Rules for deciding the number of significant figures in a measured quantity:

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1 Rules for deciding the number of significant figures in a measured quantity: (1) All nonzero digits are significant: g has 4 significant figures, 1.2 g has 2 significant figures. (2) Zeroes between nonzero digits are significant: 1002 kg has 4 significant figures, 3.07 ml has 3 significant figures. (3) Zeros left of the first nonzero digits are not significant: o C has only 1 significant figure, g has 2 significant figures.

2 (4) Trailing zeroes that are also to the right of a decimal point in a number are significant: ml has 3 significant figures, 0.20 g has 2 significant figures. (5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures. The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as: calories (3 significant figures) calories (4 significant figures), or calories (5 significant figures).

3 The accuracy of a calculated result is limited by the least accurate measurement involved in the calculation 1) In addition and subtraction,the result is rounded off so that it has the same number of digits as the measurement having the fewest decimal places (counting from left to right). For example, 100 (assume 3 significant figures) (5 significant figures) = , which should be rounded to 124 (3 significant figures). (2) In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example, 3.0 (2 significant figures ) (4 significant figures) = which should be rounded to 38 (2 significant figures).

4 Sample problems on significant figures =? =? =? =? =? / =? =? 8. (5.5) 3 =? ( ) =? =? 11. What is the average of , , , , and ? 12. What is the standard deviation of the numbers in question 11? x x 102 =? (Give the exact numerical result, and then express that result to the correct number of significant figures).

5 Answer key to sample problems on significant figures = = = = 129 (assuming that 125 has 3 significant figures) = / = = 0.87

6 8. (5.5) 3 = 1.7 x ( ) = = 1.4 x 10 2 This answer assumes that 45 has two significant figures; however, that is not unambiguous, because there is no decimal point, and because it is not expressed in scientific notation. If 45 is an exact number (e.g., a count), then the result should be 1.35 x What is the average of , , , , and ? The average of these numbers is calculated to be , which rounds to

7 12. What is the standard deviation of the numbers in question 11? The result that you get in calculating the standard deviation of these numbers depends on the number of digits retained in the intermediate digits of the calculation. For example, if you used instead of the more accurate as the mean in the standard deviation calculation, that would be wrong (don't round intermediate results or you will introduce propagated error into your calculations). The Mathworks MATLAB std function gives e -004 Microsoft Excel gives A Hewlett-Packard 50G calculator gives e-04 These results should be rounded to , which is x 10-4 (expressed in scientific notation).

8 Standard Deviation

9 Standard Deviation

10 GOOD GRAPHING RULES Use the entire graph. Choose a range for your X and Y axis scales so that your data will fill the entire page. Plot the independent variable on the X axis. (Time is always plotted on the X axis) Plot the dependent variable on the Y axis. Don t crowd axes to the edge of the page. Leave enough room to show and label your scales for X and Y axes. Have a descriptive TITLE that tells the reader what they are viewing.

11 GOOD GRAPHING RULES Use different symbols for different data sets. Draw the symbols large enough to be easily viewed. If appropriate, draw a line of best fit for each data set. (DO NOT USE POINT-TO-POINT TO DRAW A LINE) If appropriate, add error bars. Show a LEGEND defining each symbol, line, and error bars (SD or SEM).

12 The World of the Micrometer (Micron) 10-6 meters

13 The World of the Nanometer 10-9 meters