The ability to repeat an experiment and get the same results, or reproducibility, is essential in scientific research. However, it’s impossible to repeat an experiment if you don’t know how you did it. Thus, scientists keep detailed records of their experiments in lab notebooks. These records include important information like the amount of each material used and descriptions of each step of the procedure.
Equipment, like balances and volumetric or graduated glassware, is used to measure solid and liquid compounds for experiments. The precision of each measurement is limited by the equipment used. Balances and other equipment with digital readouts automatically display values at the maximum precision that they can support. The precision of graduated glassware and rulers depends on the increment, which is the amount represented by the smallest distance between two marks.
The surface of most liquids curves upwards at the edges in glass containers. This type of curved liquid surface is called a concave meniscus. When the edges of the liquid curve down, it is called a convex meniscus. When measuring liquid volumes, the reference point for the top of the liquid is the bottom of a concave meniscus or the top of a convex meniscus. To accurately identify the position of the meniscus, liquid volumes are read by looking at the liquid’s surface from the side at eye-level. Looking at the liquid from above or below will cause the liquid surface level to appear higher or lower than it actually is.
When the reference point of the meniscus touches a mark on graduated or volumetric glassware, the glassware is holding the volume defined for that mark. Volumetric glassware is designed to measure a specific volume, so it only has one mark. Graduated glassware is designed to measure a range of volumes, so it has many marks, or graduations, which are labeled at regular intervals. Both graduated and volumetric glassware may have additional manufacturer-provided information about their precision.
Manufacturers calibrate equipment to ensure that the measurements are accurate to within a specified range of uncertainty. For example, a high-precision analytical balance may read out to four decimal places (0.0000 g) with an uncertainty of ± 0.0001 g. A reading of 0.0345 g indicates that the true value is between 0.0344 g and 0.0346 g.
Glassware typically has its measurement uncertainty printed on it. To accurately record volume measurements, you must consider both the increment and the uncertainty. For example, a precise volume measurement in a 100 mL graduated cylinder with 1 mL increments and an uncertainty of ± 0.5 mL would be recorded to the tenths place (000.0 mL ± 0.5 mL). If the reference point of the meniscus falls between two marks, you should estimate a value for the tenths place; otherwise, leave it as 0. Since temperature affects volume, the temperature for which the glassware is calibrated will be printed on it.
Glassware is calibrated either to contain (TC) or to deliver (TD). TC glassware holds the specified volume of liquid when it is filled to the mark, but a small amount of liquid will be left behind when it is poured into another container. Solutions are often prepared in TC glassware because the accuracy of the solution volume affects the accuracy of the concentration. Volumetric flasks are typically calibrated to contain.
TD glassware holds a little more than the amount of liquid specified when it is filled to the mark, but it dispenses only the specified volume. Thus, TD glassware should not be emptied completely when dispensing liquids. This type of glassware is useful for transferring a precise volume of liquid to another container. Both graduated and volumetric pipettes are typically calibrated to deliver.
Unless otherwise specified, the uncertainty of a number is assumed to be on the scale of the last digit. Thus, it’s particularly important to report the uncertainty of a measurement if the uncertainty is on a larger scale. For rulers and graduated glassware without a known uncertainty, the uncertainty is estimated as half of the smallest increment. The uncertain digit is based on where the meniscus falls between two tick marks.
Significant figures are the numbers in a value that are meaningful, or essential for expressing that value with the appropriate precision. All numbers other than leading zeroes (0.001), trailing zeroes (1,000), or scientific notation multipliers (10x) are always significant.
Leading zeroes are never significant because they can be removed by rewriting the number either in scientific notation or, if it is an SI unit, in smaller units. For example, there are only three significant figures in the value 0.00123 m because it can be rewritten as either 1.23 × 10-3 m or 1.23 mm without losing any information. Note that the scaling factor between different multiples of the same SI unit does not affect the number of significant figures.
Trailing zeroes are significant when they are before or after a decimal point. For example, the measurement 100.110 mL has six significant figures and 100.0 mL has four. Trailing zeroes in a number written without a decimal point are assumed to be insignificant unless they are marked otherwise. There are several different conventions for marking significant zeroes in those cases, but to avoid ambiguity it is advisable to rewrite those values in scientific notation. If the measurement is in SI units, use larger units.
The precision of a measurement limits the precision of any values calculated from it. A calculated value has the same number of significant figures as the least precise measurement or value used in its calculation. If the result has too many significant figures, it is rounded to the appropriate precision. If the result has too few significant figures, it is extended with significant trailing zeroes to preserve the precision. The significant figures of experimentally derived values, like molar mass, density, or some unit conversion factors, must also be considered when they are used in calculations.
The number of significant figures in a calculated value is not limited by physical or mathematical constants in formulas or exact numbers, like the number of data points acquired in an experiment. For example, the formula for calculating the volume of a sphere is V = 4/3πr3. The number of significant figures in the calculated volume is only affected by the number of significant figures in the radius measurement r. 4/3 and π are constants, and 3 is simply notation for a mathematical operation.
Defined conversion factors are also treated as constants. For instance, the inch is defined as exactly 25.4 millimeters; therefore, the number of significant figures of the converted value depends only on the original measurement. Always check whether the conversion factor is exactly defined or an experimental value when converting between units.
For simple addition and subtraction of measurements, we use significance arithmetic to determine the number of significant figures. In this case, the answer has as many decimal places as the measurement with the fewest decimal places regardless of the number of significant figures. Thus, 15,643.7 mL + 0.613 mL = 15,644.3 mL.
For multi-step calculations or complex equations that you break into smaller parts to solve, you should keep at least one or two insignificant figures on the intermediate values. For example, if you calculate the values of the numerator and denominator of a fraction separately before dividing them, you should retain the insignificant figures of the numerator and denominator when you divide them. This minimizes rounding error, which is the difference from the number you would get if you entered the entire formula into your calculator at once. If you write down these intermediate values, you should note which digits are insignificant.
Harris, D.C. (2015). Quantitative Chemical Analysis. New York, NY: W.H. Freeman and Company.
