There are some measurements for which we are almost certain of the mean of a large number of repetitions. For example, we expect that the average number of heads appearing in a large number of tosses of ten unbiased coins should be 5 and that the standard deviation should be 1.58 (Note: The calculation of SD=1.58 is based on a large number of trials---and unlike the mean of 5, is not obvious). We can use such measurements to gain confidence in the reliability of the methods we have been discussing.
Design your own spreadsheet for this experiment. Use rows 1 and 2 for a title and other rows for column headings. Make sure you get enough information on the screen so that the numbers mean something. Text is as important as numbers.
Count and record the number of heads that appear when ten coins are tossed. Repeat this procedure five times. Calculate the mean, standard deviation and standard error. Repeat this experiment 15 more times (15 additional tosses of the 10 coins) and calculate the mean, standard deviation and standard error for all 20 trials. Your results to 90 % confidence should also be recorded on the blackboard, along with similar results obtained by your classmates, so that we can compare them.
Do your 90 % confidence intervals for the mean include the expected value of 5? Are you surprised that some estimates, perhaps including your own, do not?
What do you think might be the outcome if you looked at results based upon the average number of heads appearing in 100 tosses of these ten coins? Is there a better chance that the 90 % confidence interval will include 5?
The coin toss is an example of a binomial distribution. One can prove quite generally that, as the size of the sample increases (5 measurements, 10 measurements, 100 measurements), sample means tend to be distributed in Gaussian fashion, no matter what the shape of the original distribution. This is known as the Central Limit Theorem of statistics. From a more practical perspective, why should we expect our measured distributions to look like Gaussian curves?
Design a new spreadsheet for this experiment. Use rows 1 and 2 for a title and other rows for column headings. Make sure you get enough information on the screen so that the numbers mean something. Text is as important as numbers.
Use the ruler to measure the length (long dimension) of the block on the table. Use the Vernier caliper to measure the other dimension (the width). Do not measure the thickness. Do each measurement five times and compute the mean, the standard deviation and the standard error for each dimension. Report your results to 90 % confidence.
What are some sources of random error in this procedure?
Now calculate the area of the block. How do you think you would/should report a 90 % confidence interval for your area? (This will be one of the topics that we deal with in next week's lab.)