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Hooper has noted that the number of recaptures increased sharply on 1 July, the same day that E. Ford sent a letter to Kettlewell. The letter is unremarkable, and two facts militate against a finding of fraud.

First, Kettlewell finished collecting data in the wee hours of the morning and therefore could not have received the letter before collecting his data on 1 July. He markedly increased the number of moths he released on 30 June, the day before the letter was mailed, not 1 July. Additionally, as Hooper admits, he continued to release more moths after 30 June.

Not surprisingly, he also captured more moths: Indeed, Figure 1 plots recapture rate as a function of the number of moths released on any day. The line is a line of best fit constrained to pass through the origin on the assumption that no moths are recaptured if none are released.

Figure 1 shows that the recapture rate is very nearly a linear function of the number released. The square r2 of the correlation coefficient is 0. The fit improves only slightly if the line is not constrained.

Why did Kettlewell release more moths beginning on 30 June? He released both moths he had reared and moths he had captured. Because the moths were just hatching, he had limited control of the number he could release on any given day.

There is no reason to suspect that the increased numbers of releases reflect anything other than the number of moths that were available. More pointedly, if we plot his recapture rate as a function of time, as in Figure 2, we find what looks to the eye as a sudden increase.

Figure 2 omits those days, 27 June and 30 June, that were preceded by 0 releases. It is hard to make much out of a mere 8 data points, but the recapture rate certainly appears to the casual observer to increase sharply after 1 day of inactivity.

Mathematical Model Kettlewell recaptured most of his moths after they had been in the wild for only 1 day, but he recaptured some after 2 days. Let us therefore define a 1-day recapture rate R1 and a 2-day recapture rate R2 as the ratios of the numbers of moths recaptured after 1 and 2 days in the wild.

Kettlewell reported no 3-day recaptures.

We may estimate the 2-day recapture rate by looking at the 4 moths captured on days 2 and 5. R2 is very nearly equal to the square of R1, as we would expect if the model is appropriate. Our mathematical model is straightforward: The number of moths captured on any given night is equal to the number of moths released the day before times the 1-day recapture rate, plus a similar term, the number of moths released 2 days before times the 2-day recapture rate.

The results of a calculation based on this model are shown as the solid curve in Figure 3. Note that I have made no artificial assumptions, such as adjusting the recapture rates to get a good fit to the data, in constructing Figure 3. How well does the model fit the data?

To answer that question, we have to estimate the normal range of variability in the data. In statistical terms, we calculate the standard uncertainty of the data points. The standard uncertainty is a number that tells us, in this case, how much variation we might expect if we repeated the experiment many times. By way of introduction, suppose that you toss N marbles at a hole in a table. Count the number of marbles that fall through the hole, and repeat the experiment many times.

Suppose that the average number of marbles that fall through the hole is M. You will not count M marbles every time you perform the experiment; to the contrary, the number will vary about M and very possibly will never exactly equal M.

The mean number of marbles that pass through the hole is equal to Np. How much will any one toss differ from M? Assume that the number of marbles that pass through the hole is described by a binomial distribution. Then the standard deviation of M is. The uncertainty can be surprisingly large. For example, if p is 0. You can expect anywhere between 16 and 32 marbles to fall through the hole on any given toss.

You should not be especially surprised by any number unless it is much less than 12 or much more than This fact alone should militate against a charge of fraud. On any given day, Kettlewell released N moths and recaptured M moths. Because the 2-day recapture rate is small,the statistics are essentially the same as those of the marble example. The number M is highly variable, as we have seen; on another day, he might capture a substantially different number, even if conditions were unchanged.

The standard uncertainty u estimates the probable variation of M quantitatively ISO It is given by, where the recapture rate R replaces the probability p. The result is shown in Figure 2 as a series of error bars. Inasmuch as the model the solid curve passes through virtually every error bar, it may be said to be a nearly perfect fit to the data, however poor it might appear in the absence of error bars.

The points on days 7, 8, and 9 lie noticeably above the curve. If the data were completely unbiased, then we would expect about a chance that any one of those points lay above the curve.

### Peppered moth - Wikipedia

The odds that 3 consecutive points lie above the curve are 1 in 8 -- exactly the same as the odds against tossing 3 heads in a row and by no means improbable enough to base a charge of cheating. Even if 5 points lay above the curve, the odds against would be 1 in 32, again, not very impressive in its improbability.

Additionally, 2 consecutive data points lie noticeably below the curve. Moonlight The differences between the data and the curve are not statistically significant; the observed variation very probably is the result of chance. It is, however, possible that the deviations from the curve are "real" -- that is, due to some systematic effect, or systematic error, not due solely to random error.

It is very hard, unfortunately, to track down a source of systematic error when that error is itself less than the standard uncertainty of the data set; the systematic error is said to be lost in the noise. Hooper tells us that the weather was stable and could not have accounted for the increase in the number of recaptures though her description suggests somewhat variable winds.

We have, nevertheless, a strong candidate that can account for the systematic deviations of our simple model from the curve: The moon was full on 27 June that is, the night of June. By 2 July, the moon was 5 days past full but visible for only part of the night.

Thus, the total exposure to the moon -- the product of illuminance brightness and time -- was approximately one-quarter what it was during the full moon, and it dropped steadily over the next few days. Clarke and his colleagues have investigated the effect of the phase of the moon on capture rates of peppered moths in a single environment over 30 years and concluded that the moon does not affect capture rates.

Unfortunately, theirs was a retrospective study, and they did not record weather data, that is, did not control for cloudy or rainy days. They averaged the data over 5-day periods surrounding the full moon and did not use the actual exposure to the moonlight as defined above. All of these factors will reduce the correlation between capture rates and exposure to moonlight. Even so, they calculated a small but not statistically significant correlation that suggests a slight increase of capture rate around the full moon.

In addition, when they checked the new moon against the full moon, they calculated a small, barely significant increase, which they discounted. Possibly the effect is due to the presence of streetlights, to which they refer obliquely, and which may attract moths away from the stronger mercury vapor light only when the moon is dark.

At any rate, they conclude that moonlight does not affect capture rates. Kettlewell worked on clear days only; I do not think that the conclusion of Clarke and colleagues is necessarily pertinent. I made no effort to control for the elevation of the moon. The equation in Figure 4 is the equation of the line of best fit to the data.

### CB601: The peppered moth story

The daily recapture rate rises by a factor of 3 as the brightness of the moon decreases. Still, the calculation based on total captures yields much the same result as that outlined below. The result of the calculation is shown in Figure 3 as the light,dashed curve.

It demonstrates a somewhat better fit to the data than the solid curve, especially during the first few days. Hooper evidently did not consider the most likely cause of the changes she saw, exposure to moonlight, let alone realize that the change in recapture numbers began before Kettlewell could have read the letter that supposedly triggered this change.

Hooper and Sargent should have performed a careful analysis before Hooper presumptuously insinuated fraud. It did not rely on the release-recapture experiments alone.

It is also supported by at least 30 studies of different moth species that also developed melanic forms Grant, The variations in his data are no more than the uncertainties associated with sampling and other factors, possibly including exposure to the moon. It is an irresponsible leap to accuse a distinguished naturalist of fraud on the basis of a single letter and a wholly imperfect, offhand analysis of his data.

The peppered moth properly remains a valid paradigm -- no, an icon -- of evolution. Ian Musgrave provided the lunar data. I am further indebted to Pete Dunkelberg and Bruce Grant for helping me understand the uncertainties of field work in biology.

Musgrave, Laurence Cook, and Nicholas Matzke reviewed the paper and made many helpful suggestions regarding both clarity and content.

This paper may be reproduced on the Worldwide Web on condition that it be reproduced in its entirety and that the author be notified. Print or hard-copy reproduction requires the express written consent of the author.

### Talk Reason: arguments against creationism, intelligent design, and religious apologetics

He is the author of No Sense of Obligation: Dawkins, and Susannah Kahtan Forrest, Barbara, and Paul R. The Wedge of Intelligent Design. Of Moths and Men: Guide to the Expression of Uncertainty in Measurement. International Organization for Standardization.