1. Error, Accuracy, Precision, and Standard Deviation Notes

2. Errors Two types of errors: random and systematic Random errors: uncontrollable events, like air currents, temperature variations, and electrical variations. The error can be minimized by taking a large number of measurements (100 or more). Random errors are recognized by the fact that the values are BOTH above and below the true value

3. Systematic errors: controllable events. Find the error, fix the error and repeat the experiment.

4. Types of systematic errors: Equipment : the equipment is worn, out of calibration or broken, fix the equipment and repeat the measurements. Technique: some part of the procedure is incorrect. Examples: not looking at eye level at a graduated cylinder or balance, cars are not released at the same time. Bias: eliminating a number because you do not like it. Unless a known error has occurred (can eliminate then) you cannot throw out a value because it is different.

5. Systematic errors are recognized because all of the values are EITHER above or below the true value.

6. All of the numbers we measure need to be evaluated and accuracy, precision and standard deviation are the tools we use to do this.

7. Accuracy How close a value is to the true or accepted value (an average can be compared to the accepted value) Only one measurement is necessary for calculating an accuracy but many numbers is preferred and the accuracy of the average is then taken.

8. Accuracy: % error = ‌ True value – experimental value ‌ x100 True value * The experimental value can be the average Desired value is zero.

9. Precision How close a set of values are to each other. Requires at least 2 values; more are better. % difference = High – Low x 100 Average * Desired value is zero.

10. Acceptable ranges are arbitrary but for Physics we will use 0-1% Excellent 1-7% Good 7-15% Fair 15 and up Redo (Unacceptable)

11. What is the precision is high, how can it be fixed? Look to see if there is an outlier in the set and statistically try to eliminate it.

12. Standard Deviation (S) (sample standard deviation) Population Standard Deviation ( σ x on the calculator) The standard deviation of the entire population of data Sample Standard Deviation (Sx on the calculator) The standard deviation of a small sample of the whole population – this is all that we are able to collect.

13. √ Σ (x-ave) 2 n-1 √ - the square root of the entire thing Σ – sum of x – a value ave – average of all values n – the number of values

14. √ Σ (x-ave) 2 n-1 Take the value, subtract the average and square this number. (Do this for all values.) Add all of these together. Subtract one from number of values. Divide your sum by this difference. Take the square root of the whole thing.

15. Example Values 2.54 2.55 2.56 2.57 2.58 Ave = 12.80/5 = 2.560

16. Example Values value – average difference 2 2.54-.020.0004 2.55-.010.0001 2.5600 2.57.010.0001 2.58.020.0004 Ave = 12.80/5 = 2.560

17. Example Values value – averagedifference squared 2.54-.020.0004 2.55-.010.0001 2.5600 2.57.010.0001 2.58.020.0004 12.80/5 0.0010 Average = 2.560n-1 = 5-1 = 4 0.0010/4 = 0.00025 √ 0.00025 = 0.016

18. Standard deviation values are hard to interpret (2.560 + 0.016) Hard to say from the numbers whether they are good or not. Therefore, we use Relative Standard Deviation.

19. Relative Standard Deviation = s/average x 100 0.016 x100 = 0.63% 2.560 Easier to interpret: 2.560 + 0.63% very close

20. If you have a value that does not fit the set, you must statistically show if it is an outlier. Two methods to do so are: 1.2 standard deviations 2.q test

21. If have a set of values, is 2.79 an outlier? 2.54 2.55 2.56 2.79 2.57 2.58 2 Standard Deviations

22. 2.54 2.55 2.56 2.79 2.57 2.58 Average = 2.598 Sx = 0.09 2.598 +.18 = 2.782.598 -.18 = 2.42 Range of values 2.42 to 2.78 The value 2.79 would be an outlier because it is beyond 2 standard deviations from the average.

23. Q test Questionable value – closest value numerically Range of all values = q value Compare the results to the Q values, if your questionable value is larger than the 95% confidence Q value, then it is an outlier.

24. 2.79 – 2.58 = 0.84 2.79 – 2.54

25. Number of values95% confidence Q value 30.943 40.754 50.640 60.564 70.510 80.469 90.438 100.412

26. Using the Calculator for Standard Deviation Plug the values into the calculator Hit STAT button Select 1: Edit Enter the list of data Hit STAT button Select CALC menu Select 1: 1-Var Stats Hit Enter Avg 2.56 (¯x) Sum 12.8 ( Σ x) Sx = 0.0158 = 0.016 (sf of standard deviation values is first non- zero digit unless it is a one then keep 2 digits)