Uncertainty

All measurements have uncertainty. In science, we work to be forthright about the uncertainty in measurements and the conclusions we draw from them to better allow for intelligent conversation about the validity of any new findings. For example, if there were a study showing that left-handed people live, on average, 2 years longer than right handed people, but there was an uncertainty of 5 years in that average, nobody would start touting the virtues of being a lefty.

We have two types of uncertainty to deal with, each of which will need to be discussed in any lab journal entry:

Random Uncertainty

Random uncertainty is expected in all measurements. Since there are no perfect measurements, it makes sense to assume that sometimes we'll measure a value that's a little higher than the true value, and sometimes we'll measure a little low. Random uncertainty is the reason that the average of multiple measurements tends to be more reliable than any single measurement. Random uncertainty is caused by a few different factors:

• Limitations of our measuring devices. The tools we use aren't perfect. Even a very well tuned instrument has limits to how reliably it can make a measurement.

  • "The balance used to measure the mass of our pendulum bob did not give a consistent reading. It fluctuated by about 0.05 grams, so all mass measurements have an uncertainty of at least 0.05 grams."

  • "The length of the block was measured using a ruler marked down to the millimeter. All length measurements, therefor, have and uncertainty of 0.5 mm."

• Our own ability to use our measuring devices.

  • "Stopwatches were used to measure the time for the pendulum to complete one oscillation. These rely on the experimenters to push the button at the right time. Since our reaction time is roughly 0.1 seconds, all time measurements will have an uncertainty of at least 0.1 seconds."

  • "The length of the pendulum is taken to be the distance from the point at which the pendulum hangs to the center of mass of the bob. We had to estimate the location of the center of mass, so length measurements have an uncertainty of about 1 cm.

• Inconsistencies in our procedure.

  • "The pendulum was to be released from an angle of 10 degrees from vertical, but we found it difficult to measure this angle and to hold the pendulum steady at any angle. The release angle might have varied by up to 2 degrees, which may have caused some of the measured values for time to be affected as well."

  • "When the ball was dropped, we assumed it was released with zero speed. In practice, thought, the ball seemed to be moving slightly upward or downward very slightly when it was released due to the way the hand uncurled to let the ball go. The starting velocity, then, has an uncertainty of roughly 0.2 m/s."

Systemic Uncertainty

Systemic uncertainty means there was a problem with the experiment. We work very hard to find and eliminate systemic uncertainty before completing any experiment. Systemic uncertainty differs from random uncertainty in that the error will be in the same direction every time. Random uncertainty sometimes causes a measurement to be too high and sometimes too low. Systemic uncertainty causes a measurement to always be too high, or always be too low. Systemic uncertainty can not be accounted for by taking an average of multiple data points, because all of the data points will be wrong in the same direction. The average will not be close to the true value. Systemic uncertainty is caused by a few different factors:

• A faulty or incorrectly adjusted measuring tool. Don't forget to zero the scale!

• Technique error. Something we do in the experiment is consistently done in a way that skews the results.

  • "The person releasing the cart was not the same as the person timing the cart's motion. The person timing had to watch for the cart to start to move, then react to that by pushing the start button on the stopwatch. This always happened after the actual start of the movement, so the very first part of the motion was not timed, leading to all time measurements being too low."

  • "The length of the pendulum was measured from the point where the pendulum attached to the very bottom of the pendulum bob. It should have been measured to the middle of the bob. All length measurements, then, were roughly 2 cm longer than they should have been.

• Incorrect assumption. Usually, this means we didn't account for something that affects our experiment. It could also mean that we thought about it, decided it wouldn't have a noticeable effect, and were wrong in that decision.

  • "In determining the ball's fall time, we used a value for the acceleration of 9.8 m/s/s downward, which is the acceleration due to gravity in a vacuum. However, since the ball was not in a vacuum, air resistance caused the ball to accelerate downward less rapidly. This led to calculations for time that were consistently smaller than they should have been."

  • "In measuring the force applied to the cart, we measured the force of tension in the string. However, we measured this after the string had passed over a pulley. The pulley was treated as frictionless, but really there was a small amount of friction. This would cause our measured value of tension to be slightly higher than what the cart would feel."

Incidentally, these types of statements should never be the end of a lab entry. Something like this should be followed by, "we will correct this error by making the following changes to our procedure ..." and then collecting new data to analyze using those changes. The only exception to this is when there is no way to correct for the errors you identified. In this case, you should make an estimate for what the corrected values should be and add those to your conclusion discussion. Never change your data or ignore your findings, but it is OK to add to those findings when you think you're able to make better estimates for values that were affected by systemic uncertainty that you cannot correct for.

• "Using the data we collected originally, we calculated that the acceleration due to gravity at our location is 9.95 m/s/s. However, if we estimate that the pendulum was actually 2 cm shorter than our measurements in each case, we find a value of 9.81 m/s/s, which is much closer to the expected value."

Amounts of Uncertainty

There are several systems that are frequently used to track and report the amount of random uncertainty in measurements and calculated values.

• • Significant figures / significant digits

    • In this system, numbers are written in a way that not only shows the value, but also an estimate of the uncertainty. For example, the measurements "1.0 meters"and "1.00 meters" in this system would have the same value, but different amounts of uncertainty.

    • This systems will be used when we are solving numerical problems for practice, homework, and exams.

    • For the AP exam, numerical answers are expected to be written with an appropriate number of significant figures (though penalties are only imposed for egregious errors here).

    • For more information about this system, see the notes and video in the general math and science skills part of this web site.

  • Absolute uncertainty

    • In this system, measured values are followed by the +/- symbol and a second number that represents the uncertainty in the value. For example, (1.0 +/- 0.05) meters, which is ready as "one point zero plus or minus zero point zero five meters". This means that we believe the value to be 1.0 meters, but the true value might be as much as .05 meters more or less than that. We are confident that the true value is between 0.95 meters and 1.05 meters.

    • The amount of uncertainty in a measurements is sometimes determined by our measuring tool (Xplorer units will often report uncertainty this way), but sometimes must be estimated.

      • If you needed to measure the length of a piece of string, you could hold it up to a meter stick to get the length. You might find, however, that the string is slightly stretchy, and how you hold the string (pulled tight vs. held slack) can change the length of the string. If, for example, the string is 750 mm long when held slack, and 754 mm long when pulled tight, it would make sense to report the length of the string as (752 +/- 2) mm. 752 mm is the middle of the range of measured lengths, and 2 mm is one half of the range (or the difference between the middle and either extreme.

      • If you needed to measure the distance between two objects, you could put a meter stick next to the two. It might be unclear, however, where on each object we should measure to. For example, if we were calculating the gravitational force between the two objects, we would want the distance between the centers of mass of each object. We might need to estimate where the center of mass of each object would be. Depending on the shapes of the objects, this might be easy (for a sphere, for example) or more difficult (for a human being). For the human being, we might estimate our uncertainty to be +/- 10 cm. This just means that we acknowledge this measurement is somewhat of a guess, but we are confident that we measured to a spot within 10 cm of the right place.

    • In more difficult measurements, it often makes sense to make the measurement multiple times and take the average of those measurements. The average will have its own uncertainty, which can be calculated by using the equation

where R is the range of the values (the largest value minus the smallest value) and N is the number of values that you averaged.

• • • • Example: Let's use measured values of 5.0 cm, 4.8 cm, and 4.6 cm. The average of those is 4.8 cm. The range R is 0.4 cm (5.0 cm - 4.6 cm). N, then number of values averaged, is 3. Plugging R and n into the equation above gives an uncertainty of 0.12 cm. We then report our average as 4.8 cm +/- 0.12 cm..

• • Relative uncertainty / percent uncertainty

    • In this system, measured values are followed by the +/- symbol and a percent. For example, 3.0 m +/- 5% , which is read as one point zero meters plus or minus five percent. This means that we believe the value to be 1.0 meters, but the true value might be as much as 5% of 3.0 meters (.15 meters) more or less than that. We are confident that the true value is between 2.85 meters and 3.15 meters.

    • To calculate relative uncertainty from absolute uncertainty, we simply divide the absolute uncertainty by the measured value (and multiply by 100 to express as a percent). For example, (100 +/- 3) kg can be expressed as 100 kg +/- 3%, because 3 kg is 3% off 100 kg.

  • This is a useful system when analyzing the overall effect of the uncertainty of a measurement on our ability to draw conclusions. For example, most people can use a stopwatch to measure an amount of time to an uncertainty of +/- 0.10 seconds. A measurement of 1.0 seconds and 10.0 seconds would both have the same absolute uncertainty of +/- .10 seconds. However, that uncertainty is a much bigger problem for the 1.0 second measurement. Using the relative uncertainty system, we would write these measurements as 1.0 seconds +/- 10% and 10.0 seconds +/- 1% respectively. 0.10 seconds is a much bigger portion of the 1.0 second measurement than it is of the 10.0 second measurement.

Tracking uncertainty through calculations

Many of the quantities we will be interested in are not easy to measure directly, but are instead calculated from measured values. Since we are using uncertain values in our calculations, the answer will have some uncertainty, too. Following are the rules for determining the uncertainty of a calculated value obtained through common mathematical operations.

• • Addition or subtraction

    • When adding or subtracting two uncertain values, the absolute uncertainty of the sum or difference will be equal to the sum of the absolute uncertainties of the values being added or subtracted.

    • Algebraically: (A +/- Z) + (B +/- Y) = (A+B) +/- (Z+Y) - or - (A +/- Z) - (B +/- Y) = (A-B) +/- (Z+Y) where A and B are measurement values and Y and Z are absolute uncertainties for those values.

  • Example: (12 m +/- 1 m) - (5 m +/- 0.5 m) = 7 m +/- 1.5 m

• Multiplication or division

  • When multiplying or dividing two uncertain values, the relative uncertainty of the product or quotient is equal to the sum of the relative uncertainties of the values being multiplied or divided.

    • Algebraically: (C +/- X) x (D +/- U) = (CxD) +/- (X+U) - or - (C +/- X) / (D +/- U) = (C / D) +/- (X+U) where A and B are measurement values and Y and Z are relative uncertainties for those values.

  • Example: (30 m +/- 5%) / (10 sec +/- 3%) = 3 m/sec +/- 8%

• Powers

  • When taking an uncertain value to a power, the relative uncertainty of the answer is equal to the relative uncertainty of the value multiplied by the power to which it is taken..

  • Algebraically: (F +/- T) ^P = F^P +/- TP where F is a measured value, T is the relative uncertainty of F, and P is the power (with no uncertainty) to which F is being raised.

  • Example: (5 m +/- 6%) ² = 25m² +/- 12%

  • Note that this works for square roots and cube roots (which is equivalent to raising a quantity to the one half power and one third power, respectively).

• Multiplication or division with an exact value

  • We often find numerical coefficients in the calculations we complete which are not measured values with uncertainty, but are rather derived values which are exact. We will sometimes multiply or divide a measured value by an exact value. In this case, it is quite easy to write rules for both absolute uncertainty and relative uncertainty:

    • Absolute uncertainty rule

      • When multiplying or dividing an uncertain value by a certain value, the absolute uncertainty of the product is equal to the absolute uncertainty of the uncertain value multiplied or divided by the certain value.

      • Algebraically, (G +/- R) x H = GH +/- RH -or- (G +/- R) / H = G/H +/- R/H where G is a measured value, R is the uncertainty of G, and H is the exact value by which G is being multiplied

      • Example: (20.0 +/- 0.1) seconds / 10 = (2.00 +/- 0.01) seconds

    • Relative uncertainty rule

      • When multiplying or dividing an uncertain value by a certain value, the relative uncertainty of the product is exactly equal to the relative uncertainty of the uncertain value that was being multiplied or divided.

      • Algebraically, (J +/- Q) x K = JK +/- Q -or- (J +/- Q) / K = J/K +/- Q where J is a measured value, Q is the relative uncertainty of J, and K is the exact value by which J is being multiplied or divided.

      • Example: (25 meters +/- 3%) x 4 = (100 meters +/- 3%)

  • Note that we will NOT utilize this rule when calculating averages for multiple trials. For that case, use the equation for the uncertainty of an average discussed above.

We will frequently do calculations which are more complex than just a single operation. For these, we know to follow the order of operation for our calculations. We'll do the same for our uncertainty rules. As you do each operation (in order), do the associated uncertainty operation as well. Click here to see an example with notes.