2.  Consider the following "experiment" where you construct a catapult which launches a dart at a target.  The object is to hit the bulls eye. Topic 1.2 Extended B - Laboratory number crunching First TrialSecond TrialThird Trial Fourth Trial Low Precision Hits not grouped Low Accuracy Average well below bulls eye Low Precision Hits not grouped High Accuracy Average right at bulls eye High Precision Hits grouped Low Accuracy Average well below bulls eye High Precision Hits grouped High Accuracy Average right at bulls eye FYI: RANDOM ERROR is where accuracy varies in a random manner. FYI: SYSTEMATIC ERROR is where accuracy varies in a predictable manner.

3.  Having gathered your data (recorded to the most significant figures allowed by your measuring devices), you will want to analyze your data, and somehow decide on its validity.  We have already talked of precision. For our meter stick we could measure to 3 significant figures with a precision of ± 0.05 mm: Topic 1.2 Extended B - Laboratory number crunching 01 1 cm 1 mm  I estimate the length to be 1.17 cm or 11.7 mm.  Considering error, L = 11.7 mm ±.05 mm. FYI: This is the precision in a SINGLE measurement.  We call the precision of 0.05 mm the absolute error.

4. Topic 1.2 Extended B - Laboratory number crunching  There are three useful percent formulas: Fractional Error = Absolute Error Measured Value  The first formula compares absolute error to a measured value:  For example, since we measured L = 11.7 mm ±.05 mm, we have ·100% Fractional Error = 0.43% Fractional Error FYI: FRACTIONAL ERROR is a way to evaluate the ACCURACY of a measurement. Fractional Error = 0.05 11.7 ·100% FYI: Note that the SMALLER the MEASURED VALUE, the LARGER the FRACTIONAL ERROR. FYI: FRACTIONAL ERROR between 1% and 5% are considered fairly accurate. Less than 1% is very accurate, and greater than 10% is considered "rough."

5.  There are three useful percent formulas: Topic 1.2 Extended B - Laboratory number crunching Percent Difference = Difference Average  The second formula compares two measured values x 1 and x 2 : % Diff = x 1 - x 2 (x 1 + x 2 ) 1212  For example, suppose you measure the line of the previous example to be x 1 = 11.7 mm the first time, and x 2 = 11.8 mm the second time. % Diff = 11.7 - 11.8 (11.7 + 11.8) 1212 ·100% % Diff = 0.851% Percent Difference FYI: PERCENT DIFFERENCE is a way to evaluate the PRECISION in your measurements - provided they are expected to be the same.

6.  There are three useful percent formulas: Topic 1.2 Extended B - Laboratory number crunching Percent Error = Difference True Value  The third formula compares a measured value x meas to an accepted or "true" value x true :  For example, suppose you measure the diameter and circumference of a circle, and calculate pi to be 3.11. ·100% % Err = x meas - x true x true ·100% % Error = 3.11 - 3.14 3.14 ·100% % Err = 0.955% Percent Error FYI: PERCENT ERROR is a way to evaluate the ACCURACY in your measurements - provided you know what they are supposed to be.

7.  Sometimes you have lots of data that you need to compare to a known value: Topic 1.2 Extended B - Laboratory number crunching  For example, suppose you measure the freefall acceleration of a dropped ball.  We will learn that all freely falling objects should have a CONSTANT acceleration.  So we find the average, or mean value of the data. Average Value = Sum of Values Number of Values x = xixi  i=1 N Average or Mean Bob's Data a = 0 + 2 + 10 + 18 + 20 5 a = 10 m s -2  Suppose another student has this data for the same experiment: Abby's Data Bob a = 0 + 10 + 10 + 10 + 20 5 a = 10 m s -2 Abby  Who's lab results are best - Bob's or Abby's? 1N1N

8. Topic 1.2 Extended B - Laboratory number crunching Bob's Data Abby's Data a a  At first glance we may say the results are identical: After all, both have the same average acceleration of 10 m/s 2.  But instinctively, you may feel that Abby's data is better, because, after all, three of the data points are exactly equal to the average acceleration.  In technical terms, both data sets have the same ACCURACY, but Abby's data set has better PRECISION.  So how do we distinguish between the data sets (preferably with a single number)?  One way is by examining the range of the data points. Range = Max(x i ) - Min(x i ) Range  These data sets each have the same range (spread) of 20 m/s 2 - 0 m/s 2 = 20 m/s 2.  Thus range isn't the best indicator of a data set's precision. Range

9. Topic 1.2 Extended B - Laboratory number crunching Bob's Data Abby's Data a a  Another indicator of precision is deviation from the mean: i th deviation = (x i - x) Deviation from the Mean - 10 - 8 0 8 10 - 10 00 0 10 FYI: The ith deviation only tells you how precise a SINGLE data point is. To find the PRECISION of the data SET, you might find the AVERAGE of all of the deviations. But the average in both cases is ZERO! -10 + -8 + 0 + 8 + 10 5 = 0 m s -2 -10 + 0 + 0 + 0 + 10 5 = 0 m s -2

10. Topic 1.2 Extended B - Laboratory number crunching Bob's Data Abby's Data a a  A way around this unfortunate circumstance is to take the AVERAGE OF THE ABSOLUTE VALUES of the deviations: Average Absolute Deviation from the Mean 10 8 0 8 00 0 average absolute deviation =  |x i - x| 1N1N 10 + 8 + 0 + 8 + 10 5 = 7.2 m s -2 10 + 0 + 0 + 0 + 10 5 = 4 m s -2  Now we have an indicator (in the form of a single number) which tells us that Abby's data is better than Bob's.

11. Topic 1.2 Extended B - Laboratory number crunching Bob's Data Abby's Data a a  Because absolute values are tedious (you have to take the opposite of negative differences), standard deviation is the accepted way to determine precision: Standard Deviation -10 -8 0 8 10 -10 00 0 10 (-10 2 ) + (-8 2 ) + 0 2 + 8 2 + 10 2 5-1 = 9.1 m s -2 (-10) 2 + 0 2 + 0 2 + 0 2 + 10 2 5-1 = 7.1 m s -2 standard deviation =  =  (x i - x) 2 1 N-1 FYI: The standard deviation as the square root of the average of the squares of the deviations. FYI: Squaring the deviations produces POSITIVE values, just like the absolute value. FYI: For reasons we don't need to go into, we divide by ONE LESS THAN THE TOTAL NUMBER OF DATA POINTS when finding this average.