Monday, April 10, 2017

Using the Error Function to Evaluate Expected Values

  Excel's error function, ERF(x), can be used to evaluate the probabilities for the intervals in a histogram and their expected values. The probability integral, p, below is simplified by the use of a z-score eliminating the standard deviation found in a normal probability distribution. The error function is simplified even further since the factor of 1/2 is missing from the exponential so its arguments require the z-scores a and b of the interval to be divided by the square root of 2. This can be done for the deviations for the times of the Spring Equinoxes. Formulas for binomial distribution can be use to compute the expected value, k=np, for an interval once the probability is known and also the deviation, δk, for the expected value. δk2=np(1-p). The z-scores for the observed counts is found in the last column and gives us a measure of how good the counts for the intervals are. One would expect nearly all the observations to be within 3z for the interval as is the case.

So even though the Earth's orbit deviates in a complicated spiral motion from the Keplerian ellipse, the observed deviations time of the Equinoxes appears to be random.

Supplemental (Apr 10): Another way of looking at the analysis above is that we compared the distribution for the deviations in time with a normal distribution. The z-scores in the last column indicate the central peak is somewhat flattened and the "outliers" at the ends are under represented in our sample. I had a part-time job in the Physics Shop while attending university and one of my teachers used a bean machine to study the statistics. He said that the distributions he was getting were not what was expected. We considered the possibility that the balls might be gaining a little momentum as they fell and were more likely to bounce off the pins and move outwards or the pins may get bent after prolonged use. But rare events are less likely to be represented in a sample and the peak may be more likely to lose its balls than gain them if the total number of balls is small.

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