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computing standard deviationIntroduction to Standard Deviation

Standard Deviation is a mathematical method that is often used in calculations for statistical data. If an object deviates from its course, then it moves away of its initial course. If an amount or position based on standard measurements changes and then deviates from the standard, then you can calculate a measurement for the variance. A way to measure the estimated average amount of variance from an expected value is to calculate the Standard Deviation. Calculating Standard Deviation can be useful for many goals which include providing supporting information for scientific studies and determining margins of error for statistics.

Standard Deviation Formulas

The symbol used for Standard Deviation is a Greek letter, lowercase Sigma and is as follows:


Tip: Using a standard keyboard, you can use a combination of the following keys to make the Standard Deviation symbol:

ALT + 229

Note that the uppercase Sigma is Σ and represents sum. Do you find statistics to be intriguing and could find more guidance to be helpful? Perhaps, you will find a tutorial on Udemy to be exactly what you are seeking.

There are different variations of the Standard Deviation Formula, the formula used for Discrete Random Variables (with the same probability being applied to variables) is as follows:

Standard Deviation Formula

Scenario: A science class has 9 plants that they have placed in a class room; the seeds had been planted at the same type and the plants are of the same type. At the end of the period of time allotted for the study, they measured all nine plants and recorded the measurements. Your objective is to find the Standard Deviation of the plant heights, as recorded below:

Science Class Chart

Note that the entries for the plants also have the x symbol along with ordered numbers, this is to indicate that the symbol designates these objects as the individual variables.

The following is a summary of the formula and operations:

So, you have the Standard Deviation is equal to (σ=) ……

The overall Square Root of (√) …….

Working in accordance with the Order of Operations, you will simplify what is in the parentheses Equation where Formula to find Mean (μ=the mean of variables)

At this point, μ is for an unknown value.

The variables you are working with (the height of the plants) are as follows:

10, 9, 7, 10, 11, 5, 9, 8, 7

Firstly, find the mean by adding up all the values of the variables and then dividing by the total number of variables (9, in this case).

10+9+7+10+11+5+9+8+7= 76

73/9 = 8.44

Mean = 8.44 (μ=8.44)

Now you will have Step for Finding Square of Mean and as noted above; each plant is a variable, so you will perform this calculation for each variable value (or each plant height for this case).

x1 10 – 8.44 = 1.56 [then find amount of 1.56 squared ( Squared ) = 2.43]

x2 9 – 8.44 = 0.56 squared = 0.31

x3 7 – 8.44 = (-1.44) squared = 2.07

x4 10 – 8.44 = 1.56 squared = 2.43

x5 11 – 8.44 = 2.56 squared= 6.55

x6 5 – 8.44 = (-3.44) squared = 11.83

x7 9 – 8.44 = 0.56 squared = 0.31

x8 8 – 8.44 = (-0.44) squared = 0.19

x9 7 – 8.44 = (-1.44) squared = 2.07

Wonderful, you now have the values of each variable value after being squared (after the mean had been calculated in the equation) and this is going to enable you to progress to the next step of your equation!

The next step is to incorporate the Formula Step portion; which essentially states that variables will continually be added until after the last variable has been used, so you will use the values resulting from the previous step (squared results) and calculate the sum of the values.

2.43 + 0.31 + 2.07 + 2.43 + 6.55 + 11.83 + 0.31 + 0.19 + 2.07 = 28.19

At this point, you will incorporate the 1/N portion of the equation by dividing the sum of the values that resulted from the previous step (sum of values after being squared) by the number of individual square root values (9).

28.19/9 = 3.13

Superb! You have found the Variance and you are almost done with this scenario.

Recall that your Standard Deviation formula contained operations (which have been incorporated in your calculations at this point) that had been under a Square Root symbol? Thus, the following formula has been accounted for at this point, with the exception of the overall square root portion:

Standard Deviation Formula

Due to the fact that you have “solved for” (you had found the resulting answers of the problems being calculated after applying variables in the formula) the variance, you only have to find the square root of the Variance to solve this formula and find the Standard Deviation.

The Standard Deviation = √3.13

σ = 1.77

The Standard Deviation for the plant measurement study case is 1.77!

Did you realize that formulas, such as Standard Deviation, can also be very useful for developing reports? If you want to learn more about about reporting based on statistics, Udemy has comprehensive tutorials.

You had used the formula that applies the same probability for Discrete Random Variables in the plant measurement scenario. Discrete Random Variable formulas are formulas that are used for what is considered Population Standard Deviation. The formula used for Discrete Random Variables (using varying probabilities) is as follows:

Population Standard Deviation

If the scenario you had been presented with had been different and had involved sampling, then a different type of Standard Deviation formula could have been utilized. For instance; if the scenario would have involved fifty plants being planted and the height being recorded for nine random plants out of the fifty plants, the random selections would have involved a sampling method.  Explore the world of Statistics and Probabilities by taking a class on Udemy. The following is the formula used for Sample Standard Deviation:


Notable differences between the Population Standard Deviation formula and the Sample Standard Deviation formula include the one subtracted from the denominator in the fraction and the following symbol for a sample mean:

Sample Mean Symbol

The reasoning for the 1 being subtracted in the fraction, involved with finding the mean of the squared values step of the formula, is based on a concept called Bessel’s Correction. Bessel’s Correction had been developed in an attempt to produce more accurate results for Sample Standard Deviation by “correcting”, or compensating for, possible biased values that could occur with sampling.


When data for statistics is analyzed, using Standard Deviation formulas can help with the indication and maintenance of accuracy. Factors that should be considered when deciding which type of Standard Deviation formula to use include the type of data gathered (from a population or sample) and the accuracy of the initial data used before it is integrated into a formula. In many cases; not only should results that involve different values from studies be recorded, but the variance and then standard deviation of the values should also be recorded to provide a more comprehensive overview of the conclusions which result from research and analysis of data.

Page Last Updated: February 2020

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