In science every number should have an indication of the uncertainty in that number. Every number should have something like a margin of error. There are many ways to determine the uncertainty in a number. Scientists use instruments to get numbers. Each instrument has some uncertainty associated with it. A thermometer may have lines representing 0.5 ^{o}C between each line. You could tell that a temperature was between 21.0 and 21.5, but would have to guess if it was 21.2, 21.3, etc. The uncertainty would be in the first decimal place. You might think that the reading should be 21.3 and that you are confident it is between 21.1 and 21.5. You could then report the number as 21.3 ± 0.2. The "plus or minus" would tell anyone who saw the number how much confidence you have in the number. In this case it says that the actual temperature is likely between 21.3 + 0.2 = 21.5 and 21.3 - 0.2 = 21.1. If you had a better thermometer, you could report more decimal places and the "plus or minus" (±) would be smaller.

There are several mathematical ways to get the uncertainty. You may have heard of the standard deviation. It could be used to get an uncertainty number (a "plus or minus" number). A maximum differential error analysis could be performed to get the number for the uncertainty. In this class we will use a simple way of getting an uncertainty number using what is called the range for a set of data.

Consider the following set of data to illustrate the language and process of reporting numbers to be used in this class:

6.14, 6.03, 5.96, 6.11, 5.93

The sum of the numbers is 30.17 and the average is 6.034 (the sum divided by the number of data points, also called the mean). The median of a set of data is the data point that has exactly the same number of data points greater than itself as it has data points less than itself. In our case the median would be 6.03 since there are two numbers greater than 6.03 and two numbers less than 6.03.

The range is defined as the largest number minus the smallest number. In our case it would be 6.14 - 5.93 = 0.21. The range is how far it is between the high and the low.

If the median and the mean are close to the same they will be in the middle of the range. So, if half of the range is added to the average the high number would be obtained and if you subtract half of the range from the average the low number would be obtained. In our case they aren’t exactly the same, but they are close so that half of the range, 0.21/2 = 0.105, could be used as the "plus or minus". For purposes of this class always assume the average and the mean are the same.

It would seem that we should report 6.034 ± 0.105 as our result, but there is one more item to attend to. Notice that the "plus or minus" causes changes in the first decimal place. It is reported to the third decimal place, but does that five on the end matter if it changes in the first decimal place? The answer is no, the numbers beyond the first decimal place don’t matter. This means that the "plus or minus" should always be rounded to one number. In addition it should always be rounded up to make sure that all of the uncertainty is covered. In this case it should be rounded from 0.105 to 0.2, 0.11 would not be right because the one in the hundredths place doesn’t make any difference when there are changes in the tenths place.

Now that the "plus or minus" number is rounded off to the first decimal place, the number that is being reported for the result should also be rounded off to match it. The three and the four, in 6.034 of the number we are reporting, don’t matter if it is changing in the tenths place. Changing in the tenths place automatically changes the hundredths and thousandths places, so we should round the result, in the normal way, to the same decimal place as the "plus or minus", in this case to the tenths place.

The number to report would then be 6.0 ± .2, where both the reported value and the "plus or minus" number are given to the first decimal place.

The process is to calculate the number and the "plus or minus" value, round the "plus or minus" value up, and then use the normal rounding rules to round the number being reported to the same decimal place as the "plus or minus" value.

Here are a few more examples.

4.381 ± 0.062 | → 4.38 ± 0.07 |

0.682 ± 0.244 | → 0.7 ± 0.3 |

0.1388 ± 0.0038 | → 0.139 ± 0.004 |

2.867 ± 1.3 | → 3 ± 2 |