Uncertainty in Measurements (or How to Weigh Your Car)

One can purchase a tire pressure gauge for less than a dollar. Believe it or not, this crude tool can be used to measure the weight of your car. The units it measures, PSI are pounds per square inch. If we know the "footprint" of the tire being measured, we can multiply that area by the pressure and that will give the weight supported by that tire. Add up all four tires to get the total weight of the car.

Every physical measurement involves uncertainty. The main point of this lab is to give you experience in how to deal with these kinds of measurement difficulties and to understand how individual measurement contribute to your final results.

Uncertainty from Multiple Measurements
Let's start with the tire pressure gauges. These are not scientific-grade instruments. To deal with this, we will take several measurements. This is a method used even when we are using "good" tools. The idea is that if our measurements are affected by uncertainty, the high readings will partially cancel the low readings or vice-versa and we'll get a closer estimate of the true value. Often students feel that "more measurements means more error", but the opposite is actually because of the cancellation effect.

We use the average (mean) value, which is simply the sum of all of our results divided by the number of measurements. For example, if we were counting the number of jelly beans in a jar we find 1534 beans. We count four more times to complete the following set of results:

Trial	Beans
1	1534
2	1523
3	1571
4	1537
5	1517
Total	7682

In this case we would report 7682/5 = 1536.4 jelly beans. Note that the average is only one aspect of a set of numbers. Take for example this second set of numbers:

Trial	Beans
1	1534
2	1423
3	1671
4	1637
5	1417
Total	7682

While this set of jelly bean counters have concluded that they also have 1536.4 jelly beans, one can see that there is a lot more "scatter" in their individual measurements. The difference between the two sets reflects different uncertainties in the measurements. This is reflected in a parameter known as the standard error (also known as the standard deviation of the mean). The standard error is closely related to a measure that some students might already know, the sample standard deviation.

The sample standard deviation is calculated by summing the square of the differences from the mean, dividing by one less than the number of measurements, and then taking the square root. The standard error is calculated by dividing the sample standard deviation by the square root of the number of measurements. That's a complicated written explanation, but the actual procedure is fairly straightforward. Let's calculate the sample standard deviations and then standard errors for our data sets:

First we sum the squares of the differences from the mean:
(1534 - 1536.4)^2 = 5.76
(1523 - 1536.4)^2 = 179.56
(1571 - 1536.4)^2 = 1197.16
(1537 - 1536.4)^2 = 0.36
(1517 - 1536.4)^2 = 376.36
This totals to 1759.2

We need to divide this by one fewer than our measurements. Since we measured five times, we need to divide by four. 1759.2/4 = 439.8. Finally we take the square root to get 20.97. We are not finished yet. 20.97 is the sample standard deviation, to get the standard error we need to divide by the square root of five. The result is 9.378, which we can round to 9 (rule of thumb is one sig fig for error measurements, with all kinds of loopholes for "leading ones"). We can now claim that we've got 1536 ± 9 jelly beans from the first set of measurements. You should verify that the sample standard deviation from the second set is 117.64. We'd claim that we have 1540 ± 50 jelly beans in this case.

The first step in weighing the car is to use the tire pressure gauge to take five measurements at each tire to get a mean and sample standard deviation for each.

Uncertainty from Multiplication
Now that we have a pressure for each tire, if we can multiply by the area that the tire is in contact with the ground we will have a value for the weight that the tire supports. But we need to know how the uncertainty is affected by this multiplication.

Of course, there is more than one thing going on here. Not only is there an uncertainty involved in the pressure, but also one in the area. When you measure the length and the width of the footprint, each of these will have some uncertainty. Also, the tread on the tire represents only a percentage of the total area. The pressure, width, length and percentage all need to be multiplied together in order to get the weight. What is the uncertainty in this weight?

When we use multiplication, we deal with percentage uncertainties. Suppose we have a good jelly bean counter who counts 60 ± 2 jelly beans. This is a 2/60 = 3.3% uncertainty. A sloppy counter might report another sample of 50 ± 5 jelly beans for a 10% uncertainty. Percentage uncertainties will add in a quadratic fashion. This means we'll sum the squares and then take the square root. Suppose we wanted to multiply our two groups of jelly beans. We would get 60 * 50 = 3000 jelly beans. The percentage uncertainty would be the square root of the sum of the squares of each percentage uncertainty. Again, many confusing words where a quick example is more illuminating, so herešs how we would do the calculation:

[ (3.3)² + (10)² ] ^1/2 = 10.5%

10.5% of 3000 is 315, so we would report 3000 ± 315 jelly beans. Note that the quadratic nature of the operation means that you should concentrate on your largest percentage uncertainties, as they will dominate your final result.

Uncertainty in Addition
At this point each tire should have a weight assigned to it. How do we handle uncertainty and addition? Here we simply add the uncertainties in a quadratic fashion. For example, 200 ± 15 added to 250 ± 40 is 450 ± (15² + 40²)^1/2, in other words, 450 ± 34.

You should now know the total weight for your car! Compare this to the number found in your owner's manual as a double check.

You should use these uncertainy techniques in almost every lab this quarter.