Significant Figures

Unfortunately, there is no hard-and-fast rule to use when determining the precision of the coefficients of a best-fit equation.  If you actually went through all the individual calculations which are used to perform the regression, you could certainly determine the sig-figs using the standard techniques.  But, with the TI-82, you can perform a statistical regression without knowing the details of the regression algorithm.  Therefore, you must use some other standard for determining how many significant figures to keep when analyzing a regression.

Though you may not know exactly how the regression is performed, it is nevertheless clear that the it is based on the numerical data in the calculator's list variables.  So in order to determine the precision of the regression coefficients, you must turn your attention to the list elements themselves.  When analyzing a regression, follow this rule of thumb:
 

"The number of significant figures in the regression coefficients is the highest number which is less than or equal to the number of significant figures used in two thirds of the data set."

While this may seem cumbersome, it is not very difficult to apply in practice.

For example, consider the sample data set:
 
 

Sample d vs. t Data
Time 
(s)		 Distance 
(m)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50 		  0.19
 3.48
 6.30
 8.86
 9.84
11.39
12.09
11.82
10.96
 9.21
7.20

Of the 22 data points in this table:

So, 17 of the 22 have 3 or more sig-figs.  According to the rule, the regression coefficients would have 3 significant figures as well:
 
So:   a = -9.91 m/s2
       vo = 15.3 m/s
       do = 0.0181 m

Remember, the 2/3 is just a rule of thumb.  It is not necessary to calculate the fraction exactly.  The idea is, you don't want to give up a sig-fig just because a couple of  points in a large data set are less well known than the others.  On the other hand, it would be misleading to suggest that your overall answer is known with as much precision as the two or three most precise numbers you measured.
 
 
 

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