If you take a large set of data that has some amount of random error in it,
the results will tend to follow the normal curve of uncertainty, often
called the bell-shaped curve. The normal curve for the value 10 with standard
deviation s = 1 is shown below
This curve shows up no matter what the data: if the errors are
random (due to things like electrical noise and thermal motion) you will see
this spread of data.
This curve has important implications for how we handle errors. The standard
deviation, or "one-sigma" error, contains about 68% of the area of this curve
In the picture
above, value 10+-1 is shown. About 68% of the area of the curve lies from 9
-> 11, the one sigma range. If we take another measurement, there is a ~68%
chance that that measurement will fall between 9 and 11. This means that there
is a ~68% chance that the actual value that we are trying to measure will also
fall within this range. If you go out to 2s, 10+-2,
about 95% of the curve lies between 8 -> 12, so there is an 95% chance that
the true value falls in this range. A table of the sigma values and the chance
that the true value falls in the given range for the value 10+-1 is shown below:
| Average +- s | Range | Chance true value is in this area |
| 10+-1*s | 9 -> 11 | ~68.23% |
| 10+-2*s | 8 -> 12 | ~95.44% |
| 10+-3*s | 7 -> 13 | ~99.73% |
| 10+-4*s | 6 -> 14 | ~100% |
As you can see, beyond 2s the chances that the true value lies in the +- 3 or 4 sigma range is very high.
We never have enough data to say for sure that the true value is within a certain range- there's always a tiny chance that it doesn't. (The normal curve never goes completely to zero.) In reality, we have to cut off the range somewhere: most scientists use the ~95% error range as a reasonable compromise.
Example: If you are given a set of data reported as 4.86 +- 0.09, what is the range that would give you a 68% confidence interval? What is the range for 95%?
Solution: A confidence interval of 68% corresponds to one standard deviation from the norm. Since the standard deviation is +- 0.09, there is a 68% chance that the true value falls within +-0.09 of the average value, or 4.77-> 4.95. The 95% confidence interval corresponds to two standard deviations, or 2*0.09 = +- 0.18. There is a 95% chance that the true value falls within 4.68-> 5.04.
Information provided by: http://learn.chem.vt.edu