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Significant Figures

Significant Figures in Measurements

Significant figures report the precision of an experiment. Every measuring device has a scale that is used to obtain numerical values for a given property, such as, mass, length or volume. To measure length, for example, a ruler is often used. The figure below shows an object and a ruler. The length of the object falls between 6 and 7 cm. We can estimate that the length is 6.5 cm. This number contains two significant figures. The first number (‘6’) is certain, but the second one (‘5’) is not. The length may actually be 6.4 or 6.6 cm. We cannot tell for sure because the scale does not permit a more precise determination of the length. (With this particular measuring device, the measurement 6.5 cm and 6.4 cm are not statistically different and should be considered the same measurement.) It would be incorrect to report the measurement as 6.50 cm (3 significant figures) because this would imply that the length is known to two decimal places and that it is the 0 that is not certain (in other words, the value could be 6.51 but not 6.60). This is very poor technique for the precision of the measurement cannot readily be duplicated by another investigator.

Other examples of correctly reading a scale can be seen in the drawings below. The graduated cylinder, which is calibrated in tenths of a milliliter, contains a volume of liquid that is between 25 and 26 mL. From the scale, we can easily determine that there is more than 25.4 mL but less than 25.5 mL. We must estimate what that last number is. We could correctly read the volume as 25.46 mL or 25.47 mL. This scale permits us to estimate the volume to two decimal places, giving us four significant figures (25.4 are three certain numbers and the 6 or 7 is the uncertain figure). Likewise, the digital scale below gives the mass of a sample as 50.12 g. All four numbers must be recorded because they are significant (50.1 are certain digits and 2 is an uncertain digit).

Significant Figures in Calculations

When performing calculations involving measurements, the number of digits or decimals in the answer is limited by the original measurements. The rules for determining the number of significant digits in an answer vary according to the type of mathematical operation being performed.

When multiplying or dividing, the answer can only have the same number of significant figures as the factor with the least number of significant figures. For example, if you multiply 5.20 x 3.060 on a calculator, the answer that appears on the visor is 15.9120000. If you write this answer, you imply a far greater accuracy than is actually possible with the original numbers. The answer should be reported as 15.9.

When adding or subtracting, the answer cannot have any more decimal places than the figure with the least number of decimals. For example, addition of 15.9, 6.02, 5.33 and 10.263 gives a total of 37.513. The answer, however, must be reported as 37.5 because 15.9 has only one decimal place.

To determine how many significant figures there are in a number, the rules below are helpful:

  • All numbers between 1 and 9 are significant.
  • Zeros are significant if they
    • o fall between other numbers (sandwich zeros).
    • o Trailing zeros if a decimal point is present.
  • Zeros are not significant if they simply determine the power of ten to use in scientific notation, including terminal zeros and zeros to the left of the digits in decimal numbers. (Trailing zeros without a decimal point.)
  • Exact or counting numbers have an infinite number of significant figures.

The number of reported significant figures is never “uncertain”. When in doubt, always report the number of significant figures to the first digit that “may” be uncertain. This may overestimate the error associated with the measurement but, will always result in a reproducible measurement. For example, if the measurement to be reported is 250 m, the precision in the measurement can be either 250 ±1 m (3 significant figures) or 250 ±10 m (2 significant figures) depending on whether the “zero” or the “five” is the first estimated digit, respectively. If the measurement is reported to 3 significant digits (±1 m) than further measurements between 249 m and 251 m are considered statistically identical. If reported to 2 significant digits, then the measurement is (±10 m) and further measurements between 240 m and 260 m are considered identical. The fewer significant digits demonstrate lower precision and higher error but, result in a measurement whose precision can be easily reproducible by another investigator.

HINT: When in doubt, convert the measurement to scientific notation. In high school, most students were taught to use the correct number of significant digits when converting to scientific notation, although, they where probably not made aware of the significance!

Rounding off numbers to obtain the correct number significant figures is quite simple. If the first digit to be dropped is 6 or higher, increase the digit to the left by one unit. If 4 or lower, simply drop the excess digits. What should be done if a 5 is to be dropped? Standard practice is to round up. This is to be discouraged in science because it introduces human bias into the measurements (systematic error). The number 5 is equidistant between 0 and 10 therefore, 50 % of the time it should be rounded up and 50 % of the time it should be rounded down. To accomplish this, the number should always be rounded to the nearest EVEN number. Therefore, some measurements will be rounded up while others will be rounded down thus removing the systematic error from the measurements. (Remember: Significant figures are not the same as decimal places.)

Exact and counting numbers are numbers that we obtain by counting items or by defining the relationship of two units in the same measuring system. For example, if you want to know the number of students in your laboratory section, you would count them. The number obtained is certain (assuming you count correctly) and is considered to contain an infinite number of significant figures. The number of inches in a foot is defined to be exactly 12. The number of grams in a kilogram is defined to be exactly 1000. Both of these are exact numbers and also are considered to contain an infinite number of significant figures. Some words also represent counting numbers. For example, a pair of flowers is exactly two flowers. Other words of this type are: dozen (12), gross (144), ream (500) and mole (6.022x1023). Notice, the words pair, dozen, gross, ream and mole are NOT units, they are numbers.

In contrast, measured numbers are obtained by using an instrument to obtain the number. For example, if you want to know the width of this paper, you could measure it with a ruler. If the ruler is calibrated in centimeters and millimeters, you could report the number as 22.57 cm (225.7 mm). This number has a finite number of significant figures and is subject to the rules concerning significant figures. This kind of number always has some uncertainty, which is dependent on the instrument used to make the measurement. It is never exact like those discussed in the preceding paragraph.

 

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4 significant figures.

2 . 7 1 3

Note: Significant figures are highlighted in green.