Averages and Deviations

Accuracy, Consistency and Precision

Mathematics deals with pure numbers, and pure numbers are precise, or exact. In science, getting precise numbers is difficult. Sometimes you can be precise. For example, you may divide a number by 2, which is exact, or you may multiply by an exact whole number. However, most data is not exact. It might be possible to get a precise value for the number of people in a room, but if there are 50 people moving about it might be difficult. You might count them 3 or 4 times and get a different value each time. Your value might not be very accurate or precise. In science, there is often a wide range of acceptable values, and few numbers are exact.

Accuracy is a measure of how close a number is to the actual value. The further from the measurement is from the true value, the less accurate that measurement is. In many experiments, accuracy cannot be determined for sure; only an estimate of the accuracy can be found, because the actual value is not known.

When we make lots of measurements, and they are all close to the same, then our data has good consistency. If the results are all over the place, then we say they are inconsistent, or have a lot of uncertainty. Being consistent means getting roughly the same result every time. (Sometimes this is also called “precision,” but we will use that term to mean something else.) We will estimate our consistency by calculating something called the “average deviation,” described below. (Although standard deviation is technically a more accurate method of finding the error margin we will use the average deviation method because it is relatively easy to calculate.)

Here is an illustration of the difference between accuracy and consistency:

Precision is the term we will use to describe the exactness of our data. Saying, “The temperature is 87.5°C” is more precise than saying, “It’s in the eighties today.” We will also use precision when we talk about answering a question, even if no numbers are involved. A more precise answer is more exact, and gives more information. For example, if your parents ask where you’ve been, you can be precise and tell them exactly where you went, who you were with, and what you did. Or you can just say, “ I was out” (which may be accurate, yet not precise.).

Averages and Deviations

To Find Average Deviation (a measure of consistency or uncertainty):

1. Find the average value of your measurements.
2. Find the difference between your first value and the average value. This is called the deviation.
3. Take the absolute value of this deviation.
4. Repeat steps 2 and 3 for your other values.
5. Find the average of the deviations. This is the average deviation.

Sample Problem:

Eight people looked at a rope and estimated how long it was. Their estimates are shown. The average of the eight estimates was 10.m, so it is probably 10. m long, plus or minus “a bit.” How much is “a bit?” For each measurement, the deviation was calculated. (Notice that the deviation is always positive.) In this case, notice that, on average, the deviation is 1.25 m. This tells us that, on average, any given measurement will be maybe 1.25 m less, maybe 1.25 m more than the average value. In this case, we can report that the rope is estimated to be 10. +/- 1.25 m long (or, alternatively, that it is probably between 8.75 and 11.25 m in length).

EXAMPLE: Estimated Length of a Rope

Now eight people actually measure the rope with a meter stick, and they get the following data:

EXAMPLE: Measured Length of a Rope

In this case, their average is the same as before. The average deviation is much smaller, which shows that the measurements are all very close to the average. In this case, we can be pretty confident that the rope is actually 10. m long.

In this example, the estimates (see the first table) turned out to be accurate. In other words, the average was close to the correct value. The measurements (second table) were more consistent. And often this is the case: we can make reasonably accurate estimates, even though our estimates may not be consistent.

(Much of this is taken from “Average Deviation,” 15 September 2006)