Solved – How to calculate ‘times likely’ for a percentage and whole number data set

This question exemplifies poor statistical communication, so it's worth some comment.

The issues with the language are many, but the most egregious are

1. The meaning of "less likely" is ambiguous, because it depends both on how "likelihood" is expressed and on what kind of comparison "less" refers to.

2. The meaning of "more words" is ambiguous, because it is unclear whether it refers to totals or averages.

3. In a related pair of sentences it expresses a difference in two distinct ways, which is potentially confusing.

4. It confuses data (the counts) with modeled, inferred quantities (the probabilities) and speaks of them as being equivalent. This is confusing.

5. The ratio of average words spoken is not an average of ratios of words spoken, which is a potential lurking ambiguity.

Let's fix these problems and, in the process, answer your questions.

The "probability" is either an observed proportion or an estimate of a proportion. We can't tell which from the context, but the harder case to communicate is the latter, so let's assume it. What we are told, then, is that (under a scenario that is not described here) a single person is estimated to have a 0.6 chance of speaking whereas a person in a group is estimated to have a 0.8 chance of speaking. Two natural and meaningful ways to compare these are

• The difference in probabilities (between group and single people) is 0.8 – 0.6 = 0.2.

• The odds of speaking changes from 0.6:0.4 to 0.8:0.2. The odds ratio (of group vs single) therefore is (0.8:0.2) = 4 divided by (0.6:0.4) = 1.5, which is 8/3.

Regardless, the interpretation of a ratio of probabilities depends heavily on what those probabilities are. A decision maker might treat two probabilities of 0.8 and 0.6 as being very different while viewing two probabilities of 8 and 6 per million as being essentially the same–even though the ratio is the same in both cases. Thus, if you insist on reporting a ratio, you had better report the individual probabilities, too.

(The same warning holds for interpreting differences of probabilities! For instance, if a disease that never occurred before is found in 20% of a population, that would likely be far more alarming than an increase from 60% to 80% of the population. But the difference in each case is the same. Once again, it's clearest to report the numbers when you compare them.)

Assuming the problem is equating "likely" with "probability," it asks you express the difference 0.8 – 0.6 as a fraction of 0.8. That's what would go in the first blank.

The second statement asks for a similar calculation, but first we need to interpret it. The likeliest intended meaning is that on average, a single person spoke more words than a group person. For instance, if the numbers for Data 2 were 75 words per single and 25 words per group person, you could say "on average, a single person spoke three times as many words as a group person." In this case, the correct multiple is not 75/25 = 3, but rather 43/33 = 1.3. The question asks you to convert this factor into a percentage increase. For example, three times as many words is a 200% increase because 75 is 50 more words than 25 and 50 is 200% of 25. You need merely perform the analogous calculation for the numbers 43 and 33 in place of 75 and 25.

An example of an unambiguous, consistent, and straightforward statement of these results is

We estimate that in low music conditions, the chance a single adult would speak is 0.6, which is 3/4 the chance an adult in a group would speak. But in our data, the average number of words spoken by single adults is 43, equal to 1.3 times the average number of words spoken by adults in groups.