Standard Deviation in Math is a measure of the spread regarding a set of data.

Specifically, what standard deviation tells us is how far away the values of a data set are from the average/mean value.

The smaller the Standard Deviation value, then the less spread out from the average a set/group of data is.

Standard Deviation is a measure that can often be useful. As sometimes an average value can be a good fit for a set/group of data values, but there are also situations when an average value may not be a good fit.

We can observe this below in  2  examples of working out the average/mean of a set of numbers.

### Calculating the Mean/Average Examples

*(1.1) *List of  7  numbers:

**5 , 7 , 3 , 5 , 6 , 4 , 5**

Mean/average  =   \bf{\frac{5 \space + \space 7 \space + \space 3 \space + \space 5 \space + \space 6 \space + \space 4 \space + \space 5}{7}}   =   \bf{\frac{35}{7}}   =  

**5**

The average for the list of numbers is  5,  and it turns out that all values in the list are close to this value.

*(1.2) *List of  7  numbers:

**3 , 10 , 12 , 5 , 18 , 6 , 2**

Mean/average   =   \bf{\frac{3 \space + \space 10 \space + \space 12 \space + \space 5 \space + \space 18 \space + \space 6 \space + \space 2}{7}}   =   \bf{\frac{56}{7}}   =  

**8**

The average of the list of numbers here is  8,  however there are some values that are quite a distance away from that value, especially  18.

## Calculating Standard Deviation

The Standard Deviation of a set of data values, is a vale that helps to give a measure of how spread out the values in that set are from the mean/average. This is what standard deviation tells us.

The Standard Deviation of a set of data values, is in fact the square root of the ** variance** of the data values.

So in order to establish the standard deviation of a data set, we first need to know the variance number.

Variance is given by the formula:

σ  is known as the delta symbol, and it is the common symbol used as notation for Standard Deviation.

### Calculating Standard Deviation Example

*(2.1) *If we look at the same list of  7  numbers from example  (1.1):

**5 , 7 , 3 , 5 , 6 , 4 , 5**  =   \lbrace \space x_1 \space , \space x_2 \space , \space x_3 \space , \space x_4 \space , \space x_5 \space , \space x_6 \space , \space x_7 \space \rbrace

**=**

*n***6**  ,  

*μ*=

**5**

\sum_{i=1}^7 \space(x_i \space – \space \mu)^2

=  

**( x_1**−

**5 )**+

^{2} **  ( x_2**−

**5 )**+

^{2} **  ( x_3**−

**5 )**+

^{2} **  ( x_4**−

**5 )**+

^{2} **  ( x_5**−

**5 )**+

^{2} **  ( x_6**−

**5 )** +

^{2}**  ( x_7**−

**5 )**

^{2}=  

**( 5**−

**5 )**+

^{2}**( 7**−

**5 )**+

^{2}**( 3**−

**5 )**+

^{2}**( 5**−

**5 )**+

^{2}**( 6**−

**5 )**+

^{2}**( 4**−

**5 )**+

^{2}**( 5**−

**5 )**

^{2}

=  

**0**+

**4**+

**4**+

**0**+

**1**+

**1**+

**0**  =  

**10**

σ   =   \bf{\sqrt{\frac{10}{7}}}   =  

**1.195**

### What Standard Deviation tells us,

Using the Standard Deviation Value

Now considering the Standard Deviation is an indicator of how far away a set of values are from the mean/average.

Let’s see how we can make use of it with the list of  7  numbers.

For  

**5 , 7 , 3 , 5 , 6 , 4 , 5**,     the mean is  5.

While the Standard Deviation is  1.29.

We can establish how far way  1  Standard Deviation is from the mean/average in both a positive direction and a negative direction also.

**5**−

**1.195** = 

**3.805**   ,   

**5**+

**1.195** = 

**6.195**

What this means is that we should expect to see the majority of the values in the list of 7 to be between  3.805  and  6.195.

From looking at the list, this does turn out to be the case.

With only  3  and  7  being out side of this range.

## Alternative Formula for Standard Deviation

The variance does have a slightly simpler and quicker formula that can be used. This leads us to an alternative formula Standard Deviation that can also be used, which is:

Again  *μ*  is the mean/average, and  * n*  represents the number of values.

So the calculations required for example  (2.1)  from above using this formula are:

σ  =  \bf{\sqrt{{\frac{5^2 \space + \space 7^2 \space + \space 3^2 \space + \space 5^2 \space + \space 6^2 \space + \space 4^2 \space + \space 5^2}{7}} \space – \space 5^2}}   =   \bf{\sqrt{\frac{185}{7} \space – \space 25}}   =  

**1.195**

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