54

Anscombe's quartet is pretty good: All four of these sets have almost identical mean and variance for both x and y coordinates, correlation, and best-fit linear regression. But they're obviously very different!


52

Here are two well known examples: If someone tests positive for a rare disease (say its prevalence is 1 out of 100,000) with a test that has a 1% false positive rate, it is tempting to say that we are 99% sure they have that disease. This isn't true if you go through the numbers; they probably don't have that disease and are a false positive. (Bayes) If you ...


48

Actually, your definitions are backwards: the standard deviation is the square root of the variance. In other words, one defines variance first --- it has a simpler formula, and it has simpler mathematical properties. Then one defines the standard deviation as the square root of the variance. As for why to study it, there are many mathematical formulas ...


40

A book I remember has the title "the egg-laying dog". The titular dog enters a room where we placed 10 sausages and 10 eggs. After a while the dog leaves the room, and we observe, that the percentage of eggs relative to the sausages increased, so we conclude that the dog must have produced eggs. It's easy to spot the mistake in the above example, because ...


38

Sally Clark (http://en.wikipedia.org/wiki/Sally_Clark) was convicted in the UK of murdering both her infant sons, when in fact it is much more likely that they died of natural causes. The case against her was largely based on invalid statistical reasoning. The Royal Statistical Society made a statement about at at the time, which begins as follows: In ...


33

Simpson's paradox: see http://en.wikipedia.org/wiki/Simpson%27s_paradox. To summarize the Berkeley Admissions example: in 1973, 43% of men applying to graduate school at Berkeley were admitted, but only 35% of women. But, broken down across the six departments, women either did better than men, or the difference was not significant. The paradoxical result ...


24

Firstly, don't forget that your student has thought hard to come up with his answer and to be told it is wrong may be taken as invalidating his effort, or even insulting his intelligence. This might be at least a small part of his resistance to accepting your response. I would have recommended starting by asking him to explain more about his thinking, and ...


22

Percentages are a source of many, many, many common mistakes. One that is very common is believing that percentages can be added. An example: one of our presidents increased its salary by 172%; the next president decreased the presidential salary by 30%. It was commented that compared to the salary before the raise, it was still a 142% increase. Another ...


21

Multiple hypothesis testing is a common one. Let's say you run a study where you try to link some genetic marker to cancer rates. You look at perhaps 80 different genes and see if any of them have a correlation with occurrence of cancer. Lo and behold, one does! With p-value = 0.03! You conclude that there is a strong correlation (and seek to prove ...


20

Just two (now three, see below), to whet the appetite. Stating the mistakes: "Correlation implies Causation": it doesn't. The finding of statistical correlation between two variables may strengthen a pre-existing theoretical/logical argument of existing causal links. But it may also reflect the existence of an underlying third variable that affects both ...


19

To properly understand and appreciate the different rounding rules, one really needs to have some grounding in statistics and probability theory. It is a little hard to describe the full theory in detail (the least being that I don't have a entirely coherent formulation of the theory off the top of my head), but here are some examples: Naively if we ...


17

An analogy: speed is to standard deviation, as kinetic energy is to variance. Energies can be added usefully; speeds can only be added in very limited circumstances. Similarly, variances can be added across data points, or across alleged sources of the total variance; standard deviations cannot be directly added. When analysts say things like, "This ...


17

One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of numerators to sum of denominators is $\frac12$. However, we can also write $a=\frac03$ and $b=\frac22$. Now the ratio is $\frac25$, which is not equal to $\...


16

This semester I adopted Linear Algebra by Jim Hefferon for the junior-level linear algebra course. I didn't have any particular difficulty adopting it, the bookstore was able to arrange printing through some print shop and it was fairly easy given the free availability of the text. I'm not sure how many students actually purchased a copy. Once nice thing ...


15

Sometimes extreme sample bias. Here is an example (numbers made up, but realistic): In some country with a population of 100 million people, every year 100 people are bitten by poisonous snakes and 50 of these die. Every year 50 people are given treatment against snake bites, and 10 of these die (40 die without getting treatment). Your chances of dying ...


15

Number Sense: At the elementary level, estimation helps students to develop number sense. As Daniel R. Collins notes, order of magnitude estimates can be quite important. Anecdotally, I once rented an apartment. I moved out of the apartment about halfway through the month, and the landlord offered to prorate the rent. I said "Great!" However, when she ...


13

While I haven't used an open-source text for a semester-long university class, I have used an open-source calculus text for a summer program with high school students, and I'm currently modifying a number-theory text for a different summer program. Summary, with extended discussion below: Upsides: I can modify the text to suit my needs. I can modify the ...


13

I don't see any studies of this sort on prime numbers, though I'm sure you could conduct an informal one and get a good estimate relatively quickly. Instead, I tackle your final note: A good answer would be numerical data about this question or a similar one. How about: Is zero even? Citing a popular media piece: According to Dr James Grime of the ...


13

I've been teaching introductory statistics for the same amount of time at a large urban community college. I have never had this response from a class in toto. Last semester I did have one student say that privately. Every semester I definitely have sharp students who do in fact "get it". I don't know exactly what may cause this, but here's a few tidbits ...


13

It actually depends on exactly what you're asking. Or even what you SHOULD be asking. If you want the average profitability of all the 500+ operators in the Permian, you could just average all the profit margin percentages. This is taking the ratios (profit/revenue) for each company and averaging them. It corresponds to your expected (mean) profit margin ...


12

This is a preliminary, not about a distribution of means: How about having every student measure your desk? Ask them to keep their measurements secret, so they don't influence each other (an interesting issue in itself). Then use their answers as a data set. If you have enough students, you should get a normal distribution, whose mean is the proper answer. (...


12

Opening section of this document refers to a number of studies of numeracy of US population. One of the major international efforts in providing standardized, comparable-across-countries data on adult literacy and numeracy skills is OECD's Programme for the International Assessment of Adult Competencies (PIIAC). The US country report is here; I know some of ...


12

I like guest's answer. To elaborate, here is a possible question to ask them. You take two trips in your car: Trip 1 is a 100 mile drive that takes you 2 hours. Trip 2 is a 200 mile drive that takes you 1 hour. (a) What is the average speed of your car? (b) What is the average speed on an average trip? The answer to (a) is $\frac{...


11

DASL (pronounced "dazzle" and short for Data And Story Library) is an online collection of stories with matching data sets to be used for educational purposes. They are real data from real research. Searchable by statistics concept and by theme of the story. OzDASL is similar, but most of the data has an Australian or New Zealandish source. Personally I ...


10

The question we pose to students is: How far away, on average, are these values from their mean? The "natural" way to answer that question is to compute the deviations of the individual data points and compute the average. When students do this they are usually surprised to find out that the answer comes out to be precisely $0$ — that the positive ...


9

If a coin is biased to land heads with probability $p$ and $(a,b)$ is a $95\%$ confidence interval for $p$ then $p$ is in $(a,b)$ with probability $95\%$. Added in edit - While it is often argued that the difference is only philosophical, this distinction is of huge practical importance because the latter phrase is usually interpreted as if $p$ were the ...


9

Unfortunately, it is in German, but the book Angewandte Statistik: Eine Einführung für Wirtschaftswissenschaftler und Informatiker by Kröpfl, Peschek and Schneider contains many typical mistakes that you can make. My favorite example is that you can show a strong geographic correlation in Germany between the number of stork nests and the number of newborn ...


9

In my experience, there are a few key things that students misunderstand around the central limit theorem. Firstly, many students don't actually have a concept of distribution in the first place. They need the idea that anything you could record about an object/person/group has various possibilities for what the outcome could be, some of which might be more ...


9

If you succumb to the temptation of ejecting, say, a 5-sigma outlier from a n=10 sample taken from what you believe to be a normally distributed source, then you are discarding 50% of the sample's information content. Not so harmless. EDIT: I'll give it a go: Low-probability events carry more information (a.k.a. surprisal) than high-probability events. E.g....


9

Ask whether they think putting the marbles in a bag into any particular arrangement would affect the outcome. If they're ok with this, have them consider the arrangement of Bag 2 where the marbles are arranged in 10 wbb groups. They're likely to agree that the hand now has 10 equivalent groups to pick from, and that each group yields the same odds of getting ...


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