Sometimes I have seen textbooks refer to the second parameter in the normal distribution as the standard deviation and the variance. For example, the random variable X ~ N(0, 4). It’s not clear whether sigma or sigma squared equals 4. I just want to find out the general convention that is used when the standard deviation or the variance is unspecified.
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$\begingroup$ by default it is variance always. $\endgroup$– NeerajCommented Jul 1, 2018 at 5:03
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8$\begingroup$ @Neeraj: Can you back that up by some authoritative reference? $\endgroup$– kjetil b halvorsen ♦Commented Jul 1, 2018 at 11:51
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2 Answers
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There's a choice of parameterizations of the normal distribution, and none is inherently more correct. Sometimes one or another is more convenient, and arguably one or another is more conventional in a given situation.
From what I've seen, when statisticians* are writing algebraic formulas, the most common convention is (by far) $N(\mu,\sigma^2)$, so $N(0,4)$ would imply the variance is $4$. However the convention is not completely universal so while I'd fairly confidently interpret the intent as "variance 4", it's hard to be completely sure without some additional indication (often, careful examination will yield some additional clue, such as an earlier or subsequent use by the same author).
Speaking for myself, I try to write an explicit square in there to reduce confusion. For example, rather than write $N(0,4)$, I would usually tend to write $N(0,2^2)$, which more clearly implies that the variance is 4 and the sd is 2.
When calling functions in statistics packages (such as R's dnorm for one example), the arguments are nearly always $(\mu, \sigma)$. However, as usεr11852 points out, check the documentation! Of course in the worst case - missing or ambiguous documentation, unhelpful argument names - a little experimentation would resolve any dilemma about which it used.
* here I mean people whose primary training is in statistics rather than learning statistics for application to some other area; conventions can vary across application areas.
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11$\begingroup$ I would like to also add that any reasonable software package (R, MATLAB, etc.) explicitly defines what the input arguments are. There is no ambiguity there. (+1 obviously) $\endgroup$ Commented Jul 1, 2018 at 11:45
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1$\begingroup$ WinBugs is a notable exception to the std.deviation rule, but then any WinBugs user with more than five minutes' experience should know to look at the documented parameterisations! $\endgroup$– JDLCommented Jul 2, 2018 at 7:40
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From an earlier answer 7 years ago: ".... there are at least three different conventions for interpreting $X \sim N(a,b)$ as a normal random variable. Usually, $a$ is the mean $\mu_X$ but $b$ can have different meanings.
$X \sim N(a,b)$ means that the standard deviation of $X$ is $b$.
$X \sim N(a,b)$ means that the variance of $X$ is $b$.
$X \sim N(a,b)$ means that the variance of $X$ is $\dfrac{1}{b}$.
Fortunately, $X \sim N(0,1)$ means that $X$ is a standard normal random variable in all three of the above conventions! "
answered Jul 2, 2018 at 3:28
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$\begingroup$ It's more helpful if you list these in order of decreasing frequency $\endgroup$– smciCommented Jul 2, 2018 at 6:54
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$\begingroup$ @smci frequency according to what? the last is the rarest in my day-to-day experience, but if you're only doing work involving lengthscales/precision then I imagine it's more common (as noted in the comments on the cited answer, e.g.). $\endgroup$ Commented Jul 2, 2018 at 8:24
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$\begingroup$ Frequency according to how people generally use them $\endgroup$– smciCommented Jul 3, 2018 at 4:52
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$\begingroup$ @smci Some people use the first convention exclusively, some the second exclusively, and some the third exclusively. Others are more inclusive, using two conventions, and the ultra-liberal are OK with all three. The vast majority of people in the world are totally ignorant of all three conventions. As Glen_b says, lots of people use $N(\mu,\sigma^2)$ in writing text but $N(\mu,\sigma)$ when programming in R, and so each person's usage might vary from day to day. So, which frequency do you want? Your query doesn't make much sense to me. $\endgroup$ Commented Jul 3, 2018 at 13:52
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$\begingroup$ Dilip: we know that. The question is which convention is the most common? If 'most common' answer differs when the context is 'textbook' or 'literature' vs 'programming', that's fine to state as an answer. $\endgroup$– smciCommented Jul 3, 2018 at 14:24