The "A closer look at Statistics" take-away note card includes the following.

Definition: Mean - The average or mean of a collection of numbers is obtained by adding the numbers together and then dividing by the number of items.

Definition: Median - The median is the midway mark, with the same number of items above the median mark as there is below the median mark.

Definition: Mode - The mode is the number most often included in the set or most often attained (in the case of test scores).

[Conclusions to keep in mind...] An average is a useful figure to know only if:

• There is not too much variation in the figures or scores or numbers;

• The average is close to the median amount; and

• The distribution is more or less bell- shaped.

Whether from inadequate early educational experiences or owing to a long period of time since taking a math course or owing to something else, many students have an aversion to math. In turn, while the Data Detectives Student Handbook provides math-related assurances as well as a mini review of how to calculate averages and use percentages, be on the lookout for students in need of additional math-related assurances and/or review materials.

Excerpt from the Data Detectives Student Handbook:

Before looking at the materials included in the STATISTICS tutorial, please keep in mind you will NOT be called upon to make complex statistical calculations. Rather, the most complex statistical calculation you may need to perform includes adding four (or less or more) numbers and then dividing the sum by four (or less or more) to find the average of the four numbers.

As a numerical example of the above:

45 + 73 + 80 + 14 = 212 / 4 = 53

Also, as a brief review of percentage(s), a percent is another way of expressing parts of a whole. Further, 100% equals one whole and when we convert percentages to decimal numbers, 100% becomes 1.00 (and 40% becomes 0.40). To calculate percentages, see the example below.

40 Students like multiple choice tests; 77 Students do not like multiple choice tests; and 30 Students are undecided (re: multiple choice tests) -- what is the percent of students that do not like multiple choice tests?

Add 40 + 77 +30 = 147 and divide 77 by 147 = 0.52 or 52%

The "A closer look at Graphs" take-away note card includes the following.

Definition: Correlation – The degree to which two or more attributes or measurements on the the same group of elements show a tendency to vary together.

[Conclusions to keep in mind...] Graphs distort comparisons when the:

• Baseline in the graph is not zero;

• Graph uses bars;

• Spacing of the points on the axes are either expanded or contracted too far;

• Time period selected to display is not representative; and

• Reader is unclear about the data used and/or the origin of the data.

The "A closer look at Comparisons" take-away note card includes the following.

Definition: Analogies – A comparison becomes reasoning by analogy when it is part of an argument. On one side of the comparison we draw a conclusion, so on the other side we say the conclusion should be the same.

Definition: Fallacy of composition – It is a mistake to argue what is true of the individual is therefore true of the group, or what is true of the group is therefore true of the individual.

[Conclusions to keep in mind...] To Evaluate Analogies:

• Is this an argument? What is the conclusion?

• What is the comparison?

• What are the premises (the sides of the comparison)?

• What are the similarities?

• Can we state the similarities as premises and find a general principle that covers the two sides?

• Does the general principle really apply to both sides? What about the differences?

• Is the argument strong or valid? Is it good?

The "A closer look at Generalizations" take-away note card includes the following.

Definition: Generalizing – To generalize is to conclude a claim about a group, the population, from a claim about some part of it, the sample. To generalize is to make an argument. Sometimes the general claim that is the conclusion is called the generalization; sometimes we call the whole argument a generalization. The claims about the sample are called the inductive evidence for the generalization.

Definition: Representative sample – A sample is representative if no one subgroup of the whole population is represented more than its proportion in the population. A sample is biased if it is not representative.

Definition: Random sampling – A sample is chosen randomly if at every choice there is an equal chance for any of the remaining members of the population to be picked.

Definition: Sample size – For a generalization to be good, the sample has to be big enough. Generalizing from a sample that is obviously too small is called a hasty generalization based on anecdotal evidence.

Definition: Well studied sample – For a generalization to be good, the sample has to studied well . If a generalization is based on questionnaire or survey data, the questions have to be constructed without bias. In addition, questions need to constructed to check for internal consistencies (read: measures of truthfulness).

Definition: The gambler’s fallacy – The gambler’s fallacy is to reason that a run of events of a certain kind makes a run of contrary events more likely in order to even up the probabilities.

[Conclusions to keep in mind...] Premises needed for a good generalization:

• • The sample size is representative;

  • The sample is big enough; and

  • The sample is studied well.

The "A closer look at Cause & Effect" take-away note card includes the following.

Definition: Causal claims – A causal claim is a claim that can be rewritten as “X causes (caused) Y.” A particular causal claim is one in which a single claim can describe the (purported) cause and single claim can describe the purported effect. A general causal claim is a causal claim that generalizes many particular causal claims.

Definition: Normal conditions – The obvious and plausible unstated claims that are needed to establish that the relation between purported cause and purported effect is valid or strong are called the normal conditions for the causal claim.

Definition: Post hoc ergo propter hoc – (“after this, therefore because of this”) is the fallacy of claiming that there is cause and effect just because one claim became true after another.

[Conclusions to keep in mind...] Necessary conditions for cause and effect:

• The cause and effect happened (are true).

• It is (nearly) impossible for the cause to happen (be true) and the effect not to happen (be false), given the normal conditions.

• The Cause precedes the effect.

• The cause makes a difference – if the cause had not happened, the effect would not have happened, given the normal conditions.

• There is no common cause.

Common mistakes in reasoning about cause and effect:

• Tracing the cause too far back – when we trace a cause too far back, the problem is that the normal conditions begin to multiply.

• Confusing cause with effect – if reversing cause and effect sounds just as plausible as the original claim, a further investigation of the evidence is needed.

• Looking too hard for a cause – when sometimes it is sometimes just a coincidence.