# Significant Figures

See this section of your textbook.

Rules about significant figures may seem arbitrary from a theoretical standpoint, but in the laboratory you will see that they allow you to determine the precision of your measurements and calculations. When your measurement has a limited number of digits, your subsequent calculations will also have a limited number of digits.

#### Significant digits from common measurements

1. Mass – analytical balances generally give many significant digits, particularly when weighing 0.1 g or more, you get 4, 5, or 6 significant digits. For example, 0.5012 g of a substance has 4 significant digits. Higher masses give you more significant digits until you reach the capacity of the balance.
2. Volume
• Volumetric flasks are extremely precise tools for measuring volume and often give you 4 or more significant digits, depending on the size of the volumetric flask. For example, a precision 1 liter volumetric flask filled exactly to the line etched on the neck contains 1.0000 L, which is 5 significant digits. The precision is printed on the flask for reference.
• Micropipette are very precise tools for measuring extremely small volumes (less than one milliliter). The number of significant digits you get from a micropipette is printed on the pipette for your reference, but it is usually about 4 significant digits.
• Burets are very precise tools for measuring volume. Our lab is equipped with burets that measure to the nearest 0.05 mL, so a volume greater than 1 mL will have 3 significant digits, and a volume greater than 10 mL will have 4 significant digits. You always estimate one more digit than you can read from the lines.
• Graduated cylinders are the most flexible tool for measuring volume. Our lab is equipped with many different graduated cylinders, and the number of significant digits they give you depends on the exact graduated cylinder you are using and the volume you are measuring. The number of decimal places you can read is printed on the glass for your reference.  You always estimate one more digit than you can read from the lines, typically giving you three significant figures. Always use the smallest volume graduated cylinder that you can use because smaller graduated cylinders typically are more precise. Estimate volumes to the nearest 1/10th between graduations.
• 10 mL graduated cylinders let you measure volumes up to 10.00 mL to the nearest 0.01 mL.
• 25 mL graduated cylinders let you measure volumes up to 25.0 mL to the nearest 0.05 or 0.02 mL, depending on your exact cylinder.
• 50 mL graduated cylinders let you measure volumes up to 50.0 mL to the nearest 0.1 or 0.2 mL, depending on your exact cylinder.
• Repeat after me: “Erlenmeyer flasks and beakers are not designed to measure volumes,” although they can be used to get a rough estimate if the volume is not critical. For example, if you are making a hot water bath with approximately 400 mL of water (1 significant digit), then it is appropriate to use a beaker measure that volume.
3. Temperature – our red and blue alcohol thermometers are read to the nearest 0.2 °C. This means that we can only read 3 significant digits for temperatures between 10 and 99.8 °C, and we only get 2 significant digits for temperatures between 0 and 9.8 °C.You always estimate one more digit than you can read from the lines. Mercury thermometers give more significant digits, but they have fallen out of favor due to safety concerns when they (inevitably) break.

#### Significant digits from common calculations

1. Adding/subtracting: Use the least number of decimal places involved in the calculation. For example, if you measure a temperature change from 25.0 °C to 28.1  °C, 28.1 – 25.0 = 3.1 °C. See how fast you can lose significant digits in the lab?
2. Multiplying/dividing: Use the least number of significant digits involved in the calculation. For example, if you dissolve 0.3829 moles of a substance in 1.0000 L of water, then the concentration is 0.3829 moles per liter. The volume had 5 significant digits, but the number of moles only had 4 significant digits, so you are left with just 4 significant digits in your answer.
3. Averaging: We have special rules for averaging multiple measurements. Ideally, if you measure the same thing 3 times, you should get exactly the same result three times, but you usually don’t. The spread of your answers affects the number of significant digits in your average; a bigger spread leads to a less precise average. The last significant digit of the average is the first decimal place in the standard deviation. For example, if your average is 3.025622 and your standard deviation is 0.01845, then this is the correct number of significant figures for the average: 3.03, because the first digit of the standard deviation is in the hundredths place, so the last significant digit of the average is in the hundredths place.

#### Rules for rounding

Numbers between 6 and 9 round up.

Numbers between 1 and 4 round down.

5s round up or down to an even number. For example, if your average in lab is 92.5, the 5 would round down to 92, so an A- letter grade. If your average is 89.5, the 5 would round up to 90, so an A- letter grade.