Sample Standard Deviation Formula (Table of Contents)
Sample Standard Deviation Formula
In statistics, the standard deviation is basically a measure to find the dispersion of the data set values from the mean value of the data set. It measures the distance of that data point and the mean. So higher the standard deviation, the higher will be the dispersion, and data points will tend to far from the mean. Similarly, a lower standard deviation means that data points will be closer to the mean. It is very useful in comparing data sets which may have the same mean value but a different range.
Usually, calculate the standard deviation of population data, but sometimes population data is so huge that it is not possible to find the standard deviation for that. In that case, sample standard deviation is calculated, which will represent population standard deviation. So we will assume that the sample is the correct representation of the population and will focus on sample standard deviation in this article.
Suppose you have a data set X with data points {X1, X2……..Xn}.
The formula for population standard deviation is given by:
In case you are not given the entire population and only have a sample (Let’s say X is the sample data set of the population), then the formula for sample standard deviation is given by:
Where:
- Xi – ith value of data set
- Xm– Mean value of data set
- n – Total number of data points
The formula may look confusing at first, but it is really to work on. Following are the steps which can be followed to calculate sample standard deviation:
- Find the number of points in the data set, i.e. n.
- Then the next step is to find the mean value of the sample. It is basically the average of all the values.
- After that, for each data point, find the difference of that from the mean and then square it.
- Take the sum of all the values in the above step and divided that by n-1.
- The last step is to take the square root of the number calculated above.
There is another way to calculate population and standard deviation simply by using the STDEV.P () function for population standard deviation and STDEV.S () function for sample standard deviation in excel.
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Examples of Sample Standard Deviation Formula (With Excel Template)
Let’s take an example to understand the calculation of the Sample Standard Deviation in a better manner:
Sample Standard Deviation Formula – Example #1
Let’s say we have two sample data sets, A & B, and each contains 20 random data points and have the same mean. Calculate the Sample Standard Deviation for the data set A & B.
Solution:
Mean is calculated as:
- Mean of Data Set A = 51.25
- Mean of Data Set B = 51.25
Now, we need to calculate the difference between the data points and the mean value.
Similarly, calculate for all the data set of A.
Similarly, calculate it for data set B also.
Calculate the square of the difference for both the data sets A and B.
Sample Standard Deviation is calculated using the formula given below:
Sample Standard Deviation = √ [Σ (Xi – Xm)2 / (n – 1)]
So if you see here, although both the data sets have the same mean value, B has a more standard deviation than A, which means that data points of B are more dispersed than A.
Sample Standard Deviation Formula – Example #2
Let say you are a very risk-averse investor and you looking to invest money in the stock market. Since your risk appetite is low, you want to invest in safe stocks which have a lower standard deviation. Your financial advisor has suggested to you 4 stocks from which you can choose. You want to select 2 stocks among those 4, and you will decide that on the basis of lower standard deviation.
You have got information on their historical returns for the last 15 years.
Solution:
Sample Standard Deviation is calculated using the excel formula:
Based on the information and sample standard deviation, you will choose stock Y and Z to invest in since they have the lowest standard deviation.
Explanation
We discuss the meaning of standard deviation from a statistical standpoint, but it also plays a vital role if we talk about a financial point of view. It is basically the measure of risk an investment carries and how risky that investment is in finance. Based on the risk an investment has, investors can then calculate the minimum return they require to compensate for that risk.
As in the above example, since Y and Z have a lesser standard deviation, it means that there is less variability in the return of these stocks, so they are less riskier. One point while using the standard deviation tool we need to keep in mind that it is highly affected by the extreme values or outliers. These outliers can skew the standard deviation value.
Relevance and Uses of Sample Standard Deviation Formula
Standard deviation helps the investors and analysts to find the risk and reward ratio or Sharpe ratio for an investment. Basically, anyone can earn a risk-free rate of return by investing in Treasury and risk-free securities. But return over and above this is the excess return and to achieve that, what is the level of risk one needs to take is a measure of Sharpe ratio:
Sharpe Ratio = (Return on Investment – Risk Free Rate) / Standard Deviation
So as to the higher the Sharpe ratio, the better is the investment.
As we said, the standard deviation is a measure of risk, but a lower standard deviation value is not always preferred. If an investor has a higher risk appetite and wants to invest more aggressively, he will be willing to take more risk and prefer a relatively higher standard deviation than a risk-averse investor. So it all depends on what level of risk an investor is willing to take.
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