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Talk:Algorithms for calculating variance

Online algorithm in testing yields horrible results when mean=~0
Most all the tests I've seen of these algorithms add some unrealistic constant (i.e. 10^6 or larger) to the dataset to demonstrate that the suggested algorithm on this page is indeed better. I naively used this algorithm in my own work, to horrible effect. My dataset consists of a large number of discrete values, perhaps with the values -1, 0, or 1, and with an average usually between -1 and 1. I wrote the following simple test program to demonstrate the difference in results between what I'll call METHOD1 (using a running sum and sum of squares of the dataset) and METHOD2 (using a running computation of the average and the variance, which the current wiki strongly recommends).

And here sample output:

Note how the computed average using float's and method 2 fails to six digits accuracy before even 1 million trials, while method 1 using floats reproduces the double results all the way out to 17 million trials. — Preceding unsigned comment added by 173.219.85.45 (talk) 19:41, 19 October 2012 (UTC)

 * Method 1's sum += sample is exact (with sample values 1 and 0) until it has no effect, so method 1 fails catastrophically after about 18 million. Method 2's average += delta / numtrials is generally imprecise, and you can't expect single precision to be great here for numtrials in 6 or 7 digits, but it's not too far off on average, so even with single floats and these large numbers of samples the estimated variances are not too unreasonable. Dicklyon (talk) 21:18, 22 May 2018 (UTC)

Untitled
''Editorial comment: it is actually the first formula that has precision problems when dealing with limited-precision arithmetic. If the difference between measurements and the mean is very small, then the first formula will yield precision problems, as information will be lost in the (xi - µ) operation. There is no such loss of significance in the intermediate operations of the second formula. -- ScottMoonen''

''Editorial comment the second: in fact, the second formula is the one more commonly beset with problems. The first can have problems when the mean is very large relative to the variance, but this is relatively rare in practice and this problem also affects the second formula. Much more common is the situation where you have comparable mean and variance and a very large number of observations. In this case, the second formula will result in the subtraction of two very large numbers whose difference is relatively small (by a factor roughly equal to the number of observations). If you have one million observations, you lose roughly six significant figures with the second formula if you use ordinary floating point arithmetic. -- TedDunning''

''comment: The problem may occur To be precise we have to specify the instrument and the nature of the data. -- DickBeldin''
 * 1) when the deviations are very small relative to the mean or
 * 2) when they are small relative to the representational capacity of the arithmetic instrument (floating point computer, fixed point calculator, paper and pencil).

Ted, what do you mean by ordinary floating point arithmetic? Are talking about precision with repsect to register width, or is there an implicit reference to some "non-ordinary" scheme, that is, non-IEEE? Thanks.

I thought the reference was pretty explicit. I didn't mean, however, to imply the use of non-standard but still floating point arithmetic. I had in mind systems that were entirely different. A simple example is a system that uses rationals based on unlimited precision integers. Another system might be fixed point arithmetic. The first would not lose precision in the computation while the second would lose more or less precision than I quote depending on the values in question and the characteristics of the fixed point system. You could also imagine an optimizer that noted that you might be adding up large numbers of very small numbers and would re-order the addition to avoid most of the round-off errro by using log N temporary values to perform the addition. -- TedDunning

FWIW my favorite variance algorithm is the 'corrected two pass algorithm' of Chan, Golub, Leveque (American Statistician, 1983, page 242):

avg=0 for all i  avg=avg+data[i] end for avg=avg/n

sum=0 sum2=0 for all i   sum2=sum2+(data[i]-avg)^2 sum=sum+(data[i]-avg) end for var=(sum2-sum/n)/(n-1)

return var

AlexC

 * I added this algorithm (as Algorithm II), but without the sum in the second loop -- the sum of the variations from the mean is of course 0 ($$\sum_i (x_i-\mu)=\sum_i x_i-N\frac 1N \sum_i x_i=0$$). But perhaps this is a clever sort of compensation, where the errors in sum and sum2 are known to be (probably, perhaps) correlated, so that subtracting them is a benefit?  Still, though, for all datasets where $$\sigma^2 \not\ll \mu/N$$, $$\left|s\right|\ll s_2$$ (as in sum and sum2), so the gain in accuracy might be irrelevant.  And, if the second loop is meant to be the normal variance operation on preconditioned data, why isn't sum squared at the end?  Anyway, if it's a Good Thing that I've omitted, please tell me and/or add it to the article (with explanation!).  --Tardis 03:26, 9 June 2006 (UTC)

I removed the following comments from the article as they do not belong there. I have posted them here so that somebody may investigat them further. It would be nice if users use the talk page for comments and not the article. --Richss 16:12, Nov 11, 2004 (UTC)

 * Why is the second equation n(n-1)? I know two possibilities n (unbiased) or n-1 biased...See also the article Variance in wikipedia. This equations do not look good... --132.231.54.1

 * The above algorithm does not seem to be correct. The resulting variance does not equal either one of the variances whose formulas are given at the top of the page. --137.226.12.196

--- I had a look at the source for AlexCs algorithm. I believe there is a mistake: var=(sum2-sum/n)/(n-1) should read var=(sum2-sum*sum/n)/(n-1)

The result of this algorithm agrees with my graphic calculator. I think the corrected algorithm should be added to the page

-- aj504@york.ac.uk who still hasn't got a wikipedia account :)

Incorrect Algorithm?
The second algorithm doesn't look right to me. At each step it's calculating the variance of data[i] with respect to the *average so far*, not the true average.

Can someone confirm that I'm not just reading it wrong?

I have derived and added the update formulas (only the results) for the unbiased/biased variance. The clearly shown that the (old) second algorithm which was in dispute was wrong, so I removed it. I have also removed the example which just shows some numbers (was it a weak numerical proof?) and does not provide any insight on how the formulas work. It was not clear whether the algorithms were for biased or unbiased estimators, so I added comments on it. BurkayDonderici

Algorithm III does not compute
Hi, In the mathematical equation for m(new), I don't agree with (n+1) in the denominator. I believe that the denominator should be simply (n).

I didn't want to edit the entry since I am not a mathematician. CAN SOMEONE WHO IS PLEASE MAKE THE APPROPRIATE COMMENTS.

If you plug some numbers into the equation shown, and compare your results to what a spreadsheet calculator gives, you will see that there is a significant error. I tested the show equation in MS Excel. I used a population of 404 values generated by the RANDBETWEEN(0,1000) function. I used 4 methods to calculate the average at each point through the population. Method 1) (Cumulative total)/(Number of samples) [this calculation was done for each sample for tracking purposes Method 2) Previous average + new contribution. M(new) = M(old) + ( (M(old)-X(new))/n ) Method 3) Previous average + new contribution. M(new) = M(old) + ( (M(old)-X(new))/(n+1) ) Method 4) I then used the Average function on the total population.

Methods 1 and 2 produced the same results all the way through the population and matched Method 4's result for the whole population. However, Method 3 always produced a lower result, and the error (which started significatly) reduced as the number of samples grew.

Regards, Napoleon BlownApart

Algorithm III
I'm pretty sure this is wrong. From Welford's Paper:

S[n]=S[n-1]+(n-1)/n*(x[n]-mean[n-1])^2

--cfp 21:58, 15 August 2006 (UTC)

 * I'm talking rubbish it's fine as x[n]-mean[n] = (n-1)/n*(x[n]-mean[n-1]). Sorry! --cfp 22:20, 15 August 2006 (UTC)

--Gabriel (talk) 21:35, 17 July 2011 (UTC)

I think, the advantages of Welford's algorithm should be toned down. It is no more numerically stable than the direct two-pass method. See http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/ and http://www.jstor.org/stable/1267176.

Online algorithm
I removed the line about Knuth/Welford being on online algorithm. All the algorithms in this article are online algorithms, it's not worth noting. —johndburger 12:24, 18 September 2007 (UTC)
 * Well, algorithms I and III only scan the list of data once, whereas both versions of algorithm II scan it twice. Hence, since I/O is by several orders of magnitude slower than calculations, that means that, unless the whole data set is read into memory, algorithm II will be twice as slow, and it won't work if the data file is not seekable. --Army1987 16:00, 28 October 2007 (UTC)

The Welford algorithm *is* an on-line algorithm, because it needs to see only one data item at a time! It does *not* need to store the complete set of data, i.e., it does not need to look at any data item more than once. — Preceding unsigned comment added by Snoopy67 (talk • contribs) 21:30, 17 July 2011 (UTC)

 * The Welford algorithm as written calculates the sample variance in the finalize routine, and nowhere in the description does it specifically call that out. — Preceding unsigned comment added by 192.91.171.34 (talk • contribs)

Parallel algorithm
Is there an error in $$M_{2,AB}$$? Chan et al. in its paper has another relation for combining arbitrary sets $$A$$ and $$B$$: $$M_{2,AB} = M_{2,A} + M_{2,B} + \frac{n_A}{n_B n_{AB}} \cdot \left( \frac{n_B}{n_A} \bar x_A - \bar x_B \right)^2 $$, which yelds to $$M_{2,AB} = M_{2,A} + M_{2,B} + \frac{1}{n_{AB}} \cdot \left( \bar x_A - \bar x_B \right)^2 $$ if $$n_A$$ and $$n_B$$ are equal.

Algorithm IV
Doesn't the final step in West's algorithm have a typo? Instead of:

Variance = S * sumweight * n / (n-1)

Shouldn't it actually be:

Variance = S / sumweight * n / (n-1)

Looking at West's paper, he has "mathematical description" and "computational description" side by side. The math for the final line looks like: $$ S^2 = \frac{T}{(n-1)/n \sum w_i} $$ And if you work through it, $$T$$ should be the variance times the (incremental) weight for the recurrence to work. West's "computational description" is as its presented here, but I think its a typo in West. Shauncutts (talk) 06:15, 29 January 2008 (UTC)

 * Actually, I'm pretty sure the last line should be: Variance = S * n / ((n-1) * sumweight)

 * (I just implemented this algorithm based on the original paper, and have been manually checking results - I'm going to edit the main page to reflect this)

 * Gorm (talk) 12:28, 19 February 2008 (UTC)

Incidentally, would it not make sense to alter algorithm 4 by adding the temporary variables used in the original paper. This will save at least three operations per loop, at the cost of making it less readable. Specifically the algorith would then look like this:

n = 0 foreach x in the data: if n=0 then n = 1 mean = x      S = 0 sumweight = weight else n = n + 1 temp = weight + sumweight Q = x-mean R = Q*weight/temp S = S + sumweight*R*Q mean = mean + R      sumweight = temp end if end for Variance = S * n / ((n-1) * sumweight) // if sample is the population, omit n/(n-1)

Gorm (talk) 12:54, 19 February 2008 (UTC)

 * Agreed. I've now added the intermediary variables Q and R to the formula as per West's original paper The imp (talk) 10:35, 30 September 2009 (UTC)

Minor spelling and grammar fixes
OSFTactical (talk) 22:59, 6 December 2007 (UTC)

Suspecting a formulea error
On 17/11/2007 Hi,

I suspect that the two end formulaes given for the third algorithm :

variance_n = sqrt(M2/n) variance = sqrt(M2/(n - 1))

are not the "variance" but the "standard deviation" (Ecart-type in French)

Can someone check it ?

Pascal P. 90.37.121.130 (talk) 13:52, 17 November 2008 (UTC)

Division by zero problem?
The online algorithm should work with n=1 since it needs to progress through that stage to get to n=2, n=3, etc. Therefore, it would be incorrect to calculate the variance as

variance = M2 / (n-1)

A possible fix is:

if n > 1 then variance = M2 / (n-1) else variance = 0 end if

Does any one have any objections to me updating this? JBrusey (talk) 14:06, 9 April 2009 (UTC)

 * The sample variance is not defined for samples of size 1. So while it is good to avoid dividing by 0, the value 0 is wrong too. I suggest writing

if n > 1 then variance = M2 / (n-1) else variance = undefined end if
 * Btw, I don't understand your comment at the beginning since the problematic division is outside the loop. McKay (talk) 01:02, 10 April 2009 (UTC)
 * It should not say "else variance = 0". That implies the sample variance is 0 when n = 1.  That is incorrect.  The sample variance is undefined in that case. Michael Hardy (talk) 01:18, 10 April 2009 (UTC)

Easiest online algorithm yet.
Hi all,

I was looking at Murray Spiegel's Probability and Statistics (1994), and it lists this formulation for variance:

$$Var(X) = E[X^2]-(E[X])^2.$$

Mathworld had a similar formulation for standard deviation. This all means that, as long as you keep running tabs of:

- sum of all samples

- sum of squares of all samples

- number of samples

then you have an easy way to compute the variance as you go. You also have an easy way to combine the variances of two populations. Just add the sums (and sums of squares) and the counts. No divide-by-zeros. Very few math operations. What more can you ask for? :)

 * That is already in the article (the first algorithm), and the problem with it is stated there. McKay (talk) 02:14, 2 May 2009 (UTC)

 * Ah, well the connection between the two wasn't spelled out to the satisfaction of my (possibly caffeine-deprived) mind. And, as far as I can tell, the online algorithm does not easily merge statistics from multiple populations, but rather updates the statistic one sample at a time. Correct me if I am wrong, or please add a suitable method to the page. 209.237.23.36 (talk) 22:56, 13 May 2009 (UTC)

 * The connection to the second equation of the "naive algorithm" follows immediately from the definition of $$E[X^2]$$ and $$E[X]$$. The "parallel algorithm" is an online algorithm that easily merges statistics from multiple population (as noted in the text), though a better heading might make this clear. The key point is that a parallel algorithm can be implemented as a repeated serial algorithm, and the merging part is identical in both. Markjamesabraham (talk) 22:45, 26 May 2010 (UTC)

Pseudocode
The numerically stable mean and variance computation works, but attempting something similar for skewness and kurtosis using the incremental moments computation code in the later section seems to fail. Is something missing in it ? Are there any reference implementations ? Shyamal (talk) 16:17, 14 July 2009 (UTC)
 * I've implemented the Terriberry moment-based method for skewness and kurtosis in the incremental case (i.e. B={x}) and it worked fine. Markjamesabraham (talk) 22:36, 26 May 2010 (UTC)
 * Some published code is at http://lib.stat.cmu.edu/apstat/52, which also indicates its original source. Melcombe (talk) 12:13, 27 May 2010 (UTC)
 * Actually I subsequently found that Chan's online-merging algorithm does fine for higher-order moments, but calculates the mean very badly. When A and B have large and comparable size, $$\bar x_X = \bar x_A + \delta\cdot\frac{n_B}{n_X}$$ for $$\delta\! = \bar x_B - \bar x_A$$ is not as numerically stable as when one of A and B are very much larger than the other. The error contribution from the subtraction that forms $$\delta$$ is scaled only by about 1/2, whereas when $$n_A >> n_B = 1$$ the error contribution from the subtraction is divided by $$n_A + 1$$. Instead, prefer to use $$\bar x_X = \frac{n_A \bar x_A + n_B \bar x_B}{n_A + n_B}$$. Main page updated accordingly Markjamesabraham (talk) 14:54, 30 May 2010 (UTC)

Compensated variant
I cannot make sense of the "compensated" algorithm. It does not appear to be using ordinary Kahan summation. Bkalafut (talk) 01:19, 9 July 2010 (UTC)

Hey
Actually I just wanted to ask that on the page :Algorithms for calculating variance is the pseudocode that is given for calculating variance correct?

variance = E(X**2) - (E(X))**2 Computational formula for the variance

Should'nt it be mean*mean as opposed to Sum * mean?

Thanks...

```` —Preceding unsigned comment added by Nk28 (talk • contribs) 06:59, 12 October 2010 (UTC)

Pseudocode is valid Python, but gets tripped up by integer division
Would anyone object to a note pointing out that the pseudocode can be run verbatim in a Python intepreter, but that in Python < 3.0 you'll need a  statement. (Can you tell I just got caught out by this? :D) Alexsdutton (talk) 15:05, 23 December 2010 (UTC)
 * I think it would be inappropriate. The whole idea of pseudocode is that it's not language specific.  -- RoySmith (talk) 18:52, 25 December 2010 (UTC)

 * I'm torn. Perhaps the text should say it is Python 3.0? I am loathe to add that line to an otherwise beautiful looks-like-pseudocode example, but as you say, the gotcha is easy to fall into... —Ben FrantzDale (talk)

Divide by n vs n-1?
Can somebody explain why there are two different flavors of the formula for variance, one where you divide by n, the other where you divide by n-1? Obviously, for large values of n, they approach each other, but why the difference at all? -- RoySmith (talk) 18:55, 25 December 2010 (UTC)
 * Wikipedia's mathematics reference desk would probably be a better place for this sort of question.
 * Notice that
 * The rule that
 * var(X + Y) = var(X) + var(Y)
 * when X and Y are independent random variables applies when you divide by n and not when you divide by n − 1.
 * Dividing by n − 1 is done when one is trying to estimate the variance of a population by using a random sample. It should never be done when one has the whole population and the variance can be computed exactly.  It's effect is to make the estimate unbiased.  That's not really all that good a reason to do it, but that's why it's done.  See Bessel's correction for an account of some of the mathematical theory.
 * This is standard textbook material. Maybe more of it should be included in the article.
 * Michael Hardy (talk) 19:28, 25 December 2010 (UTC)
 * Thanks for the pointer to Bessel's correction, that explains this better than I've ever understood it before. And, yes, my intent in asking the question was not so much for my own knowledge (although that's part of it), but as a suggestion that something along these lines be included in this article.  -- RoySmith (talk) 03:28, 28 December 2010 (UTC)
 * I am confused by the distinction of sampling from a population of size N vs sampling from a smaller subset n<N. The formulas to compute the variance, and its sample estimate, have exactly the same form (for finite N and n). This probably results from some authors referring to "population variance" when actually meaning "variance of the distribution" from where the independent samples originate (for example see Navidi, William (2006) Statistics for Engineers and Scientists, McGraw-Hill, pg 14.).

 * The formulas in the article implicitly assume samples to be i.i.d random variables (Independent_and_identically_distributed_random_variables). The variance of such population of size K (either n or N) involves dividing by K. An unbiased estimate of this variance involves dividing by K if the mean of the distribution is known, or dividing by K-1 if the mean is also estimated from the data.

 * This is consistent with the articles:
 * Bessel's correction
 * Variance

 * And the references:
 * Knuth, Donald Erwin, Ronald L. Graham, and Oren Patashnik. "Concrete mathematics." Adison Wesley (1989), p373.
 * Cramér, Harald. "Mathematical Methods of Statistics". Princeton university press (1961), p351.

 * I believe the main article should be corrected to reflect the probabilistic nature of the population (i.i.d samples) and drop mention of N vs n.

Mpbrigham (talk) 02:35, 26 January 2017 (UTC)

Weighted variance pseudocode correctness
The weighted variance pseudocode seems correct to me; I just did a back-of-the-envelope check (but feel free to check it more thoroughly!). I've removed the misleading comment in the pseudocode (#[WARNING] This seems wrong. It should be moved before M2 assignment). This comment doesn't seem to make much sense; one can easily show that it yields the wrong result for the trivial case in which the weights are all equal to one. Theaucitron (talk) 11:07, 21 August 2012 (UTC)

Online algorithm for covariance
Is this algorithm right? I'm not an expert, but when I compare it to the online variance computation, it's different, and if I replace y with x, I should get cov(x,x) = var(x), so could anyone have a look please? 131.227.95.222 (talk) 18:03, 27 February 2013 (UTC)
 * Alright, I found a paper talking about this and now I understand that the formula is talking about co-moments and not covariances, I will add the note to the article. 131.227.95.222 (talk) 10:31, 28 February 2013 (UTC)

Kurtosis algorithm biased for normal distribution?
The wikipedia page for kurtosis claims that unbiased estimation of kurtosis is possible for normally distributed inputs. However, when I run the code my results are clearly biased.

Here's the code I'm running:

208.77.214.129 (talk) 21:02, 3 June 2013 (UTC)

Weighted Incremental Algorithm Alternative Incorrect
I did the math to see if the stated update formula is indeed correct with the alternative mentioned in the comment; I have found that it is not. As such, I propose the alternative comment be removed.

If anyone would like me to do show the math of why there are not equivalent, let me know. Basically, the alternative confuses the sumweight of the previous iteration and the sumweight of the current iteration (which is updated after M2 is calculated). — Preceding unsigned comment added by Anishpatel0 (talk • contribs) 13:11, 29 April 2014 (UTC)

If anyone else would confirm that I am right, I'll remove the alternative comment in the article. Anishpatel0 (talk) 13:14, 29 April 2014 (UTC)

Ok, boss. Here's your math. Really wonder what they teach you folks these days...

What we have

temp = weight + sumweight delta = x - mean R = delta * weight / temp mean = mean + R

What we want to show

sumweight * delta * R                    = weight * delta * (x−mean) sumweight * delta * R                    = weight * delta * (x−(x-delta+R)) sumweight * delta * delta * weight / temp = weight * delta * (x−(x-delta+R)) sumweight * delta * delta * weight / temp = weight * delta * (x−x+delta-R))       delta * weight * sumweight * delta / temp = weight * delta * (delta-delta * weight / temp)        weight * delta * sumweight * delta / temp = weight * delta * (delta-delta * weight / temp)        weight * delta * sumweight * delta / temp = weight * delta * delta * (1 - weight / temp)        weight * delta * delta * sumweight / temp = weight * delta * delta * (1 - weight / temp)        weight * delta * delta * sumweight / temp = weight * delta * delta * (1 - weight / temp)        weight * delta * delta * sumweight / temp = weight * delta * delta * (temp - weight) / temp        weight * delta * delta * sumweight / temp = weight * delta * delta * sumweight / temp

--204.120.71.252 (talk) 06:05, 16 July 2014 (UTC)

What language are the algorithms in?
There are algorithms in the article which appear to be in some computer language or logical/mathematical notation. What is the language? Its syntax differs from programming languages I studied back in the day. It is not self-evident and self-explanatory. A link to some article dealing with the conventions and assumptions of the language would be helpful. None of the examples in Algorithm or in Algorithm examples are quite like the ones here, whose syntax appears to require they start with "DEF" and end with "RETURN." These algorithms have "do loops" which do not include an index telling it how many times to execute the statement in the loop (such as "for i = 1 to n", or a test at the end of each loop execution to determine if all the data items have been processed. Edison (talk) 18:01, 26 May 2014 (UTC)
 * Looks like Python_(programming_language) to me -- RoySmith (talk) 00:56, 30 May 2014 (UTC)
 * Python - style pseudocode. The details you mention are omitted here to highlight the main points, and are omitted in practice in many programming languages. --Hroðulf (or Hrothulf) (Talk) 13:32, 23 July 2014 (UTC)

Why "naive?"
What makes the first algorithm "naive?" I see no explanation as to how it is lacking, or what rules are broken, or what complications are not taken into consideration. Edison (talk) 18:11, 26 May 2014 (UTC)

 * It is naive because it is the obvious application of the formula that defines variance.
 * As for things that are lacking, those are discussed further down the article. For example, it is highly vulnerable to underflow, loss of significance , and it could be more efficient.
 * Feel free to edit the article to make these things more clear.
 * --Hroðulf (or Hrothulf) (Talk) 13:27, 23 July 2014 (UTC)

Usually the term naive is used if it is the definition formula and its calculation using it is more laborious than doing it in other way. The formula called naive in the article is none of them. It is not the formula that defines variance and it is not laborious but simplifies a lot the calculation. The only problem with the formula is actually a problem of the float point arithmetic limitations. A small change by computing shifted data fixes the the problem. (see reference 2 T.F.Chan, G.H. Golub and R.J. LeVeque "3 Computations with shifted data"). In this article (ref 2) the formula (called textbook) is recommended in certain conditions and variations. So the use of the algorithm is far away from "should not be used in practice" as stated in the Wikipedia article. Elxala (talk) 23:17, 23 August 2014 (UTC)

Merge discussion
There was a merge tag added at Standard deviation to this article. I think it is quite reasonable to have that section be a summary of this article. I must admit I was rather surprised I couldn't find a version of the $$\operatorname{E}\left[(X-\operatorname{E}(X))^2\right] = \operatorname{E}\left[X^2 \right] - (\operatorname{E}[X])^2$$ formula in the standard deviation article, I've put a bit in its talk page about that and that that article is slow referring to variance and never links to it.

For this article as old speedup calculation method is at assumed mean. Dmcq (talk) 09:19, 17 May 2013 (UTC)

I think what is written here is much more clear than the article that the merge tag refers to. Hence I propose to leave it as is. — Preceding unsigned comment added by 194.105.120.70 (talk) 14:50, 11 July 2013 (UTC)

 * Oppose merge Too big for Standard deviation article per WP:SUMMARYSTYLE and qualifies as notable under WP:GNG per its references. Brycehughes (talk) 04:41, 9 November 2013 (UTC)

 * I think we all agree with that the Algorithms article is too big to merge into the Standard deviation one. The merge tag suggests a move in the other direction.

 * I think the sensible thing is to use Summary Style, as suggests. The detailed algorithms article need not change very much, but the section at Standard deviation should be rewritten as a summary of the algorithms article.

 * --Hroðulf (or Hrothulf) (Talk) 10:17, 24 July 2014 (UTC)

Algorithms by (uppercase roman) number mapped to names
Revisions of this page up to referred to algorithms by number. Next revision improved the titles like so:

 * I. Naïve algorithm
 * II. Two-pass algorithm
 * III. On-line algorithm
 * IV. Weighted incremental algorithm

Revision of further "corrected" titles by eliminating roman numerals. Today's revision of replaced roman numerical references from the text according to the above map. ale (talk) 10:32, 29 February 2016 (UTC)

Issues with the section called "Welford's online algorithm". 1. I don't see the point of these equations: {\displaystyle s_{n}^{2}={\frac {n-2}{n-1}}\,s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}=s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}-{\frac {s_{n-1}^{2}}{n-1}},\quad n>1} {\displaystyle \sigma _{n}^{2}={\frac {(n-1)\,\sigma _{n-1}^{2}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})}{n}}=\sigma _{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})-\sigma _{n-1}^{2}}{n}}.} {\displaystyle \sigma _{n}^{2}={\frac {(n-1)\,\sigma _{n-1}^{2}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})}{n}}=\sigma _{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})-\sigma _{n-1}^{2}}{n}}.} They are just scaled versions of the equation that follows. I recommend removal.

2. The comment "These formulas suffer from numerical instability, as they repeatedly subtract a small number from a big number which scales with n." is confusing. It looks to me like some of the formulas repeatedly add a small number to a large number. In this respect they are no worse than the equation that follows.

3. The advantage of the {\displaystyle M_{2,n}} recurrence is that it is simpler and more computationally efficient because there are fewer multiplications and no divisions.

4. The last two equations should be {\displaystyle {\begin{aligned}s_{n}^{2}&={\frac {M_{2,n}}{n-1}}\\[4pt]\sigma _{n}^{2}&={\frac {M_{2,n}}{n}}\end{aligned}}} ProfRB (talk) 18:51, 22 March 2019 (UTC)

 * 2. I think the comment "These formulas suffer from numerical instability, as they repeatedly subtract a small number from a big number which scales with n" is actually wrong. Maybe there is an accuracy issue, but I don't see why there should be an instability here. A reference would be most welcome. Natchouf (talk) 14:08, 20 March 2023 (UTC)

Typo in "Computing shifted data" section
The second shown formula in this section does not compute the population variance $$\sigma^2$$, it rather computes the sample variance $$s^2$$. This can easily be seen when comparing the (wrong) formula divisor $$n-1$$ with the respective divisor $$n$$ as used in the section "Naive algorithm" just above (where capital $$N$$ is used instead of $$n$$).

The comments in the corresponding code snippet below makes the situation a bit clearer. Maybe one could write the code explicitly as

... // for sample variance use variance = (Ex2 - Ex**2 / n) / (n - 1) // for population variance use // variance = (Ex2 - Ex**2 / n) / n ...

141.249.133.134 (talk) 06:17, 10 April 2024 (UTC)


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Demographics of Antigua and Barbuda

This article is a demography of the population of Antigua and Barbuda including population density, ethnicity, religious affiliations and other aspects of the population.

Population
According to the 2011 census the estimated resident population of Antigua and Barbuda was 86,295. The estimated population of is.

Ethnic groups
The population of Antigua and Barbuda, is predominantly black (91.0%) or mixed (4.4%). 1.9% of the population is white and 0.7% East Indian. There is also a small Amerindian population: 177 in 1991 and 214 in 2001 (0.3% of the total population). The remaining 1.6% of the population includes people from the Middle East (0.6%) and China (0.2%).

The 2001 census disclosed that 19,425, or 30 per cent of the total population of Antigua and Barbuda, reported their place of birth as a foreign country. Over 15,000 of these persons were from other Caribbean states, representing 80 of the total foreign born. The main countries of origin were Guyana, Dominica and Jamaica. Approximately 4,500 or 23 per cent of all foreign born came from Guyana, 3,300 or 17 per cent came from Dominica and 2,800 or 14 per cent came from Jamaica. The largest single group from a country outside the region came from the United States. Of the total of 1,715 persons, nine per cent of the foreign born, came from the United States while three per cent and one per cent came from the United Kingdom and Canada, respectively. Many of these are the children of Antiguans and Barbudans who had emigrated to these countries, mainly during the 1980s, and subsequently returned.

Other demographics statistics
Demographic statistics according to the World Population Review in 2019.

 * One birth every 360 minutes
 * One death every 720 minutes
 * One net migrant every Infinity minutes
 * Net gain of one person every 720 minutes

Demographic statistics according to the CIA World Factbook, unless otherwise indicated.

 * Population:
 * 99,287 (Feb. 2022 est.)

 * Nationality:
 * Noun: Antiguan(s), Barbudan(s)
 * Adjective: Antiguan, Barbudan

 * Ethnic groups:
 * African descent 87.3%, mixed 4.7%, Hispanic 2.7%, white 1.6%, other 2.7%, unspecified 0.9% (2011 est.)
 * note: data represent population by ethnic group

 * Age structure:
 * 0-14 years: 22.52% (male 11,243/female 10,871)
 * 15-24 years: 16.15% (male 7,891/female 7,961)
 * 25-54 years: 41.68% (male 18,757/female 22,167)
 * 55-64 years: 10.74% (male 4,693/female 5,848)
 * 65 years and over: 8.91% (male 3,736/female 5,012) (2020 est.)

 * Median age:
 * Total: 32.7 years Country comparison to the world: 106th
 * Male: 30.7 years
 * Female: 34.4 years (2020 est.)

 * Birth rate:
 * 15.3 births/1,000 population (2021 est.) Country comparison to the world: 113th

 * Death rate:
 * 5.63 deaths/1,000 population (2021 est.) Country comparison to the world: 177th

 * Total fertility rate:
 * 1.96 children born/woman (2021 est.) Country comparison to the world: 117th

 * Net migration rate:
 * 2.06 migrant(s)/1,000 population (2021 est.) Country comparison to the world: 49th

 * Population growth rate:
 * 1.17% (2021 est.) Country comparison to the world: 84th

The island of Antigua is home to approximately 97% of the population; nearly the entire population of Barbuda lives in Codrington
 * Population distribution:

 * Languages:
 * English (official), Antiguan creole

 * Religions:
 * Protestant 68.3% (Anglican 17.6%, Seventh Day Adventist 12.4%, Pentecostal 12.2%, Moravian 8.3%, Methodist 5.6%, Wesleyan Holiness 4.5%, Church of God 4.1%, Baptist 3.6%), Roman Catholic 8.2%, other 12.2%, unspecified 5.5%, none 5.9% (2011 est.)

 * Life expectancy at birth:
 * Total population: 77.6 years Country comparison to the world: 92nd
 * male: 75.4 years
 * female: 79.9 years (2021 est.)

 * Urbanization:
 * urban population: 24.4% of total population (2020)
 * rate of urbanization: 0.55% annual rate of change (2015-20 est.)

 * Dependency ratios:
 * Total dependency ratio: 45.3 (2020 est.)
 * Youth dependency ratio: 31.8 (2020 est.)
 * Elderly dependency ratio: 13.6 (2020 est.)
 * Potential support ratio: 7.4 (2020 est.)

 * Education expenditures:
 * 2.5% of GDP (2009) Country comparison to the world: 160th

 * Obesity - adult prevalence rate:
 * 18.9% (2016) Country comparison to the world: 113rd

Definition: age 15 and over has completed five or more years of schooling
 * Literacy:
 * Total population: 99%
 * Male: 98.4%
 * Female: 99.4% (2015)

 * School life expectancy (primary to tertiary education):
 * Total: 15 years
 * Male: 14 years
 * Female: 16 years (2012)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Demographics of Antigua and Barbuda


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Politics of Antigua and Barbuda

The politics of Antigua and Barbuda takes place in a framework of a unitary parliamentary representative democratic monarchy, wherein the sovereign of Antigua and Barbuda is the head of state, appointing a governor-general to act as vice-regal representative in the nation. A prime minister is appointed by the governor-general as the head of government, and of a multi-party system; the prime minister advises the governor-general on the appointment of a Council of Ministers. Executive power is exercised by the government. Legislative power is vested in both the government and the two chambers of the Parliament. The bicameral Parliament consists of the Senate (seventeen-member body appointed by the governor-general) and the House of Representatives (seventeen seats; members are elected by proportional representation to serve five-year terms).

Antigua and Barbuda has a long history of free elections, three of which have resulted in peaceful changes of government. Since the 1951 general election, the party system has been dominated by the Antigua and Barbuda Labour Party (ABLP), for a long time was dominated by the Bird family, particularly Prime Ministers Vere and Lester Bird. The opposition claimed to be disadvantaged by the ABLP's longstanding monopoly on patronage and its control of the media, especially in the 1999 general election. The opposition United Progressive Party (UPP) won the 2004 election, and its leader Winston Baldwin Spencer was prime minister of Antigua and Barbuda from 2004 to 2014.

The elections to the House of Representatives were held on 12 June 2014. The Antigua and Barbuda Labour Party government was elected with fourteen seats. The United Progressive Party had three seats in the House of Representatives. ABLP won 15 of the 17 seats in the 2018 snap election under the leadership of incumbent Prime Minister Gaston Browne.

Constitutional safeguards include freedom of speech, press, worship, movement, and association. Antigua and Barbuda is a member of the eastern Caribbean court system. The Judiciary is independent of the executive and the legislature. Jurisprudence is based on English common law.

Executive branch leadership
As head of state, King Charles III is represented in Antigua and Barbuda by a governor-general who acts on the advice of the prime minister and the cabinet.

Legislative branch
Antigua and Barbuda elects on national level a legislature. Parliament has two chambers. The House of Representatives has 19 members: 17 members elected for a five-year term in single-seat constituencies, and 2 ex officio members (president and speaker). The Senate has 17 appointed members. The prime minister is the leader of the majority party in the House and conducts affairs of state with the cabinet. The prime minister and the cabinet are responsible to the Parliament. Elections must be held at least every five years but may be called by the prime minister at any time.

There are special legislative provisions to account for Barbuda's low population relative to that of Antigua. Barbuda is guaranteed one member of the House of Representatives and two members of the Senate. In addition, there is a Barbuda Council to govern the internal affairs of the island.

Administrative divisions

The country is divided into six parishes, Saint George, John, Mary, Paul, Peter, and Phillip which are all on the island of Antigua. Additionally, the islands of Barbuda and Redonda are considered dependencies.

Judicial branch
Antigua and Barbuda is a member of the Eastern Caribbean Supreme Court. This court is headquartered in Saint Lucia, but at least one judge of the Supreme Court resides in Antigua and Barbuda, and presides over the High Court of Justice. The current High Court judges are Jennifer Remy and Keith Thom.

Antigua is also a member of the Caribbean Court of Justice, although it has not yet acceded to Part III of the 2001 Agreement Establishing a Caribbean Court of Justice. Its supreme appellate court therefore remains the British Judicial Committee of the Privy Council. Indeed, of the signatories to the Agreement, as of December 2010, only Barbados has replaced appeals to Her Majesty in Council with the Caribbean Court of Justice.

In addition to the Eastern Caribbean Supreme Court, Antigua and Barbuda has a Magistrates' Court, which deals with lesser civil and criminal cases.

Movements

 * Republicanism in Antigua and Barbuda

Political pressure groups and leaders

 * Antigua Trades and Labour Union
 * People's Democratic Movement

International organisation participation

 * Organisation of African, Caribbean and Pacific States
 * ALBA
 * Caribbean Community
 * Caribbean Development Bank
 * Community of Latin American and Caribbean States
 * Commonwealth of Nations
 * United Nations Economic Commission for Latin America and the Caribbean
 * Food and Agriculture Organization
 * Group of 77
 * International Bank for Reconstruction and Development
 * International Civil Aviation Organization
 * International Criminal Court
 * International Confederation of Free Trade Unions
 * International Red Cross and Red Crescent Movement
 * International Fund for Agricultural Development
 * International Finance Corporation
 * International Federation of Red Cross and Red Crescent Societies
 * International Labour Organization
 * International Monetary Fund
 * International Maritime Organization
 * Intelsat (nonsignatory user)
 * Interpol
 * International Olympic Committee
 * International Telecommunication Union
 * Non-Aligned Movement (observer)
 * Organization of American States
 * Organisation of Eastern Caribbean States
 * OPANAL
 * United Nations
 * United Nations Conference on Trade and Development
 * UNESCO
 * Universal Postal Union
 * World Confederation of Labour
 * World Federation of Trade Unions
 * World Health Organization
 * World Meteorological Organization
 * World Trade Organization

title: Politics of Antigua and Barbuda


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Telecommunications in Antigua and Barbuda

Telecommunications in Antigua and Barbuda are via media in the telecommunications industry.

Telephone
Telephones – main lines in use: 37,500 (2006)
 * country comparison to the world: 168

Telephones – mobile cellular: 110,200 (2006) (APUA PCS, Cable & Wireless, Digicel)
 * country comparison to the world: 177

Telephone system: domestic: good automatic telephone system international: 3 fiber optic submarine cables (2 to Saint Kitts and 1 to Guadeloupe); satellite earth station – 1 Intelsat (Atlantic Ocean)

Radio
Radio broadcast stations: AM 4, FM 6, shortwave 0 (2002)

Radios: 36,000 (1997)

Television
Television broadcast stations: 2 (1997) (including ABS-TV)

Televisions: 31,000 (1997)

Internet
Internet Service Providers (ISPs): Cable & Wireless, Antigua Computer Technologies (ACT), Antigua Public Utilities Authority (APUA INET)

Internet hosts: 2,215 (2008)
 * country comparison to the world: 140

Internet users: 60,000 (2007)
 * country comparison to the world: 158

Country codes: AG


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Antigua and Barbuda Defence Force

The Antigua and Barbuda Defence Force (ABDF) is the armed forces of Antigua and Barbuda. The ABDF has responsibility for several different roles: internal security, prevention of drug smuggling, the protection and support of fishing rights, prevention of marine pollution, search and rescue, ceremonial duties, assistance to government programs, provision of relief during natural disasters, assistance in the maintenance of essential services, and support of the police in maintaining law and order.

The ABDF is one of the world's smallest militaries, consisting of 245 personnel. It is much better equipped for fulfilling its civil roles, as opposed to providing a deterrence against would-be aggressors or in defending the nation during a war.

Organisation
The ABDF consists of five major units:
 * Regiment – comprises four line companies and is the infantry unit and fighting arm of the defence force.
 * Service and Support Unit – provides administrative, logistic and engineer support to the rest of the defence force.
 * Coast Guard – the maritime element of the defence force, and is divided into four units:
 * Commanding Officer's Office
 * Engineer Unit
 * Administration Unit
 * Flotilla – the flotilla is the operational part of the Coast Guard, and consists of the following water-craft:
 * 1 Swiftships Shipbuilders 19.81-metre Commercial Cruiser-class patrol boat (P-01 Liberta), in service since 1984
 * 1 SeaArk Boats Dauntless-class patrol boat (P-02 Palmetto), in service since 1995
 * 1 Point-class cutter (P-03 Hermitage), transferred from the U.S. Coast Guard in 1998
 * 1 Defender 380X-class all-weather interceptor (D-8), date of acquisition unknown
 * 2 Boston Whaler 8.23-metre launches (071 and 072), in service since 1988
 * 1 Zodiac Marine & Pool 8.23-metre Hurricane-type rigid-hulled inflatable boat in service since 1998
 * Air Wing
 * Cadet Corps

Defence Board
The Defence Board is in charge of overseeing the leadership, management, and discipline of the Antigua and Barbuda Defence Force as well as all other matters pertaining to it. The Defence Board has the authority to govern its work, how it will carry out its duties, and the responsibilities of its members. It can also assign any member of the Board any authority or duty, by publishing a notice in the Official Gazette; consult with non-members as appropriate, including officers commanding units of the Force, regarding matters pertaining to their units; and the officers are required to attend the meetings as the Board requests; decide on the protocol to be followed in conducting its business; and provide for any other matter that it deems necessary or desirable for achieving the better performance of its functions.

Officers
A person cannot be appointed to the force unless they have received a recommendation from a board, known as the Commissions Board in the Defence Act. This board is made up of the Chairman, who is appointed by the Chief of Defence Staff; the Chairman of the Public Service Commission, or in his absence, the Vice Chairman of the Public Service Commission; and an individual appointed by the Defence Board for a duration determined by the Board. His Majesty has the authority to appoint people to the Antigua and Barbuda Defence Force, and the Governor-General may act in that capacity. A commission may be awarded for a predetermined amount of time or for an unlimited amount of time.

Reserve forces
There are two classes in the Antigua and Barbuda Defence Force Reserve. The soldiers enlisted, deemed to be enlisted, or re-engaged in accordance with this Part for service in that class; the Reserve soldiers of the second class who have, upon written application to the appropriate military authority, been accepted by that authority for service in the first class; and the soldiers transferred to the first class in accordance with section 31 of the Defence Act comprise the first class. The soldiers who are members of the second class by virtue of Part IV of the Defence Act, the officers who are appointed or transferred to that class, and the soldiers who are enlisted, deemed to be enlisted, or re-engaged in accordance with Part IX of the Defence Act for service in that class are all included in the second class.

Each officer and soldier in the first class of the Reserve is required to report for duty at the location and for the duration determined by the Defence Board. They are also required to meet all training-related requirements. Subject to any general directives from the Defence Board, the requirements of the related section may be disregarded in whole or in part with regard to any unit of the first class of the Reserve, as well as with regard to any individual officer or soldier of the first class of the Reserve, by his commanding officer. Nothing stops a Reserve officer or soldier from participating in optional training in addition to any mandated training.

The first class of the Reserve, or as many officers and soldiers of that class as the Board deems necessary, may be called out on temporary duty by the Defence Board when needed. Under section 203, officers and soldiers who are called out for duty are not required to serve for more than ninety days at a time. Because the worker is a Reserve member who is called out in accordance with the section, no employer may fire or give notice of termination to any worker. When an employer violates, they are guilty of a crime and face a maximum fine of $5,000, a maximum sentence of two (2) years in jail, or both. This applies even in cases of summary conviction.

The Governor-General may, on the advice of the Prime Minister, by proclamation order that the Reserve, or any class thereof, be called out on permanent service in the event of a state of war, insurrection, hostilities, or public emergency; the Defence Board will then take appropriate action. When called out on permanent duty, every Reserve officer and soldier is eligible to stay on permanent duty until told otherwise. Every officer and soldier belonging to such a class, as the case may be, to the part of any class so called out, shall attend in person at the designated location whenever the whole or any part of the first class of the Reserve is called out on temporary or permanent service; or whenever the whole or any part of the second class of the Reserve is called out on permanent service. The Defence Board shall cause every officer or soldier subject to such call-out to be served with a notice requiring him to attend at the time and place specified in the notice in the event that the first class of the Reserve is called out on temporary service or the Reserve is called out on permanent service.

An officer of the Antigua and Barbuda Defence Force may file a summarily complaint against a person who willfully takes away pawns, wrongfully destroys or damages, carelessly loses anything issued to him as an officer or soldier of the Reserve, or willfully refuses or fails to deliver up anything issued to him as an officer or soldier as a debt owed to the Crown. This applies even if the amount exceeds the normal monetary limit on a magistrate's jurisdiction.

A Reserve soldier may be released at any point during the duration of their service in the Reserve by the appropriate military authority under the prescribed conditions.

Enlistment and terms of service
A recruiting officer will not enlist anyone in the Force unless they are satisfied that the potential recruit has received, comprehended, and wishes to be enlisted after receiving a notice in the prescribed form from anyone who wants to join the regular Force. A recruiting officer is not allowed to enlist a minor under the age of eighteen into the regular force unless written consent has been obtained from at least one parent, from any parent who may have parental rights and powers over the minor, from any person whose whereabouts are known or can be determined with reasonable effort, or from any person who is in fact or legally responsible for the minor. For the purposes of the Defence Act, an individual who is willing to enlist will be considered to have reached, or not reached, the age of eighteen if the recruiting officer is satisfied—either through the production of a certified copy of an entry in the register of births, or by any other evidence that seems sufficient to them. The period of time that an individual may enlist in the regular force is as follows: in the case of an individual who is 18 years of age or older at the time of enlistment, a term of colour service that does not exceed 12 years as prescribed; in this case, the term of service in the Reserve will apply to the portion of the term that is prescribed as a term of colour service, and the remaining portion will apply to a term of service. In the case of an individual who is younger at the time of enlistment, the period of time that is prescribed will begin on the date that the individual reaches the age of eighteen years or a term not to exceed 12 years, as specified, starting on the day he reaches that age and consisting of the portion of it that is prescribed as a term of colour service and the remaining portion as a term in the reserve.

Any soldier of good character who has served their full term of colour service or who will be serving in the Reserve in two years may, with permission from the appropriate military authority, re-engage for an additional period of colour service and Reserve duty as prescribed; however, the additional period of colour service, when combined with the original period of colour service, may not exceed a total continuous period of 22 years colour service from the date of the soldier's original attestation or the date he became eighteen years old, whichever comes first. Any soldier who has completed 22 years of colour service may, if he so chooses and with the consent of the appropriate military authority, continue to serve in all capacities as if his colour service term had not yet expired. However, on the date that he notifies his commanding officer that he wishes to be discharged, he may be eligible for discharge at the end of the three-month period. Any soldier whose term of colour service expires during a public emergency, war, insurrection, hostilities, or other exigency of duty may be kept in the Force and have their service extended for an additional period of time as directed by the Defence Board and the appropriate military authority.

Every regular force soldier who is eligible for discharge will be released as soon as possible, subject to the Defence Act. However, until they are released, they are subject to military law as stipulated by the Act. When a regular force soldier is serving outside of Antigua and Barbuda at the time of his discharge, he has two options: if he wants to be discharged in Antigua and Barbuda, he will be sent there at no cost as soon as possible, and he will be released either when he arrives in Antigua and Barbuda or, if he agrees to a delay in discharge, six months after arriving; else, he will be released where he is currently serving. Releasing a soldier from the regular force requires authorization from the appropriate military authority, unless they are being released in accordance with a court-martial sentence. Upon their release from active duty, all regular force soldiers will receive a certificate of discharge that includes the necessary information. Every regular Force soldier who is scheduled to be transferred to the Reserve may do so in accordance with the Act, but they will remain subject to military law until they are transferred. When a regular Force soldier serving outside of Antigua and Barbuda is scheduled to be transferred to the Reserve, he will be sent there at no cost to him and will be transferred to the Reserve upon arrival, or within six months of his arrival if he agrees to a delay in transfer; he may, however, choose to be transferred to the Reserve without having to return to Antigua and Barbuda. When a regular Force soldier is being transferred to the Reserve, the appropriate military authority has the authority to immediately discharge him without providing a reason.

A court-martial sentence, a Defence Board order, or an order from an officer not lower than Major or a corresponding rank who has been given permission by the commanding officer may be the only ways that a warrant officer or non-commissioned officer's rank may be lowered. If a regular force warrant officer is demoted to the ranks, he or she may request to be discharged, barring a state of war, insurrection, hostilities, or public emergency. Anytime during the duration of the soldier's term of engagement, the competent military authority may release a regular force soldier for the prescribed reasons. Regular Force soldiers have the right to request their discharge at any point within three months of the date of their first attestation. If they do so, they will be released as soon as possible after paying a sum not to exceed EC$500. However, they will still be subject to military law until their discharge under the Act. A soldier of the regular Force who was a member of a Commonwealth Force at any point within three months prior to the date of his first attestation is exempt from this.

Former deployments

 * In 1983, fourteen men of the Antigua and Barbuda Defence Force were deployed to Grenada during the Operation Urgent Fury.
 * In 1990, twelve soldiers were sent to Trinidad and Tobago after a failed coup attempt by a radical group against the constitutionally elected government headed by Prime Minister A.N.R. Robinson.
 * In 1995, members of the Antigua and Barbuda Defence Force were deployed in Haiti as a part of Operation Uphold Democracy.

Current deployments

 * Multinational Security Support Mission in Haiti

Alliances

 * 🇬🇧 – Mercian Regiment

title: Antigua and Barbuda Defence Force