are added and divided by their weight, and in geometric mean, we multiply the numbers together, take the Nth root and subtract it with one.read more (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.

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: –