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If I have a coin that is not necessarily expected to turn up heads half of the time, then is it correct to call the coin biased? Or does bias in statistics only mean the bias of an estimator in which case it would be inappropriate to describe the coin itself as biased?

This question does not mean to be about issues like this, so maybe a different example would have been better.

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    $\begingroup$ A coin that turns up Heads with probability $\frac{1}{2}$ is often called a fair coin, so I suppose you could call your coin an unfair coin (or a coin that is not fair if you don't want to sound pejorative) if you wish to reserve the adjective biased for estimators only. $\endgroup$ Jan 11, 2012 at 22:26
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    $\begingroup$ Interesting fact: it's really hard to make an unfair coin. We did an experiment for first year stats, were we made a bunch - one cupped, one bent (about 20°), one with lots of holes drilled in one side (ie. reduced weight). After 200 flips, none of them came up with a p<0.05, although one was close (I think the bent one). You're better off learning to flip accurately, if you want to win at coins, and even then, you have to flip low, with few rotations, and it's gonna be pretty obvious what you're doing to anyone watching. $\endgroup$
    – naught101
    Apr 17, 2012 at 5:36
  • $\begingroup$ Technically no asymmetric coin (e.g. one that has heads and tails) is likely to be unbiased, as this requires the probability of a head being exactly 0.5. So the test for the un-biasedness of a coin is one where the null hypothesis is known to be false from the outset, and if you toss the coin enough you will always be able to demonstrate that it is biased, no matter how small the "effect size". But while we can't have an unbiased coin, it is (as naught101 suggests - +1) difficult to make a significantly biased coin, without it having heads on both sides. $\endgroup$ Jan 16, 2013 at 10:52

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From my connotation the coin is the data generating process (DGP) therefore it cannot be biased - but rather be unfair as Dilip Sarwate noted correctly. The expected value of the estimator should match the true value of the DGP if it does not we call it biased.

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I think it depends on the audience. In everyday language, we would call a coin "biased"; it's only in a strict statistical sense that bias means something else.

It's one of many terms that statisticians use to mean something slightly different from what it means to everyone else.

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    $\begingroup$ So if you were writing a textbook titled An Introduction to Probability for Scientists and Engineers, it would be a biased coin, while if the book were titled Statistics: An Introductory Course it would be an unfair coin? I wonder what terminology the numerous Probability and Statistics for Scientists and Engineers books use. $\endgroup$ Jan 12, 2012 at 15:46
  • $\begingroup$ +1; @DilipSarwate, a good book would have a couple of sentences that would introduce / explain the terminology and differentiate it from similar--but not identical--colloquial uses that might otherwise lead to confusion. (I don't know how many texts meet my definition of 'good', though.) $\endgroup$ Jan 12, 2012 at 17:35

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