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Physics LibreTexts

1.10 Precision and Significant Figures

Scientists record information according to the accuracy of the measurement. The number of digits used to write a measurement indicates the precision of the measurement — it is also called the number of significant figures. Suppose we quote the population of the Earth as 6,300,000,000 people. This estimate has only two non-zero digits or significant figures and implies a measurement of low precision. Alternatively, we might quote the population of the Earth as 6,356,908,417 people. This estimate has ten significant figures and implies that we counted every person!

Clearly, the way scientists write a measurement carries an implication of the degree of certainty about the measurement. The non-zero digit that is farthest to the left is the least significant digit. The non-zero digit that is farthest to the right is the most significant digit — it indicates the precision of the measurement. In our population example, the most significant digit of the number 6,300,000,000 is 3, so we infer that the uncertainty in the estimate is one unit in the most significant digit, or 100,000,000. With two significant figures, we see that the accuracy of the estimate is 1 part in 63 or about 1 to 2%. The most significant digit in the number 6,356,908,417 is 7, so the implied accuracy of the number is 1 part in 6 billion or about 0.00000001%. Since someone is born and dies on the Earth every second or so, this highly accurate estimate is very implausible! A better estimate of the Earth's population might be 6,356,910,000 - a precision of six significant figures.
 


The amount this bill is worth is $100.We know each unit precisely, so it has three significant figures. Click here for original source URL.

Here are a few more examples of measurements and their precision. Remember to count a zero just like any other digit. For example, 9004 has four significant digits. There are two types of exception where 0 is not counted as a significant digit. The first is in a decimal fraction, where the leading zeros are just placeholders behind the decimal point. The second is the fact that some numbers just happen to be round numbers. For example, a hundred-dollar-bill represents exactly $100 and has three significant figures.

• 65,400 K has 3 significant figures

• 0.00002 g/cm3 has 1 significant figure

• 980,014 km has 6 significant figures

• 7.5 × 105 pc has 2 significant figures

• 1.007 × 10-8 kg has 4 significant figures

• 3.3206 × 106 W has 5 significant figures

The precision of a number relates to the implied accuracy of the measurement. We will use the diameter of the Earth — measured to be 12,756 km — as an example. Notice that it is always possible to quote a measurement with lower precision by rounding the number. Here is how to round off a number: if the most significant digit is 4 or less, you set it to 0, and if the most significant digit is 5 or higher, you set it to 0 and increase the next less significant digit by 1.
 


Explication of the equation E=mc2. Click here for original source URL.

Consider the speed of light, which is one of the most accurately determined constants in nature. It is a good example of a physical constant that just happens to be nearly a round number in the metric system. It's measurement is 299792.458 km/s, with a precision of �9 significant figures and an error of �0.001 km/s, so accurate to 1 part in 108 (a 10-6% error). 

The precision of a number — how many significant figures we quote — has an implication about the accuracy of the measurement. Accuracy reflects how far the measurement deviates from the true value. Therefore, precision has little meaning if we know nothing about the amount of observational error. To return to our first example, it is meaningless to quote the population of the Earth with a precision of 9 or 10 significant figures, since no one could possibly count people that accurately. Scientists talk about observational error, but a better term is uncertainty. Measurements are not usually wrong or in error, but they are uncertain due to unavoidable limitations in the measuring apparatus.