%error and chi2 deviation tutorial
When we calculate % error, we are trying to compare data that we have collected from a lab with the
data that has been tested and accepted as a true and consistent value that can be used as a point of
comparison. The equation we use is:
% error = accepted value – experimental value x 100
Accepted value
If we look at an example: A student calculates the density of a piece of Pb to be 11.33g/ccm. He asked
to calculate the %error of his calculation. The accepted density of Pb is 11.34 g/ccm.
If we plug into our equation the values that we have, our equation will look like this:
% error = 11.34 – 11.33 x 100
11.34
Solving for % error:
% error = 0.01 x 100
11.34
% error = 0.000881 x 100
% error = 0.0881%
If we
Round this % error = 0.09%. This is really a very good % error. On our level we could ignore any
mistakes made that are reflected in the error.
Let’s take a look at another problem. A student calculates the boiling point of an alcohol. The data
collected shows the boiling point to be 65.1o . What is the% error if the boiling point of this alcohol is
64.7o?
Okay if we substitute all the values we know into the equation we get:
% error = 64.7 - 65.1
64.7
x 100
Solving for % error:
% error =
% error =
0.4
64.7
x 100
-0.00618
x 100
% error = 0.00618 x 100
% error = 0.618 %
Now you try one:
A student records the following data for his lab calculations for the density of a pine block.
Trial #
Density Calculated
1
0.69
2
0.65
3
0.68
Calculate the % error for each trail. The accepted value for the density of pine is 0.638 g/ccm
Chi2 Deviation
Chi2 deviations compare all of the results collected to the accepted value. The formula we use is:
Chi2 = ∑
d2
E
Where d is the deviation from the expected value and e is the expected value.
Let’s take a look at an example.
A coin is tossed 100 different times. The following are the recorded results: 53 heads 47 tails
Because a coin has two sides, the probability is that heads would come up 50 times and tails would
come up 50 times. What is the deviation?
With respect to the 53 heads, this deviates +3 from the expected 50 times heads should come up.
With respect to the 47 tails, this deviates -3 from the expected 50 times tails should come up.
If we plug these values into our formula, we get:
X2 =
+32/50
X2= 9/50 +
+
-32/50
9/50
X2 = 0.18 + 0.18
X2 = 0.36
Now that we have the deviation we need to compare it to the chi2 table of probabilities for one degree
of freedom.
Highly insignificant
0.2
1.6
Insignificant
0.1
2.7
0.05
3.8
Significant
0.01
6.6
Highly insignificant
0.001
10.8
Where does our value of 0.36 fall? Less than 1.6 which means that the deviation is highly insignificant.
This means that the tosses measured followed the laws of probability and are considered random.
Values greater than 6.6 would suggest that the deviation is too large. This would mean that something
caused the large deviation, such as inaccurate observations or inaccurate measurements.
Try this example:
A lab team does a lab on the genetic for the color of corn kernels. Their results are: total counted 100
Yellow 68 kernels, reddish brown 32 counted. What is the chi2 deviation?
Are their results according to the expected results?
Can their results be considered significant or insignificant according to one degree of freedom?