Making Measurements

Making Measurements

In this lesson we will explore measurements, precision vs accuracy, as well as significant figures.

Measurements

Information that informs the hypotheses, theories and laws of science.

Measurements provide much of the information that informs the hypotheses, theories, and laws describing the behavior of matter and energy in both the macroscopic and microscopic domains of chemistry.

Every measurement provides three kinds of information: 

  • the size or magnitude of the measurement (a number)
  •  a standard of comparison for the measurement (a unit)
  • and an indication of the uncertainty of the measurement. 

While the number and unit are explicitly represented when a quantity is written, the uncertainty is an aspect of the measurement result that is more implicitly represented and will be discussed later.

Units

Units, such as liters, pounds, and centimeters, are standards of comparison for measurements. Without units, a number can be meaningless, confusing, or possibly life-threatening.

Suppose a doctor prescribes phenobarbital to control a patient’s seizures and states a dosage of “100” without specifying units. Not only will this be confusing to the medical professional giving the dose, but the consequences can be dire: 100 mg given three times per day can be effective as an anticonvulsant, but a single dose of 100 g is more than 10 times the lethal amount.

Some of the Fundamental units in science are:

  • Length: measured in meters and represented by the symbol, m.
  • Mass: commonly measured in grams represented by g.
  • Time: measured in seconds, s, minutes, min, hours, hr, etc. depending on the length of time.
  • Temperature: measured in Celsius, ⁰C.  The common unit in the United States is Fahrenheit, ⁰F. 
  • Volume: measured in liters, L.

Everyday measurement units are often defined as fractions or multiples of other units. Milk is commonly packaged in containers of 1 gallon (4 quarts), 1 quart (0.25 gallon), and one pint (0.5 quart). This same approach is used with the scientific units, but these fractions or multiples are always powers of 10. Fractional or multiple units are named using a prefix and the name of the base unit. 

Some of the common prefixes used and the powers to which 10 are raised are:

Milli, m, is 10‾³.  A millimeter, mm. is 10‾³ (or 11000) of a meter.

Centi, c, is 10‾². A centimeter, cm, is 10‾² (or 1100) of a meter.

Kilo, k, is 10³. A kilometer, km, is 10³ (0r 1000) meters.

We will see some others later.

Derived Units

Several common units are derived from the other units.

Volume

Volume is the measure of the amount of space occupied by an object. The volume of a regular object can be calculated by multiplying the length times the width times the height. The derived unit for volume is then the cubic meter (m³).  

A more commonly used unit of volume is the cubic centimeter (cm³);  the volume of a cube with an edge length of exactly one centimeter. The abbreviation cc (for cubic centimeter) is often used by health professionals. A cubic centimeter is equivalent to a milliliter (mL) and is 1/1000 of a liter.

Figure 1.25 (a) The relative volumes are shown for cubes of 1 m³, 1 dm³ (1 L), and 1 cm3 (1 mL) (not to scale). (b) The diameter of a dime is compared relative to the edge length of a 1-cm³ (1-mL) cube.

Density

We use the mass and volume of a substance to determine its density. Thus, the units of density are defined by the units of mass and length.

The density of a substance is the ratio of the mass of a sample of the substance to its volume. The density can be expressed in kilogram per cubic meter (kg/m3). For many situations, however, this is an inconvenient unit, and we often use grams per cubic centimeter (g/cm3) for the densities of solids and liquids, and grams per liter (g/L) for gases. 

Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway. The result of such a counting measurement is an example of an exact number. By counting the eggs in a carton, one can determine exactly how many eggs the carton contains. Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.

Significant Figures

Measurements that are not exact.

The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.

The bottom of the meniscus, in this case, clearly lies between the 21 and 22 markings, meaning the liquid volume is certainly greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid’s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.

This concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs about 6.72 grams, with a nominal uncertainty in the measurement of ± 0.01 gram. If the coin is weighed on a more sensitive balance, the mass might be 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some uncertainty, which depends on the device used (and the user’s ability). All of the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows “120,” then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.

A measurement result is properly reported when its significant digits accurately represent the certainty of the measurement process. But what if you were analyzing a reported value and trying to determine what is significant and what is not? Well, for starters, all nonzero digits are significant, and it is only zeros that require some thought. We will use the terms “leading,” “trailing,” and “captive” for the zeros and will consider how to deal with them.

Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement unless the last digit is a trailing zero lying to the left of the decimal point.

Captive zeros result from measurement and are therefore always significant. Leading zeros, however, are never significant—they merely tell us where the decimal point is located.

The leading zeros in this example are not significant. We could use exponential notation and express the number as 8.32407 × 10‾³; then the number 8.32407 contains all of the significant figures, and 10‾³ locates the decimal point.

The number of significant figures is uncertain in a number that ends with a zero to the left of the decimal point location. The zeros in the measurement 1,300 grams could be significant or they could simply indicate where the decimal point is located. In cases where only the decimal-formatted number is available, it is assumed that all trailing zeros are not significant.

Significant Figures in Calculations

A second important principle of uncertainty is that results calculated from a measurement are at least as uncertain as the measurement itself. Take the uncertainty in measurements into account to avoid misrepresenting the uncertainty in calculated results. One way to do this is to report the result of a calculation with the correct number of significant figures, which is determined by the following three rules for rounding numbers:

    1. When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places (the least certain value in terms of addition and subtraction).
    2. When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures (the least certain value in terms of multiplication and division).
    3. If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, “round down” and leave the retained digit unchanged; if it is more than 5, “round up” and increase the retained digit by 1. If the dropped digit is 5, and it’s either the last digit in the number or it’s followed only by zeros, round up or down, whichever yields an even value for the retained digit. If any nonzero digits follow the dropped 5, round up. (The last part of this rule may strike you as a bit odd, but it’s based on reliable statistics and is aimed at avoiding any bias when dropping the digit “5,” since it is equally close to both possible values of the retained digit.)

The following examples illustrate the application of this rule in rounding a few different numbers to three significant figures:

    • 0.028675 rounds “up” to 0.0287 (the dropped digit, 7, is greater than 5)
    • 18.3384 rounds “down” to 18.3 (the dropped digit, 3, is less than 5)
    • 6.8752 rounds “up” to 6.88 (the dropped digit is 5, and a nonzero digit follows it)
    • 92.85 rounds “down” to 92.8 (the dropped digit is 5, and the retained digit is even)

Precession & Accuracy

When conducting experiments there is a difference between precession and accuracy.

Accuracy is a measurement that yields a result that is very close to the true or accepted value.

Precision is a measurement that yields very similar results when repeated in the same manner.

Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or accepted value. Precise values agree with each other; accurate values agree with a true value.

Figures A through C each show targets with holes where the arrows hit. The archer in figure A was both accurate and precise as all 3 arrows are clustered in the center of the target. In figure B, the archer is precise but not accurate, as all 3 arrows are clustered together but to the upper right of the center of the target. In Figure C, the archer is neither accurate nor precise as the 3 holes are not close together and are located both to the upper right and right of the target.

(a) These arrows are close to both the bull’s eye and one another, so they are both accurate and precise. (b) These arrows are close to one another but not on target, so they are precise but not accurate. (c) These arrows are neither on target nor close to one another, so they are neither accurate nor precise.