Uncertainty and Measurements There are errors associated with any measurement. Random error Random error – These errors can be caused by a variety of sources: 1.Inability to read a measurement beyond the smallest division 2.Limited accuracy of measuring instruments 3.Fluctuations in temperature/pressure Systematic errorHuman error

Errors in Measurements Systematic errors – Usually caused by a faulty instrument or improper instrument calibration. This results in measurements that are always too large or always to small. Human error – These are errors caused by carelessness or simply making a mistake. These errors are completely UNACCEPTABLE!! We need to account for random error by recognizing the limitations of our measuring instrument. We need to spot systematic errors and attempt to correct the problem.

Accuracy vs. Precision Accuracy is an indication of how close our measurement is to the known accepted value or the “right answer”. Precision is an indication of the repeatability of a measurement. Precise Not Accurate Not Precise Not Accurate

Very Precise Very Accurate Percent error is a measure of accuracy. Accepted value – Experimental value Accepted value x 100% error =

Percent Difference We use percent difference as a measure of precision. This gives us an indication of how close two experimental values are to one another. Exp value 1 – Exp value 2 Exp value 1 + Exp value 2 2 x 100% Difference =

Significant Figures We use a certain number of digits when we report the results of a measurement. The number of digits tells us the degree of precision or how close a particular instrument can measure. The last digit in a number is always assumed to be uncertain. If we measure something and get cm, we are really saying that we are sure about the 10.3 cm, but we are estimating the 4. So our measurement is between cm and cm. Or, 10.34cm (+/-) 0.01 cm

Sig Fig Rules 1. Leading zeros are never significant. 2. Zeros that occur between non- zero digits are significant. 3. Trailing zeros are only significant when there is a decimal point. How many significant figures??? 1,156,

More Sig Fig Rules When you multiply or divide, the result must contain the same number of significant figures as the number with the least number of significant figures used in the calculation. (2.45)(2.5) (2.786) = What your calculator says The number with the least number of sig figs is “2.5” (2 sig figs). Therefore, answer can only have 2 sig figs So, we round the answer to 2.2

More Sig Fig Rules When you add or subtract the result can only contain as many decimal places as the number with the fewest number of decimal places used in the calculation – = What the calculator says 4.5 only has one decimal place, so the answer can only have one decimal place. So, we round the answer to 10.3 Note: In this example, the answer has 3 sig figs and 4.5 only has 2 sig figs.

When doing calculations, it is always a good idea to keep a few extra sig figs in the calculations, and round off the final answer to the correct number of significant figures or decimal places. When rounding numbers, if the last number is a 4 or below, round down. For a 5 or above, round up = 3.46 (3 sig figs) = 3.44 (3 sig figs)

Scientific Notation Scientific notation is a shorthand way to express very large or very small numbers using the exponent 10 raised to some integer power. Remember any number raised to the zero power is one = 1 To express a number in scientific notation, write the number as a number between 1 and …. and then multiply the number by 10 raised to some integer power. Positive exponents are for big numbers >1 and negative exponents are for small numbers <1.

234,357,432 = Express numbers in scientific notation using 3 significant figures 2.34 x 10 8 Moved 8 decimal places to the left. So I need to multiply the new number (2.34) by 10 8 so it is equivalent to the original number =1.23 x Moved 9 places to the right

Metric Prefixes Table 2-2 on page 35 of the text has complete table. There are several that we use over and over and over…. PrefixAbbreviationValue mega kilo centi milli micro MkcmMkcm Greek letter “mu”

Scientific Notation on the Calculator DO NOT USE THE MULTIPLICATION KEY “x” and the number 10 when you enter a number in scientific notation. You will almost always make a mistake and be off by a factor of 10. Most calculators have an “EE” key. Some calculators have an “EXP” key. 1. Enter the number between 1 and (ex ) and then push the EE key. 2. Your calculator will show 2.36E 3. Enter the exponent (ex. 6). 4. Your calculator will show 2.36E6 5. This is equivalent to: 2.36 x 10 6 =