Chapter 2 Data Handling

Accuracy and Precision Accuracy can be defined as the degree of agreement between a measured value and the true or accepted value. As the two values become closer, the measured value is said to be more accurate.   Precision is defined as the degree of agreement between replicate measurements of the same quantity.

Assuming the correct or accepted value is represented by the center of the circles below, if all values occurred within, for example, the red circles, results are precise but not accurate. If all values occurred within the yellow circle, results are both accurate and precise. If results were scattered randomly, results are neither precise nor accurate.

Example   The weight of a person was measured five times using a scale. The reported weights were 84, 83, 84, 85, and 84 kg. If the weight of a person is 76 kg weighed on a standard scale , then we know that the results obtained using the first scale is definitely not accurate. However, the values of the weights for the five replicate measurements are very close and reproducible. Therefore measurements are precise. Therefore, a measurement could be precise but not accurate.

Significant Figures At the most basic level, Analytical Chemistry relies upon experimentation; experimentation in turn requires numerical measurements. And measurements are always taken from instruments made by other workers. Significant figures are concerned with measurements not exact countings.

Some information about measurements 1) Examples we will study include the metric ruler, the graduated cylinder, and the scale. 2) Because of the involvement of human beings, NO measurement is exact; some error is always involved. This means that every answer in science has some uncertainty associated with it. We might be fairly confident we have the correct answer, but we can never be 100% certain we have the EXACT correct answer.

3) Measurements always have two parts - a numerical part (sometimes called a factor) and a dimension (a unit). The reason for this is that we are measuring quantities - length, elapsed time, temperature, mass, etc. Not only do we have to tell how much there is, but we have to tell how much of what. Measuring gives significance (or meaning) to each digit in the number produced. This concept of significance, of what is and what is not significant is VERY IMPORTANT.

The concept of significant figures (or significant digits) is important. A measurement can be defined as the comparison of the dimensions of an object to some standard. The dimensions of an object refer to some property the object possesses. Examples include mass, length, area, density, and electrical charge. Dimensions are often called units.

Identifying significant digits The following rules are helpful in identifying significant digits 1. Digits other than zero are significant. e.g., 42.1m has 3 sig figs. 2. Zeroes are sometimes significant, and sometimes they are not. 3. Zeroes at the beginning of a number (used just to position the decimal point) are not significant. e.g., 0.025m has 2 sig figs. In scientific notation, this can be written as 2.5*10-2m

4. Zeroes between nonzero digits are significant. e.g., 40.1m has 3 sig figs 5. Zeroes at the end of a number that contains a decimal point are significant. e.g., 41.0m has 3 sig figs, while 441.20m has 5. In scientific notation, these can be written respectively as 4.10*101 and 4.4120*102 6. Zeroes at the end of a number that does not contain a decimal point may or may not be significant. If we wish to indicate the number of significant figures in such numbers, it is common to use the scientific notation.

e.g., The quantity 52800 km could be having 3, 4, or 5 sig figs—the information is insufficient for decision. If both of the zeroes are used just to position the decimal point (i.e., the number was measured with estimation ±100), the number is 5.28×104 km (3 sig figs) in scientific notation. If only one of the zeroes is used to position the decimal point (i.e., the number was measured ±10), the number is 5.280×104 km (4 sig figs). If the number is 52800±1 km , it implies 5.2800×104 km (5 sig figs).

Exact Numbers Exact numbers can be considered as having an unlimited number of significant figures. This applies to defined quantities too. e.g., The rules of significant figures do not apply to (a) the count of 47 people in a hall, or (b) the equivalence: 1 inch = 2.54 centimeters. In addition, the power of 10 used in scientific notation is an exact number, i.e. the number 103 is exact, but the number1000 has 1 sig fig. It actually makes a lot of sense to write numbers derived from measurements in scientific notation, since the notation clearly indicates the number of significant digits in the number.

Look at the following example

The general rule for estimation of the last digit is to record to 1/10th of the smallest division of the measuring device. So, for the common centimeter-ruler, in which the smallest division is 1mm, the estimate for the last digit can be to 0.1mm which is 1/10th, of the smallest division.

Suppose a student records a length of 26 Suppose a student records a length of 26.2mm with a ruler (between the red marks). In his/her judgment, the length is greater than 26.1mm but less than 26.3mm, and so the best estimate is 26.2mm. The measurement can be written as 26.2±0.1 mm. The number 26.2mm contains three significant figures. Although the last digit, 2, is an estimate, it is considered to be a significant figure for the measurement.

How do you read it?

Math With Significant Figures Addition and Subtraction In mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement.

For addition and subtraction, look at the decimal portion (i. e For addition and subtraction, look at the decimal portion (i.e., to the right of the decimal point) of the numbers ONLY. Here is what to do: 1) Count the number of significant figures in the decimal portion of each number in the problem. (The digits to the left of the decimal place are not used to determine the number of decimal places in the final answer.) 2) Add or subtract in the normal fashion. 3) Round the answer to the LEAST number of places in the decimal portion of any number in the problem.

Find the formula weight for Ag2MoO4 given the following atomic weights: Ag = 107.870, Mo = 95.94, O = 15.9994. The number with the least number of digits after the decimal point is 95.94 which has two digits for expression of precision. Also, it is the number with the highest uncertainty. The atomic weights for Ag and O have 3 and 4 digits after the decimal point. Therefore if we calculate the formula weight we will get 375.6776. However, the answer should be reported as 375.68 ( i.e. to the same uncertainty of the least precise value.

Multiplication and Division In mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement.

The following rule applies for multiplication and division: The number having the least number of significant figures is called the KEY NUMBER. The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer. This means you MUST know how to recognize significant figures in order to use this rule. In case where two or more numbers have the same least number of significant figures, the key number is determined as the number of the lowest value regardless of decimal point.

2.5 x 3.42. The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why? 2.5 is the key number which has two significant figures while 3.42 has three. Two significant figures is less precise than three, so the answer has two significant figures.

How many significant figures will the answer to 3.10 x 4.520 have? You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundredth's place is not recognized as significant when, in fact, it is. 3.10 is the key number which has three significant figures. Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.