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Hi Science Line, I had my secondary science student teachers conduct a toxicology investigation following a standard protocol. They investigated the effects of a toxin of their choice (diet Pepsi, nail polish remover, etc.) on the germination and growth of lettuce seeds. They used six different concentrations of the toxin, three plates per concentration, five seeds per plate. They counted the number of seeds germinated per treatment and measured the length of the seeds' radicle in mm. My questions: (1) What is the appropriate number of significant figures to use when calculating the average germination and average radical length? (2) If a seed hasn't germinated or has germinatd but has no radicle, do you include a 0 for radicle length when calculating the average or do you leave it out of the calculation all together? Thanks much for your help!
Answer 1:

1) I believe that the appropriate number of significant figures to use when calculating averages would be the same number of significant figures as used in the original measurements. In other words, if your student teachers were estimating radicle lengths to the nearest 0.1mm, then the averages should be expressed in the same units, to the nearest 0.1mm. Conservatively, the average must be reported with no more significant figures than the least precise measurement taken. The precision or significant figures, of a measurement are the smallest decimal place to which the measurement is made, or estimated. In the case of combining measurement through addition, subtraction, or averaging, you are not concerned with a number of significant figures per se. Instead, you are looking for the smallest decimal place of the least precise measurement. An example is provided below.

Typically, the number of significant figures in a measurement is equal to the number of digits known with a high degree of confidence, plus an additional digit that is an estimate or approximation. The number of digits known with a high degree of confidence is usually limited by the measurement device or by limits to human perception and judgment. For instance, when using calipers that have tick marks for every 0.1mm, a scientist can only measure to the nearest 0.1mm with a high degree of confidence, but can then usually estimate sizes by eye to the nearest 0.01mm. The rationale behind the conventions for significant figures is to ensure that the results of any calculations reflect the same degree of precision as the measurements upon which the calculations were based. For a good review on significant figures and measurement precision, see the following website:

click here

Example:
Five radicle length measurements = 13.0, 12.54, 9.4, 12, 7.5
Average = (13.0 + 12.54 + 9.4 + 12 + 7.5) / 5 = 10.888
Least precise measurement is 12 a rounding to the nearest ones digit,
Answer = 11

When combining measurements using addition or subtraction, the final result must have no more significant digits (i.e., decimal places) than the original measurements. In other words, the final result should have the same degree of precision as the least precise measurement. In this example, the least precise measurement is 12 because it is only measured to the nearest ones digit, so the ones is the least precise decimal place. Therefore the rounded answer, in significant figures, is 11.

2) The answer to your second question depends upon how you intend to interpret the average radicle lengths, which will in turn depend upon what questions you are asking. There are three options: 1) you can include zeros for seeds that did not produce radicles but exclude seeds that did not germinate; 2) you can exclude all seeds that did not germinate or did not produce radicles; and 3) you can include zeros for seeds that both did not germinate and did not produce radicles.

Given that you are measuring both the number of seeds that are germinating andthe length of the radicles that are growing, it seems to me that you are askingtwo independent questions. The first question is probably: Does the proportionof seeds that germinate differ among toxins? The second question may be one oftwo choices #1) Does the growth of the seeds that germinate differ amongtoxins? Or #2) Does the growth of radicles produced by seeds differ amongtoxins? To answer #1, you need to know the average growth only of the seedsthat germinate and can leave out any seeds that did not germinate. In other words, if you want the average radicle length to indicate the mean length to which radicles grow in plants that have germinated, then you should include a0 for any seed that has germinated but has not produced a radicle, but you should omit/exclude any seeds that did not germinate (i.e., option 1 in the paragraph above).

If, however, your second question is choice #2, then you need to know the average growth only of seeds that produce radicles. Therefore, you should omit all seeds that did not produce radicles as well all seeds that did not germinate. This is the same as option 2 in the first paragraph above. I think that both of these approaches would be equally valid, but have distinct interpretations.

The third option, to include a 0 both for seeds that did not germinate and for seeds that did not produce radicles, does not seem a viable option because itwould confound your second question, comparing growth, with your first question, comparing germination. The answers of the two questions would no longer be independent since the average growth calculations would depend in part upon the proportion of seeds that germinate. In other words, if one question deals with comparing the proportion of germinating seeds among treatments, and another question deals with comparing the growth of the seeds (but not th


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