Standard Deviation Examples
The following standard deviation example provides an outline of the most common scenarios of deviations. 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.
- xi = Value of the ith 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 the volatility and performance trends about a data set. Let’s understand the concept of standard deviation using some examples:
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.
In the financial sector, the standard deviation is a measure of ‘risk’ that is 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. between markets, financial securities, commodities, etc. Lower standard deviation means lower risk and vice versa. Also, the risk is highly correlated with returns, i.e., with low risk comes lower returns.
E.g., Let’s say a financial analyst analyzing the returns of Google stock and wants to measure the risks on returns if investments are made in the particular stock. He collects the data of the historical returns of google for the last five years, which are as follows:
|Returns (%) (xi)||27.70%||36.10%||10.50%||6.80%||-4.60%|
Thus standard deviation (or risk) of Google’s stock 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 as compared to Doddle because the standard deviation of Doodle is very high as compared to the returns it provides while Google provides rather lower returns than Doodle but with very low exposure to risks.
Investors are risk-averse. They wanted to get compensated for taking higher risks.
#2 – The Empirical Rule:
States that 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., nearly all (99.7%) of the data falls within three standard deviations of the mean, 95% of data fall within 2 SD, and 68% fall 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 till 0.09% and can rise up to 32.91%.
John and his friend Paul arguing about the heights of their dogs to properly categorize them as per rules of a dog show where various dogs will compete with different heights based on categories. John and Paul decided to analyze the variability in heights of their dogs using the concept of standard deviation.
They have 5 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 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 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. .
Outliers can artificially inflate standard deviation, so identify them and remove them from the better analysis.
For example, 20 students of a mathematical class graded with an average of 60% marks on a practice test. The teacher seems concerned with the poor results, so she decides to calculate the std deviation of marks to check whether students score far or close to mean marks.
As per the calculation of standard deviation, the std deviation is 22.26%, which he thinks is very high. Let’s examine the teachers’ concerns.
- Using an empirical concept, he finds 95% of student’s marks fluctuate between ( x + 2 σ ) e.15.5% and 100%. I.e., few students are failing in the subject if passing marks are 30%.
- On closely analyzing the marks, he found a very very low scoring student, roll n.6, who scored only 10%.
- Roll no. 6 is actually an outlier which disturbs the analysis by artificially inflating the std 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 finds 95% of student’s 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.
In 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., it informs how tightly various data points are clustered around the mean in a normally distributed set of data. If the data points are closely bunched near the mean, then the standard deviation will be a small figure, and 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. (average) or median may mislead the user due to the presence of extreme data points, but standard deviation educates the user about how far the data point’s lies from the mean. Also, it is helpful in the comparative analysis of two different data sets if the averages are the same for both the data sets.
Hence they present a complete picture where basic mean can be misleading.
This has been a guide Standard Deviation Examples. Here we discuss its examples along with 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