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Calculating Standard Deviation,
What Standard Deviation tells us


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:

Variance 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.

51.195  =  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:

Alternative formula that can help find Standard Deviation.

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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