All measurements are subject to error. Error can be classified as human error or technical error. Perhaps you are transferring a small volume from one tube to another, and you don't quite get the full amount into the second tube because you spilled it: this is human error. Or perhaps you are using a scale that is improperly calibrated and reads 0.5 g with nothing on it. All measurements from that scale would therefore be overestimated by 0.5 g. This is a technical error.
It is important to both minimize error and to report the uncertainty associated with all measurements. The reason we care so much about significant figures is because they allow us to honestly reflect the uncertainty of measured values. We can convey uncertainty using something called absolute error. Absolute error is the difference between the measured value and the true (or accepted) value. Scientists use the absolute error in a measurement to quantify its uncertainty. This is often called the absolute uncertainty and is reported as ± with the measurement.
The absolute uncertainty is the number which, when combined with a reported value, gives the range of values in which the “true value” of the measurement sits. For example, a length may be reported as 5.3 mm ± 0.2 mm. In this case, the reported value is 5.3 mm, and the absolute error is 0.2 mm. The range of true values is 5.1 mm to 5.5 mm.
Let's Watch
Watch this video to learn more about absolute error.
You may want to use the study guide to follow along. If so, click below to download the study guide.
The reason we care so much about significant figures, is because they allow us to be more honest and accurate when reporting measurements that we take. But there are additional tools we have in order to convey our level of uncertainty.
The method that we’ll see most often in this course is called absolute error.
Absolute error is when we give a window for how much the measurement might be off. If you are using a measuring instrument, like a scale or a thermometer, the absolute error of that instrument will usually be written on the item itself.
To see how we use this, imagine we are measuring temperature. We measure the initial temperature of a substance to be negative 2.0 degrees Celsius, but given the constraints of our thermometer, that measurement might be off by plus or minus 0.2 degrees Celsius. This plus or minus says that, while we measured negative 2.0 degrees Celsius, the actual temperature might be 0.2 degrees higher or lower than that number. So the actual value lies between negative 2.2 and negative 1.8 degrees Celsius.
Now let’s say that this substance that we’re measuring the temperature of undergoes what’s called an exothermic chemical reaction, where chemical potential energy is converted into thermal energy, and temperature increases.
When we measure the temperature of the substance after this reaction, we get a reading for T-final of 12.0 degrees Celsius, with the same absolute error of plus or minus 0.2 degrees, because, of course, we’re using the same thermometer. This means that the actual final temperature lies somewhere between 11.8 and 12.2 degrees Celsius.
So what was the change in temperature, which we are going to write as delta-T, for this reaction? Well, based on our measurements, the temperature increased by 14.0 degrees Celsius, but we have to include an absolute error here as well. When we are finding the change in a value, we have to add the initial and final absolute errors from the two measurements together. Well, for both of those measurements, our absolute error was 0.2 degrees Celsius.
That means that the delta T that accurately conveys our level of uncertainty would be 14.0 degrees Celsius plus or minus 0.4 degrees Celsius. We interpret this the same way, that the actual change in temperature lies between 13.6 degrees and 14.4 degrees. An actual change of 13.6 degrees corresponds with the temperature increasing from negative 1.8 to 11.8 degrees, while a change of 14.4 degrees corresponds with the temperature increasing from -2.2 degrees to 12.2 degrees.
Let's Practice
How well can you use absolute error to reflect the uncertainty of measured values? Find out by completing this activity. Determine the range in which the true value of each reported measurement sits. Then click the measurement to check your answer.
14.65 grams ± 0.05 grams
14.65 grams ± 0.05 grams is 14.60 to 14.70 grams
The range of values is determined by subtracting 0.05 from 14.65 (14.60) and adding 0.05 to 14.65 (14.70). The ± symbol indicates error below and above the measured value.
35 seconds ± 3 seconds
35 seconds ± 3 seconds is 32 to 38 seconds
The range of values is determined by subtracting 3 from 35 (32) and adding 3 to 35 (38). The ± symbol indicates error below and above the measured value.
