UnderstandingHow to Read, Measure and Analyze Data

Significant Figures

"Lesson 4: Measurement." Lesson 4: Measurement. N.p., n.d. Web. 20 Sept. 2013.

Introduction

If you have not had much experience in a lab, you are probably used to thinking of numbers as absolutes. When a recipe calls for 3 eggs, you put in precisely 3 eggs, not 3.1 or 2.8. If I ask how many fingers you have, you would answer "10", not "9.75" or "10.1". However, in the laboratory, most numbers are the result of measurements. Any number resulting from measurement is not an absolute number, since it is only as exact as the measuring device can register. For example, when you use a thermometer to measure air temperature, you cannot know what the absolute temperature is. Instead, you can measure the air temperature to the nearest degree. Using the thermometer below, for example, you might measure the temperature to be 66°F. However, the absolute temperature might be 65.681290°F.

We use significant figures to show how many of the digits in a number were measured and how many of them were just guessed or rounded off.Significant figures are the digits in a number which are known precisely, plus one estimated digit. In our temperature example, we are sure that the temperature is somewhere in the 60's, so the first "6" is a measured number and is a significant figure. The second "6" is an estimate, so it is also a significant figure. In our flow example from the last section, we were only sure of the initial "5". All of the trailing zeroes are merely place-keepers which were not measured or estimated, so 500,000 has only one significant figure.

Rules for Significant Figures

Rules: / Examples (# of Significant Figures):
All non-zero numbers are significant. / 251 / 3
13.49 / 4
Zeroes between significant digits are significant / 305 / 3
42003 / 5
If there is no decimal point, then trailing zeroes are not significant. / 470 / 2
10 / 1
If there is a decimal point, then all trailing zeroes are significant. / 41.00 / 4
10. / 2
If a number is less than one, then the first significant figure is the first non-zero digit after the decimal point. / 0.009 / 1
0.01060 / 4
Addition/Subtraction Problems: Your calculated value will have the same number of digits to the right of the decimal point as that of the least precise quantity. / This is what your calculator spits out:
7.939 + 6.26 + 11.1 = 25.299
What is your final answer? ______
Multiple/Division Problems: The number of sig figs in the final calculatedvalue will be the same as that of the quantity with the fewest number of sig figs used in the calculation. / This is what your calculator spits out:
(27.2 x 15.63) / 1.846 = 230.3011918
What is your final answer? ______
Combined addition/subtraction and multiplication/division: First apply the rules for addition/subtraction, and then apply the rules for multiplication/division.

Practice Problems

Provide the number of significant figures (sig figs) in each of the following numbers:

(a)0.00000055 g ______(c) 1.6402 g ______(e) 16502 g ______

(b)3.40 x 103 mL ______(d) 1.020 L ______(f) 1020 L ______

Perform the operation and report the answer with the correct number of significant figures (sig figs):

(a)(10.3) x (0.01345) = ______(b) (10.3) + (0.01345) = ______

(c)[(10.3) + (0.01345)] / [(10.3) x (0.01345)] = ______

Accurate versus Precise Measurements

"Accuracy and Precision." Accuracy and Precision. N.p., n.d. Web. 23 Sept. 2013

Definition:

Accuracy: Is how close a measured value is to the actual (true) value.

Precision: Is how close the measured values are to each other.

Example of Precision and Accuracy:

______Accuracy
______Precision / ______Accuracy
______Precision / ______Accuracy
______Precision

So, if you are playing soccer and you always hit the left goal post instead of scoring, then you are not accurate, but you are precise!

Bias (don't let precision fool you!)

If you measure something several times and all values are close, they may all be wrong if there is a "Bias"

Bias is a systematic (built-in) error which makes all measurements wrong by a certain amount.

Examples of Bias

  • The scales read "1 kg" when there is nothing on them
  • You always measure your height wearing shoes with thick soles.
  • A stopwatch that takes half a second to stop when clicked

In each case all measurements will be wrong by the same amount. That is bias.

Degree of Accuracy

Accuracy depends on the instrument you are measuring with. But as a general rule: The degree of accuracy is half a unit each side of the unit of measure

Examples:

If your instrument measures in "1"s
then any value between 6½ and 7½ is measured as "7" /
If your instrument measures in "2"s
then any value between 7 and 9 is measured as "8" /