Chemistry Case

Accuracy

Accuracy refers to how close a measurement agrees with a known value of that measurement. If measurements were compared to shots at a target, the measurements would be the holes and the bullseye, the known value. This illustration shows holes fairly close to the center of the target, but scattered widely. This set of measurements would be considered accurate.

Precision

Precision refers to how well the measurements compare to each other. In this illustration, the holes are clustered closely together. This set of measurements is considered to have high precision. Note that none of the holes are near the center of the target. Precision alone is not enough to make good measurements. It is also important to be accurate. Accuracy and precision work best when they work together.

Significant Figures

When making a measurement, a scientist can only reach a certain level of precision, limited either by the tools being used or the physical nature of the situation. The most obvious example is measuring distance.

Consider what happens when measuring the distance an object moved using a tape measure (in metric units). The tape measure is likely broken down into smallest units of millimeters. Therefore, there's no way that you can measure with a precision greater than a millimeter. If the object moves 57.215493 millimeters, therefore, we can only tell for sure that it moved 57 millimeters (or 5.7 centimeters or 0.057 meters, depending on the preference in that situation).

In general, this level of rounding is fine. Getting the precise movement of a normal sized object down to a millimeter would be a pretty impressive achievement, actually. Imagine trying to measure the motion of a car to the millimeter, and you'll see that in general this isn't necessary. In the cases where such precision is necessary, you'll be using tools that are much more sophisticated than a tape measure.

The number of meaningful numbers in a measurement is called the number of significant figures of the number. In our earlier example the 57 millimeter answer would provide us with 2 significant figures in our measurement.

Zeroes & Significant Figures

Consider the number 5,200.

Unless told otherwise, it is generally the common practice to assume that only the two non-zero digits are significant. In other words, it is assumed that this number was rounded to the nearest hundred.

However, if the number is written as 5,200.0, then it would have five significant figures. The decimal point and following zero is only added if the measurement is precise to that level.

Similarly, the number 2.30 would have three significant figures, because the zero at the end is an indication that the scientist doing the measurement did so at that level of precision.

Some textbooks have also introduced the convention that a decimal point at the end of a whole number indicates significant figures as well. So 800. would have three significant figures while 800 has only one significant figure. Again, this is somewhat variable depending on the textbook.

Following are some examples of different numbers of significant figures, to help solidify the concept:

One significant figure

4

900

0.00002

Two significant figures

3.7

0.0059

68,000

5.0

Three significant figures

9.64

0.00360

99,900

8.00

Mathematics with Significant Figures

* Addition and Subtraction

When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures). Sometimes this is considered to be the number of digits after the decimal point.

Example

32.01 m

5.325 m

12 m

Added together, you will get 49.335 m, but the sum should be reported as '49' meters.

* Multiplication and Division

When experimental quantities are multiplied or divided, the number of significant figures in the result is the same as that in the quantity with the smallest number of significant figures. If, for example, a density calculation is made in which 25.624 grams is divided by 25 mL, the density should be reported as 1.0 g/mL, not as 1.0000 g/mL or 1.000 g/mL.

Significant Figures Rules

1. All digits between two non-zero digits are significant.

321 = 3 significant figures

6.604 = 4 significant figures

10305.07 = 7 significant figures

2. Zeros at the end of a number and to the right of the decimal point are significant.

1.00 = 3 significant figures

88.000 = 5 significant figures

3. Zeros to the left of the first nonzero digit are NOT significant

0.001 = 1 significant figure

0.00020300 = 5 significant figures

4. Zeros at the end of a number greater than 1 are NOT significant unless the decimal point is present.

2,400 = 2 significant figures