Question 80712
need some help on this question 
Use the following sample data to find the mean, median, mode and standard deviation 
2,3,3,6,9,9,9,15 
thank you for your time and help Dave.

<pre><font size = 5><b>

Mean:

There are 8 numbers. Add them 

2+3+3+6+9+9+9+15 = 56
                                  _
Divide that by 8 and get the mean x = 7

----------------------------------------------------

Median:

Arrange them in order from smallest to largest:

2,3,3,<font color = "red">6</font>,<font color = "red">9</font>,9,9,15

(Remember that by "A median is in the MIDDLE
of a road")

If there had been an odd number of them you
just select the one right in the MIDDLE and
that would have been the median.  However there
is an even number of them here, since 8 is
an even number. So there is not just one number
in the MIDDLE, but there are two in the MIDDLE,
as you see, the two red one above, 6 and 9.
So we add those MIDDLE two, 6+9 = 15, and divide
that by 2, and that gives up 7.5 for the median.

-------------------------------------

Mode;

(Remember that the first two letters of "MOde" are
the same as the first two letters of the word
"MOst")

The MOde is the number, if any, in the data that appears
the MOST number of times.  There is sometimes more than
one mode because two or more numbers appear the same
number of times.  However if every number only occurs
once, we don't call them all modes, we just say, there is
no mode.

In this data list, 9 occurs 3 times, and this is the MOst
number of times any number occurs, so the MOde is 9.

-----------------------------------------

Standard deviation:

2,3,3,6,9,9,9,15

Here is what we normally do. We start out with
this table: 

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x -  )  (x -  )² f(x -  )²
                                             
                                         
                                           
                                           
_________________________________________  
sums:                                                                                               

Under x we list each number only ONCE, even if it
appears more than once.  (We take care of the number
of times it appears in the next column.)

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x -  )  (x -  )² f(x -  )²
 2   
 3     
 6     
 9    
<u>15                                       </u>
sums:  


Under the f column we list the frequency, which means
the number of times the number listed in x appears
in the set of data.

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x -  )  (x -  )² f(x -  )²
 2     1     
 3     2     
 6     1      
 9     3     
<u>15     1                                 </u>
sums:      

Then under fx we multiply each number under x by the
corresponding number in f.
                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x -  )  (x -  )² f(x -  )²                 
 2     1      2     
 3     2      6    
 6     1      6      
 9     3     27      
<u>15     1     15                          </u>
sums:

Now we add the two columns which have "f" in the headings: 
                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x -  )  (x -  )² f(x -  )²                 
 2     1      2     
 3     2      6    
 6     1      6      
 9     3     27      
<u>15     1     15                       </u>
sums:  8     56                          

We divide the sum of the fx column by the sum of the
f column, and get 56÷8 or 7. This of_course is the mean
which we call "x-bar", and write as x.
                        _
The we substitute 7 for x in the headings and write
them under the present headings:

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x - 7)  (x - 7)² f(x - 7)²
 2     1      2   
 3     2      6    
 6     1      6    
 9     3     27     
<u>15     1     15                       </u>
sums:  8     56                          
                            _
Now we form the column (x - x)  by subtracting 7
from each value in the x column


                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x - 7)  (x - 7)² f(x - 7)²
 2     1      2     -5       
 3     2      6     -4       
 6     1      6     -1        
 9     3     27      2      
<u>15     1     15      8                   </u>
sums:  8     56                      
                                        _
Next we square every member of the_(x - x)
column and write it in the   (x - x)² column

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x - 7)  (x - 7)² f(x - 7)²
 2     1      2     -5       25        
 3     2      6     -4       16        
 6     1      6     -1        1          
 9     3     27      2        4        
<u>15     1     15      8       64          </u>
sums:  8     56                       

To form the last column,_we multiply each
     number in the (x - x)² by the corresponding
           value in the f column:

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x - 7)  (x - 7)² f(x - 7)²
 2     1      2     -5       25        25
 3     2      6     -4       16        32
 6     1      6     -1        1         1 
 9     3     27      2        4        12
<u>15     1     15      8       64        64</u>
sums:  8     56                       


Then we form the sum of the last column. Notice
that we have only summed the columns which have an
 "f" in their headings.

                       _        _         _
 x     f     fx   (x - x)  (x - x)² f(x - x)² 
                  (x - 7)  (x - 7)² f(x - 7)²
 2     1      2     -5       25        25
 3     2      6     -4       16        32
 6     1      6     -1        1         1 
 9     3     27      2        4        12
<u>15     1     15      8       64        64</u>
sums:  8     56                       134

Next we divide the sum of that last column by
ONE LESS than the sum of the f column. One
less than 8 is 7, so we divide 134 by 7 and get
134÷7 = 19.14285714.

This number has a name. It is called the
"variance", and is denoted by s².

To get the standard deviation, which we
call s, we take the square root of the
variance, and get:

standard deviation, s = 4.375255095

Edwin</pre>