Populations & Samples

All peices of data are referred to as the population, a subset of that is taken to form a sample.

 

Greek letters differ based on whether the test refers to the Population or a sample. (not able to be shown here is a small bar over the top of the x as an indicator of the Arithmetic mean of a Sample).

 

                               Population        /        Sample

 

Arithmetic mean

μ

x-

Standard deviation

s

s

Correlation coefficient

r

r

Regression coefficient

b

b

 

Information may be grouped into classes where the number of classes to use is 2 raised to the square root of the number of classes (k) and 2^K equals the number (or less than) of the sample.

 

Relative frequency = frequency / total observations

 

 

Calculations for estimating the arithmetic mean        

     
 

and the variance from a frequency distribution

     
               

Midpoint

Frequency

           

(x)

(f)

fx

(x-µ)

(x-µ)²f

     
               

210

23

4.830

         (146)

                                     490.268

     

330

18

5.940

           (26)

                                       12.168

     

450

8

3.600

            94

                                       70.688

     

570

6

3.420

          214

                                     274.776

     

690

4

2.760

          334

                                     446.224

     

810

1

810

          454

                                     206.116

     
 

60

21.360

 

1.500.240

     
               

Arithmetic mean from frequency distribution:

=C16/B16

          356

 = 21.360 / 60

 

Actual arithmetic mean:

   

=MITTELWERT(Dataset!E11:E71)

          358

Sales

(millions)

               

Variance from frequency distribution.:

 

=E16/B16

      25.004

 = 1500240 / 60

Standard deviation from frequency distribution:

=WURZEL(E21)

          158

   
               

Actual variance:

   

=VARIANZEN(Dataset!E11:E70)

      24.878

Sales

(millions)

Actual standard deviation:

   

=STABWN(Dataset!E11:E70)

          158

   
               
           

 

Percentage

Growth

Actual Sales

   

Year

Change

factors

($ billions)

   

 

(1)

(2)

 

       C8      D8

20X0

25%

=1+B9

=D8*C9

1,25

         18,75

20X1

-30%

=1+B10

=D9*C10

0,7

         13,13

20X2

40%

=1+B11

=D10*C11

1,4

         18,38

20X3

-20%

=1+B12

=D11*C12

0,8

         14,70

           
           

Arithmetic mean of growth factors

D15 = MITTELWERT(C9:C12)

          1,04

 

Geometric mean of growth factors

D16 = GEOMITTEL(C9:C12)

          0,99

 

Average growth rate

 

D17 = D16-1

         (0,01)

 
           

Year 20X3:

         

Applying arithmetic mean

 

=D8*D15^4

         17,38

 

Applying geometric mean

 

=D8*D16^4

         14,70

 
           
 
 

In determination of the relative variability of a set of data to another, compare their coefficent of variation. That is the ratio of the standard deviation to the mean.

 

sample 1 CV = 0,5%

sample 2 CV = 2,5%

 

Sample 2 is twice as variable a sample 1.

 

There may be uncontrollable factors which affect samples such as state of the economy, geopolitical events, weather, interest rates, and other. So when studying investments, these factors may affect final outcomes despite historical rates and consideration of risk.

 

The coefficient of variation is useful if investments have the same standard of deviation. Looking to the CV, one can build a better picture of risk.

 

A sample proportion = x/n, x=sample event, n=sample population.