Back to EAS133: Drifting Continents Home
How to calculate the mean:
(The mean or specifically, the arithmetic
mean, is the "average" that most people know about.)
1. Add up all the individual measurements to get the sum.
Say we have measurements 2, 3, and 4. The sum is 9.
2. Divide the sum by the number of measurements.
We have 3 measurements, so the sum 9 divided by 3 equals
3.
3. The result is the mean.
Our mean or average is 3.
How to calculate the standard error
(sigma):
(Also called the the standard error of
the mean, or standard deviation of the mean.)
We first need to calculate the sample standard deviation (s):
1. Subtract each individual measurement from the mean that you
got above.
You will of course end up with some positive and some negative numbers.
Our mean was 3, so we have:
3 - 2 = 1
3 - 3 = 0
3 - 4 = -1
2. Square each of the numbers you got in no. 1.
1 squared = 1
0 squared = 0
-1 squared = 1
3. Add up all the squared numbers you got in no. 2
1 + 0 + 1 = 2
4. Divide the sum you got in no. 3 by the total number of
measurements minus 1.
We have 3 measurements. 3 - 1 = 2. So 2 / 2 = 1
5. Get the square root of the number in 4 above. The result is
the sample standard deviation.
The square root of 1 is 1.
(Note: So far, all our numbers have been fortuitously simple
integers. This may not be so with another
set of initial measurements.)
6. The standard error is then equal to the standard
deviation divided by the square
root of the number of measurements.
The square root
of the number of measurements (3) is 1.73...
So the standard error is 1 / 1.73 =
0.578.... or rounded off, = 0.6.
A 95% confidence interval on a mean is given by = mean ± 2 standard errors. In this case we have the mean and a 95% confidence interval as 3.0 ± 1.2.
Note that the standard deviation does not change significantly
even if the number of measurements, N, is increased. That is because the
standard deviation is a property of the technique for measuring. The standard
error, however, decreases with increasing N. This should make sense; one can
decrease the size of the confidence interval by doing more measurements.
____________
Note that the steps laboriously described above are
summarized by mathematicians as: