# Standard Deviation Formulas

*Deviation just means how far from the normal*

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:

** 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:

### 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:

\frac{9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4}{20}

=
\frac{140}{20} = **7**

And 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:

So what is *x _{i}* ? They are the individual x values 9, 2, 5, 4, 12, 7, etc...

In other words *x _{1}* = 9,

*x*= 2,

_{2}*x*= 5, etc.

_{3}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

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):

Which means: Sum all values from (x_{1}-7)^{2} to (x_{N}-7)^{2}

We already calculated (x_{1}-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):

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):

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.

### 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**:

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 (called "x-bar") for
**sample mean**, instead of**μ**for the population mean, - And the answer is
**s**(for sample standard deviation) instead of**σ**.

But they do not affect the calculations. **Only N-1 instead of N changes the calculations.**

OK, let us now use the **Sample Standard Deviation**:

### Step 1. Work out the mean

### Example 2: Using sampled values 9, 2, 5, 4, 12, 7

The mean is (9+2+5+4+12+7) / 6 = 39/6 = 6.5

So:

x = 6.5

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

### Example 2 (continued):

(9 - 6.5)^{2} = (2.5)^{2} = 6.25

(2 - 6.5)^{2} = (-4.5)^{2} = 20.25

(5 - 6.5)^{2} = (-1.5)^{2} = 2.25

(4 - 6.5)^{2} = (-2.5)^{2} = 6.25

(12 - 6.5)^{2} = (5.5)^{2} = 30.25

(7 - 6.5)^{2} = (0.5)^{2} = 0.25

### 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**.

But hang on ... we are calculating the **Sample** Standard Deviation, so instead of dividing by how many (N), we will divide by **N-1**

### Example 2 (continued):

Sum = 6.25 + 20.25 + 2.25 + 6.25 + 30.25 + 0.25 = **65.5**

Divide by **N-1**: (1/5) × 65.5 = **13.1**

(This value is called the "Sample Variance")

### Step 4. Take the square root of that:

### Example 2 (concluded):

s = √(13.1) = **3.619...**

DONE!

## Comparing

Using the whole **population** we got: Mean = **7**, Standard Deviation = **2.983...**

Using the **sample** we got: Sample Mean = **6.5**, Sample Standard Deviation = **3.619...**

Our Sample Mean was wrong by 7%, and our Sample Standard Deviation was wrong by 21%.

## Why Take a Sample?

Mostly because it is easier and cheaper.

Imagine you want to know what the whole country thinks ... you can't ask millions of people, so instead you ask maybe 1,000 people.

There is a nice quote (possibly by Samuel Johnson):

*"You don't have to eat the whole animal to know that the meat is tough." *

This is the essential idea of sampling. To find out information about the population (such as mean and standard deviation), we do not need to look at **all** members of the population; we only need a sample.

But when we take a sample, we lose some accuracy.

Have a play with this at Normal Distribution Simulator.

## Summary

The |
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The Sample Standard Deviation: |