Question 320246
<pre><b>
On your TI-83 or TI-84
 
Press CLEAR
Press STAT
Press 1
 
Enter the 12 numbers in L1
 
Press STAT
Press the right arrow key to highlight CALC
Press 1
Press ENTER
                 _
Read the mean as x = 34.33333333
Read the standard deviation as Sx=7.784989442
 
Chebyshev's theorem states that
 
{{{(1-1/k^2)*"100%"}}} of the data will lie within {{{k}}} standard
deviations of the mean, where {{{k>1}}}.
 
Using k=1.1, and the mean and standard deviation above, Chebyshev's 
theorem tells us that AT LEAST 17.3% of the data lies between 25.77 
and 42.897.  In fact 83.3% of it does! 
 
Using k=1.5, and the mean and standard deviation above, Chebyshev's 
theorem tells us that AT LEAST 55.5% of the data lies between 22.66 
and 46.01.  In fact 83.3% of it does!
 
Using k=2, and the mean and standard deviation above, Chebyshev's 
theorem tells us that AT LEAST 75% of the data lies between 18.76 
and 49.90.  In fact 100% of it does!
 
-----------------------------
 
The empirical rule says that if a histogram of the data is 
approximately bell-shaped, like this:
 
{{{drawing(400,200,5,15,10-.5,10+1.5, graph(400,200,5,15,10-.5,10+1.5, exp(-(x-10)^2/2)+10),  locate(10-.2,10-.05,34&1/3),
line(11,10+.05,11,10-.05),
 line(12,10+.05,12,10-.05), 
 line(10,10+.05,10,10-.05), 
line(9,10+.05,9,10-.05),
 line(8,10+.05,8,10-.05),locate(14.7,10-.05,x),
locate(4.8,-.01,z),locate(4.8,.2,z),
line(4,10-.05,16,10-.05)  
)}}}

then 

1. approximately 68% of the data will fall between 1 standard deviation
below the mean and 1 standard deviation above the mean.

That is, if the given data is approximately bell-shaped, then 68% of
the data should fall between 

34.33333333 - 7.784989442 or 26.54834389
and
34.33333333 - 7.784989442 or 42.11832278

Actually 10 of the 12 data values fall between these, so that's {{{83&1/3}}}%

Also
2. approximately 95% of the data will fall between 2 standard deviations
below the mean and 2 standard deviations above the mean.

That is, if the given data is approximately bell-shaped, then about 95% of
the data should fall between 

34.33333333 - 2(7.784989442) or 18.76335445
and
34.33333333 + 2(7.784989442) or 49.90331222

Actually all 12 of data values fall between these, so that's {{{100}}}%.

Edwin</pre>