Data Summary

 The z score

We define the z-score as

                      observation - mean  
        z   =                                       
                      Standard Deviation

Example

The mean exam score is a 85 with a standard deviation of 10, then if a student scores a 93 her z-score is

          93 - 85
                            =  0.8   
             10 

The Empirical Rule

Chevychev's Rule is a worst case scenario.  In reality the probabilities are much higher.   Typically we have the following table called the Empirical Rule which usually works:

 

z	1	2	3
P	68%	95%	97%

Application:   

Suppose that you collect data for your fresh fish restaurant and find out that there the average number of fish dishes ordered is 50 with a standard deviation of 5.  How many fish should you stock if you want run out of fresh fish no more that 5% of the time, i.e. you will have enough fish 95% of the time?   

Solution: 

1. If you are worried about the worst case distribution, then use Chevychev's Rule to solve:
   
                            x - 50
           4.472  =                  
                              5
   
           22.36  =  x - 50
   
           x  =  72.36
   
      So stock 73 fresh fish.
   

2. If you expect the distribution to be closer to bell shaped, then use the Empirical:
   
                      x - 50
           2  =                 
                        5
   
           10 = x - 50
   
           x = 60
   
   So stalk 60 fresh fish.
   

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