Average deviation

You know when performing an experiment you have to repeat the same measurements several times (at least three times). You then need to look at them, identify and investigate anomolies and then calculate an average value.

But still need to know how “good” this average value is.

One measure of this is called the average deviation.

The average deviation of a data set is the average value of the absolute value of the differences between the individual data numbers and the average of the data set.

A Hockey Captain wants to determine the average deviation for the number of goals per game that her team scored this season.

Her data set is 5, 2, 7, 3, 1 and 4.

STEP 1: Calculate the mean by adding up all the goals, then dividing by the total number of games, which is six.

5 + 2 + 7 + 3 + 1 + 4 = 22 goals

22 / 6 = 3.6 average goals/game

Step 2: Calculate the deviation, which is the value of each goal in the data set, from the mean.

5 - 3.6 = 1.4

2 - 3.6 = -1.6

7 - 3.6 = 3.4

3 - 3.6 = -0.6

1 - 3.6 = -2.6

4 - 3.6 = 0.4

The sum of the deviations is then calculated:

1.4 -1.6 + 3.4 -0.6 -2.6 + 0.4 = 0.4

The the average deviation is the sum of all the deviations divided by the number of games:

0.4 ÷ 6 = 0.06

This means that the average deviation from the mean regarding the number of goals that her team scored during the season is 0.06.

Since the goals all stayed within a close range to the average deviation, it shows that she kept consistent scores throughout the season and the team's performance per game can be expressed as

3.6 ± 0.06 goals

Rule 0 - Numerical and fractional uncertainties.

The uncertainty in a quantity can be expressed in numerical or in fractional forms.

Thus in the above example, ± 0.06 goals is a numerical uncertainty, but we could also express it as a fraction or percentage of the average value:

0.06/3.6 = 0.0166

Therefore as a percentage it is 0.0166 x 100 = ± 1.7%

Rule 1: Addition and subtraction of uncertain numbers.

If you are adding or subtracting two uncertain numbers, then the numerical uncertainty of the sum or difference is the sum of the numerical uncertainties of the two numbers.

For example, if A = 3.4 ± 0.5 m and B = 6.3 ± 0.2 m, then A + B = 9.7 ± 0.7 m, and A - B = - 2.9 ± 0.7 m.

NB the numerical uncertainty is the same in these two cases, but the fractional uncertainty is very different.

You cannot add the percentage uncertainties!

You have to add the numerical ones and calculate the new percentage ones if you need them.

A = 3.4 ± 15% m

B = 6.3 ± 3.2% m

A + B = 9.7 ± 7.2%

A - B = -2.9 ± 24%

Rule2: Multiplication and division.

If you are multiplying or dividing two uncertain numbers, then the fractional uncertainty of the product or quotient is the sum of the fractional uncertainties of the two numbers.

For example, let A = 3.40 ± 0.5 m, and B = 0.334 ± 0.006 sec,

A = 3.40 ± 14.7% and B = 0.334 ± 1.80%

then

A/B = 10.2 m/s ± 16.5%, or A/B = (10.2 ± 1.68) m/s.

and

AB = 1.1356 ms ± 16.5% or (1.1356 ± 0.187) ms

Notice that the fractional uncertainties are the same, but the numerical uncertainties are different (even different units).

Rule 3: Raising to a power

If you are raising an uncertain number to a power n, (squaring it, or taking the square root, for example), then the fractional uncertainty in the resulting number has a fractional uncertainty n times the fractional uncertainty in the original number.

So, if you are using this equation:

x = ½at²

and have found t = 2.36 ± 0.04 s,

t = 2.36 ± 1.695%

so t² = 5.57± 3.39%

So the uncertainty in t² will be ± 3.39%.

Rule 4: Complicated expressions.

In cases other than the above, you can do a numerical calculation to find the numerical uncertainty.

We could call this the “max/min” method.

For example, if you are calculating Fx = F cos θ , where θ = 34.6° ± 0.2 ° , and F = 13.2 ± 0.7 N, then do it by a two- step calculation:

Step 1: calculate the uncertainty in cos θ by comparing cos (34.6) to cos(34.6 + 0.2).

The uncertainty in cos θ is taken as the difference between these numbers.

cos 34.8° - cos 34.4° = 0.00396

So, cos (34.6° ± 0.2°) = 0.8231 ± .00396

Step 2: Use rule 2 above to calculate the uncertainty in F cosθ.

The result is F cosθ = 10.865 N ± 5.78% or (10.865 ± 0.628) N.

Rule 5: Significant figures

Once you get to a final result (no more computations are to be made with the quantity), then we usually keep one significant figure in the value of the uncertainty, and round off the quantity itself accordingly.

It is pointless adding more figures after the one that you have already calculated as being 'uncertain' - so don't!

The final result in the Rule 4 example should be expressed as 10.9± 0.6 N, and the final result in the examples of Rule 2 are (10 ± 2) m/s and (1.1 ± 0.2) ms.

This page was extracted from a pdf written by CCJ 1/8/98 for Union College - Schenectady, N.Y.