Question

Formula for standard deviation in terms of mean

Answer 1

Answer:Standard Deviation

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

You might like to read this simpler page on Standard Deviation first.

But here we explain the formulas.

The symbol for Standard Deviation is σ (the Greek letter sigma).

This is the formula for Standard Deviation:

square root of [ (1/N) times Sigma i=1 to N of (xi - mu)^2 ]

Say what? Please explain!

OK. Let us explain it step by step.

Say we have a bunch of numbers like 9, 2, 5, 4, 12, 7, 8, 11.

To calculate the standard deviation of those numbers:

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!

The formula actually says all of that, and I will show you how.

The Formula Explained

First, let us have some example values to work on:

rose

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

In the formula above μ (the greek letter "mu") is the mean of all our values ...

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

The mean is:

9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+420

=  14020 = 7

So:

μ = 7

 

Step 2. Then for each number: subtract the Mean and square the result

This is the part of the formula that says:

(xi - mu)^2

So what is xi ? 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

Example (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 ...

And we get these results:

4, 25, 4, 9, 25, 0, 1, 16, 4, 16, 0, 9, 25, 4, 9, 9, 4, 1, 4, 9

 

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

The handy Sigma Notation says to sum up as many terms as we want:

Sigma Notation

Sigma Notation

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

Example (continued):

sigma i=1 to N of (xi - mu)^2

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

But that isn't the mean yet, we need to divide by how many, which is done by multiplying by 1/N (the same as dividing by N):

Example (continued):

(1/N) times sigma i=1 to N of (xi - mu)^2

Mean of squared differences = (1/20) × 178 = 8.9

(Note: this value is called the "Variance")

 

Step 4. Take the square root of that:

Example (concluded):

square root of [ (1/N) times Sigma i=1 to N of (xi - mu)^2 ]

σ = √(8.9) = 2.983...

DONE!

 

Sample Standard Deviation

But wait, there is more ...

... sometimes our data is only a sample of the whole population.

rose

Example: Sam has 20 rose bushes, but only counted the flowers on 6 of them!

The "population" is all 20 rose bushes,

and the "sample" is the 6 bushes that Sam counted the flowers of.

Let us say Sam's flower counts are:

9, 2, 5, 4, 12, 7

We can still estimate the Standard Deviation.

But when we use the sample as an estimate of the whole population, the Standard Deviation formula changes to this:

The formula for Sample Standard Deviation:

square root of [ (1/(N-1)) times Sigma i=1 to N of (xi - xbar)^2 ]

The important change is "N-1" instead of "N" (which is called "Bessel's correction").

The symbols also change to reflect that we are working on a sample instead of the whole population:

The mean is now x (for sample mean) instead of μ (the population mean),

And the answer is s (for Sample Standard Deviation) instead of σ.

But that does not affect the calculations. Only N-1 instead of N changes the calculations.