Number of trials for desired accuracy?

Question

I am conducting an investigation in which I am measuring the period of a pendulum. I know that my timing equipment has an accuracy of plus or minus 33 milliseconds. The pendulum's I am testing usually have a period on the order of one or two seconds. How do I figure out how many trials I have to do with each pendulum to get a mean value within x% accuracy?

Answer 1

You said that your timing equipment has an accuracy of +- 33 milliseconds. In statistics, I can assume that the +- 33 milliseconds is a 95% confidence interval of the mean period of the pendulum. So, since, the critical value in a normal distribution for a 95% confidence interval is 1.96, therefore, your standard deviation is about 0.0168367 s. Working shown below,

1.96 * stddev = 0.033
stddev = 0.033/1.96 = 0.0168367

So, using the Central Limit Theorem, the standard deviation of the estimated mean is stddev/sqrt(n), where n is the number of measurements and sqrt is the square root.

So the accuracy (95% confidence interval) of the period would be
1.96 * stddev/sqrt(n)
= 1.96 * 0.0168367 / sqrt(n)
= 0.033 / sqrt(n)

So as you can see the accuracy drops by a factor of 1/sqrt(n)

So, if you do 2 measurements, you get an accuracy of plus minus 0.033/sqrt(2) = 0.023 s, ie plus minus 23 milliseconds

If you do 3 measuremenets, you get an accuracy of plus minus 0.033/sqrt(3) = 0.019 s, ie plus minus 19 milliseconds

Answer 2

4

Answer 3

Calculate the moments.

Answer 4

minimum 3 trials,
maximum upto u