Standard Deviation and variance

Definitions

Standard Deviation

The Standard Deviation is a measure of how spread out numbers are:

The symbol is σ

Formula: it is the square root of the Variance.  

Variance

The Variance is defined as: 

The average of the squared differences from the Mean.

 To calculate the Standard Deviation follow these steps:

1. Work out the Mean (the simple average of the numbers)
2. Then for each number, subtract the Mean and square the result.
3. Then work out the Mean of those squared differences.  
4. Take the Square Root of that and we are done!

I will show you:

Standard Deviation Formula 

Say what? Please explain!

Example:

Sam has 20 Rose Bushes


The number of flowers on each bush is:

9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4 

Work out the Standard Deviation. 

Step 1: 

Work out the Mean:

9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4

Mean:

9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4  

______________________________________________

20 

= 140 
  ____
    20   

= 7

μ = 7

Step 2:

For each number: subtract the mean and square the result.

This is the part of the formula that says:

So what is x_i ? They are the individual x values 9, 2, 5, 4, 12, 7, etc...

In other words x₁ = 9, x₂ = 2, x₃ = 5, etc.

So it says "for each value, subtract the mean and square the result", like this:

(continued):

(9 - 7)² = (2)² = 4

(2 - 7)² = (-5)² = 25

(5 - 7)² = (-2)² = 4

(4 - 7)² = (-3)² = 9

(12 - 7)² = (5)² = 25

(7 - 7)² = (0)² = 0

(8 - 7)² = (1)² = 1

... etc ...

Step 3:

Then work out the mean of those squared differences:

To work out the mean, add up all the values then divide by how many.

First add up all the values from the previous step.

But how do we say "add them all up" in mathematics? 

We use "Sigma": Σ

We want to add up all the values from 1 to N, where N=20 in our case because there are 20 values:

Which means: Sum all values from (x₁-7)² to (x_N-7)²

We already calculated (x₁-7)²=4 etc. in the previous step, so just sum them up:

= 4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9 = 178

We need to divide by how many, which is simply done by multiplying by "1/N":

Variance = Mean of square differences: (1/20) × 178 = 8.9

So, Variance = 8.9

Step 4:

Take the square root of that:

σ = √(8.9) = 2.983...

Standard Deviation is 2.983...

DONE!

Now, Let's try some exercises!

References:

https://www.mathsisfun.com/data/standard-deviation.html 
https://www.mathsisfun.com/data/standard-deviation-formulas.html