STANDARD DEVIATION

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

The formula is easy : it is the square root of the Variance. So now you ask, “What is the variance?”

The Variance

Variance is defined as : the average of the squared differences from the Mean.

To calculate the variance follow these steps: 

•  Work out the Mean (the simple average of the numbers)
•  Then for each number: subtract the Mean and square the result (the squared difference)
•  Then work out the average of those squared differences.

Example:

You and your friends have just measured the heights of your cats (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Find out the Mean, the Variance and the Standard Deviation.

Your first step is to find the mean:

So the mean (average) height is 394 mm.

Now we calculate each cat’s difference from the Mean:

To calculate the variance, take each difference, square it, and then average the result:

So the Variance is 21, 704

And the standard deviation is just the square root of Variance, so:

And the good thing about the Standard Deviation is that it is useful. 

Now we can show which heights are within one standard Deviation (147mm) of the mean:

So, using the Standard Deviation we have a “standard” way of knowing what is normal, and what is extra-large or extra small.

Example 2:

In the case where we have a sample size of 5 pirates, therefore we will be using the standard deviation equation for sample of a population.

Here are the amounts of gold coins the 5 pirates have:

4, 2, 5, 8, 6,

Now, let’s calculate the standard deviation.

Questions that you can try:

1. Consider the following three data sets A, B and C. 

A = {9,10,11,7,13} 

B = {10,10,10,10,10} Find 

C = {1,1,10,19,19} 

a) Calculate the mean of each data set. 

b) Calculate the standard deviation of each data set. 

c) Which set has the largest standard deviation? 

d) Is it possible to answer question c) without calculations of the standard deviation?

2.The frequency table of the monthly slaries of 20 people is shown below. 

salary(in $)	frequency
3500	5
4000	8
4200	5
4300	2

a) Calculate the mean of the salaries of the 20 people. 

b) Calculate the standard deviation of the salaries of the 20 people.

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