3.2. Unit Conversions, Error, and Uncertainty http://www.ck12.org
lines, so each one must represent 1 mL. As we will see next, the space between these marks represents an area of
uncertainty with regard to the estimate.
When the volume in our previous example was reported to be 52.9 mL, theuncertaintyassociated with this estimate
also needed to be reported. For example, we know for certain that the true value for the volume must be between 52
mL and 53 mL. However, there is uncertainty regarding how close the value is to 52 or 53. We estimated the volume
to be 52.9, but some students may have reported 52.8 or 53.0. These would be accurate estimates because they
fall within the acceptable uncertainty of the device. All measurements have an uncertainty equal to one half of the
smallest difference between reference marks. For our graduated cylinder, there is 1 mL between consecutive marks,
so the uncertainty is one half of that value, 0.5 mL. To be rigorous about our certainty regarding this measured value,
the estimate of 52.9 mL should be reported as 52.9±0.5 mL.
Accuracy
In measuring quantities we always aim for highaccuracy. Estimates that fall within the range of uncertainty for a
given instrument are said to be accurate. In our previous example, all of the values between and including 52.4 mL
and 53.4 mL would be considered accurate. Estimates that fall outside this range are inaccurate. Accuracy describes
how close an estimate is to a known standard.
Precision
Precisiondescribes how close estimates are to one another. Estimates that are relatively close to one another are
precise. Let’s assume that ten different students made an estimate of the volume shown inFigure3.3, and the values
were: 52.9, 52.8, 52.9, 52.9, 53.3, 52.0, 52.8, 52.9, 53.0, 52.8. We can determine how precise these data are by
analyzing how close they are to an average. The average could be the mean, median or mode. The most common
understanding of the average is the mean. This value is calculated by adding up all the numbers and then dividing
by the total number of values. Other terms that can refer to the average are the median and the mode. The median is
the middle value in a numerically ordered list of numbers. The mode is the value that occurs most often in a set of
numbers. If no number is repeated, there is no mode for the list. Here are the calculated averages:
Mean=
52. 9 + 52. 8 + 52. 9 + 52. 9 + 53. 3 + 52. 0 + 52. 8 + 52. 9 + 53. 0 + 52. 8
10
= 52. 8
Median= 52. 0 , 52. 8 , 52. 8 , 52. 8 ,[ 52. 9 ], 52. 9 , 52. 9 , 52. 9 , 53. 0 , 53. 3 = 52. 9
Mode= 52. 0 , 52. 8 , 52. 8 , 52. 8 ,[ 52. 9 , 52. 9 , 52. 9 , 52. 9 ], 53. 0 , 53. 3 = 52. 9
Based on this analysis, we see the value 52.8 was the mean, and 52.9 was the median and mode. Therefore, values
that are relatively close to these averages would be considered precise. We can also calculate the standard deviation
for these data, which is a more refined way of determining the precision of estimates. However, we will not concern
ourselves with standard deviation at this point.
Accuracy vs. Precision
As we just saw, accuracy describes how close a given set of data is to the “real” value, while precision describes how
close the data points are to one another. These concepts are illustrated inFigure3.4.
Target A represents the best possible "data". All of the data point points are clustered in the center, close to the
"actual" value and close to one another. This data is both accurate and precise. In target B, the set of data has
good accuracy overall if the points are averaged together. However, the three points are not very close to each other
making the imprecise. Target C, on the other hand, shows precise but inaccurate data. The three data points are close
together, making them precise, but are far from the center of the target, giving low accuracy. Target D represents the
worst possible "data". The data points are far from the center of the target lacking any accuracy, as well as being far
apart from each other, lacking precision.