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Standard Deviation and Variance By Hand: Steps to Calculate
Let's Define Standard Deviation
The standard deviation of a population (by population, I mean a set of numbers), provides information of how much variation exists from the average value or the mean.
So, the standard deviation tells us about the variation in some population or set of data.
The standard deviation of a population is mostoften represented by the lowercase Greek symbol sigma, σ.
Equation for Standard Deviation
What does the standard deviation formula mean?
In order to calculate the standard deviation of some population, we must understand the notation in the equation.
Let's have a look at the chart below:
Standard Deviation Formula Notation:
Symbol
 Meaning


Σ (sigma)
 the sum across values

x
 a number in the population

x̅
 the average of the population

n
 amount of data in population

How to Calculate the Standard Deviation:
Overview of Steps


1. Calculate the mean of all numbers

2. Calculate the difference between each value and the mean

3. Square numbers from step 2

4. Take sum of squares

5. Solve for SD formula

Standard Deviation Example
Step 1: Find the mean  To find the average of a population, you must first take the sum of the numbers and divide your result by the amount of numbers you added up (the amount of numbers in your population.
Example)
Here's a set of values: {12, 6, 5, 8, 9}
Find the sum of the set by adding all the values together: (12 + 6 + 5 + 8 + 9) =40
Notice the number of values in the population: n = 5
Take the sum of the values and divide by the numbers in the set: (40 / 5) = 8
Average = 8
Step 2: Difference between each value and the mean  In this step, you subtract the average found in step 1 from every value in your population. Some numbers may be negative.
Example)
Here's our set: {12, 6, 5, 8, 9}
Average: 8
Difference between values and mean:
(12  8) = 4
(6  8) = 2
(5  8) = 3
(8  8) = 0
(9  8) = 1
Step 3: Take the squares Now, take the squares of all the values found in step 2.
Example)
Values found in step 2: (4, 2, 3, 0, 1)
Square all values: [4^{2}, (2)^{2}, (3)^{2}, 0^{2}, 1^{2}] = (16, 4, 9, 0, 1)
Step 4: Take sum of the squares  In this step, we take all the values we squared from step 3 and add them together.
Example)
Values from step 3: (16, 4, 9, 0, 1)
Sum the squares: (16 + 4 + 9 + 0 + 1 ) = 30
Step 5: Solve the standard deviation formula  At this point, all the information needed to obtain the standard deviation has been found. Simply, we must plug in these values into the equation.
Example)
Sum of squares from step 3: 30
Number of values in set: n = 5
Standard deviation: sq. rt. ( 30/ (51))= sq. rt. (30/4) = 2.7386
Do you understand how to calculate the standard deviation?
Khan Academy: Standard Deviation
Standard Deviation Online Calculator
To make sure you've performed the standard deviation correctly, use the standard deviation calculator provided in the link below.
This is a good tool to use if you have a large set of numbers for instance, if you have a population of 20 values, I recommend using a standard deviation calculator, or set it up to calculate the standard deviation in Excel or a program similar. If there's still any confusion the Khan Academy video at the side clearly demonstrates how to perform standard deviation it's a useful tool from a trusted, wellestablished source.
Also, another great computational tool you can find at Wolfram Alpha's Site. You can calculate averages, standard deviations, variances and much more.
Point is there is a plethora of resources available to you just by a simple Google search to find the standard deviation, if you are the type of person that needs to see the problem be walked through.
Definition of Variance
Variance is a measure of how far a set of numbers is spread out, or how varied the set is! Once you find the standard deviation, you only have one more step to complete and you've found the variance! It's so easy and simple!
How to find the variance
Step 6: Square the standard deviation We find the variance by squaring the standard deviation.
Example)
Square the standard deviation: (2.7386)^{2}= 7.50
Variance = 7.50
Khan on Sample Variance
Variance Online Calculator
Here is a calculator that calculates everything to do with standard deviation. The variance is the fourth one down. Here is the variance calculator.
Any questions? Leave them below.
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