Significant Figures
Significant figures can be hard to understand. Usually, students are provided with a list of rules to follow for significant figures. Those rules are tough to understand and apply without meaning behind them. Let’s use an example to understand how it is determined if a number in a measurement is significant or not.
Let’s say that we have two scales that measure grams (g), one with a dial and one that is digital. The digital scale is great because it says, “Hey, I can only measure to the 0.1 or tenth of a gram. I’m going to give you a number with only one decimal point because that’s all that I can ensure with confidence.” When you take the measurement, you record 2.5 g. Adding zeros to this 2.5000000 g doesn’t change the measurement, but it says that you were able to measure those zeros with confidence. These zeros are not significant and you can’t just add them on.
The same principle can be applied to the rotary scale; Let’s assume that the dial is between 2 and 3 g; We can say that with some pretty high confidence; It appears as though the dial is about halfway between the two and we could say with pretty high confidence that the measurement is 2.4 g, 2.5, g, or 2.6 g; As with the digital scale, we could slap on some zeros at the end of that number, but we were not able to measure those with any confidence, much like the digital scale; Therefore, they are not significant.
When taking measurements, significant figures are the integers you can assure with relatively high confidence.
Let’s use this ruler, as you might use later in this lab. Let’s say that you are measuring the length of this mummichog (Fundulus heteroclitus) or the fish on the top of the ruler. That side of the ruler has numbers that designate centimeters. We can easily see that the mummichog measures between 7 and 8 centimeters, but this ruler allows us to add more precision to our measurement. This ruler has marks on it that denote millimeters and we can use these to make our measurement more precise; Since there are 10 millimeters in a centimeter, these would be tenths of a centimeter, or the number that should be in the first place to the right of the decimal. The mummichog is 7.8 centimeters.
But wait! This ruler allows us to add one more significant digit to this measurement. You can add another decimal place, estimating to what extent the paper clip covers the area between the two marks. What do you think? It’s an estimate, so your interpretation could be different than that of other people. I think that the mummichog’s measurement falls right on the 7.8 cm mark. I would record this measurement as 7.80 cm.
Measuring something in inches is a bit more complicated. Note that on the inches side of this ruler, you are given some of the increments like ¼ and ½. This means that you might have to calculate the fractions for the marks between these. Go for it. I prefer the metric system.
The same thing can be applied to using measuring glasses and things like graduated cylinders and beakers in the lab. These round containers are used to measure volumes in metric units. When you pour a liquid into a cylinder, the top of the liquid isn’t a straight line across; the surface dips down. The lowest portion of the convex dip of the liquid as it sits in the graduated cylinder is known as the meniscus. This is where you would take the measurement of a cup, a liter, or a milliliter of a liquid. Always record the measurement that corresponds to the lowest portion of the meniscus.
When you use a digital scale or a digital pHA measure of hydrogen ion concentration in a solution. meter that gives you numbers like 2.54 or 6.7, you assume that your scale is using the same principles as you do with the ruler. That last digit is somewhat of an estimate.
Rules for Using Significant Figures
Rule 1: All nonzero digits are significant.
1.234 g has 4 significant figures,
1.2 g has 2 significant figures.
Rule 2: Zeroes between nonzero digits are significant.
4008 kg has 4 significant figures,
2.02 kg has 3 significant figures.
Rule 3: Zeroes to the left of the first nonzero digits are not significant.
0.002 g has only 1 significant figure,
0.048 m has 2 significant figures.
Rule 4: Zeroes to the right of a decimal point in a number are significant.
0.023 mL has 2 significant figures,
0.200 g has 3 significant figures.
Rule 5: When a number ends in zeroes left of a decimal point, the zeroes might not be significant.
190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures.
The potential ambiguity in the last rule can be avoided by the use of standard exponential, or ”scientific,”
notation (follow the link to the page on scientific notation). For example, depending on whether 3, 4, or 5 significant figures is correct, we could write
30,2000 calories as:
3.02 × 104 g (3 significant figures)
3.020 × 104 g (4 significant figures), or
3.0200 × 104 g (5 significant figures).
Rules for Calculations with Significant Figures
Rule 1: In addition and subtraction, the result is rounded off to the last common digit occurring furthest to
the right in all components.
For example, 100 (assume 3 sig figs) + 23.643 (5 sig figs) = 123.643, should be rounded to 124 (3 sig figs).
Rule 2: In multiplication and division, the result should be rounded off so as to have the same number of
significant figures as in the component with the least number of significant figures.
For example, 3.0 (2 sig figs) 12.60 (4 sig figs) = 37.8000 which should be rounded off to 38 (2 sig figs).
Rule 3: In general, your calculations can’t be more precise than your original measurements.
For example, 5 pancake heights were measured in centimeters: 1.4, 1.6, 2.0, 1.8, 1.5. The average height is: 1.7 cm
For example, you use your ruler to measure centimeters and your ruler dictates that you generate your measurements with two decimal points like 2.54 and 2.60, and 2.37. Then, you go to calculate the average of your measurements and the calculator spits back at you 2.503333333333333. You must decide how many significant figures there should be, based on your knowledge of the tool used to take the measurement. Your average must have three significant figures and is 2.50 cm. Yes, the last digit (0) must remain there; you can’t record your measurement as 2.5 cm. Zero is a number, just like any other. Had you calculated the average to be 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, you would not even question leaving that last digit on there. Just because the number is a zero doesn’t mean it is not significant.
Rules for Rounding with Significant Figures
Rule 1: If the digit to be dropped is greater than 5, the last retained digit is increased by one.
For example, 12.6 is rounded to 13.
Rule 2: If the digit to be dropped is less than 5, the last remaining digit is left as it is.
For example, 12.4 is rounded to 12.
Rule 3: If the digit to be dropped is 5, and if any digit following it is not zero, the last remaining digit is
increased by one.
For example, 12.51 is rounded to 13.
List of terms
- pH