I'm about to introduce the standard normal table in my introductory statistics class, and that got me wondering: who created the first standard normal table? How did they do it before computers came along? I shudder to think of someone brute-force computing a thousand Riemann sums by hand.
Solved – Who created the first standard normal table
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Related Solutions
Here is one made specially for you. Note that the density of a distribution symmetric about $0$ is the same for positive and negative values.
density cumprob
-3.5 0.0008726827 0.0002326291
-3.4 0.0012322192 0.0003369293
-3.3 0.0017225689 0.0004834241
-3.2 0.0023840882 0.0006871379
-3.1 0.0032668191 0.0009676032
-3 0.0044318484 0.0013498980
-2.9 0.0059525324 0.0018658133
-2.8 0.0079154516 0.0025551303
-2.7 0.0104209348 0.0034669738
-2.6 0.0135829692 0.0046611880
-2.5 0.0175283005 0.0062096653
-2.4 0.0223945303 0.0081975359
-2.3 0.0283270377 0.0107241100
-2.2 0.0354745928 0.0139034475
-2.1 0.0439835960 0.0178644206
-2 0.0539909665 0.0227501319
-1.9 0.0656158148 0.0287165598
-1.8 0.0789501583 0.0359303191
-1.7 0.0940490774 0.0445654628
-1.6 0.1109208347 0.0547992917
-1.5 0.1295175957 0.0668072013
-1.4 0.1497274656 0.0807566592
-1.3 0.1713685920 0.0968004846
-1.2 0.1941860550 0.1150696702
-1.1 0.2178521770 0.1356660609
-1 0.2419707245 0.1586552539
-0.9 0.2660852499 0.1840601253
-0.8 0.2896915528 0.2118553986
-0.7 0.3122539334 0.2419636522
-0.6 0.3332246029 0.2742531178
-0.5 0.3520653268 0.3085375387
-0.4 0.3682701403 0.3445782584
-0.3 0.3813878155 0.3820885778
-0.2 0.3910426940 0.4207402906
-0.1 0.3969525475 0.4601721627
0 0.3989422804 0.5000000000
0.1 0.3969525475 0.5398278373
0.2 0.3910426940 0.5792597094
0.3 0.3813878155 0.6179114222
0.4 0.3682701403 0.6554217416
0.5 0.3520653268 0.6914624613
0.6 0.3332246029 0.7257468822
0.7 0.3122539334 0.7580363478
0.8 0.2896915528 0.7881446014
0.9 0.2660852499 0.8159398747
1 0.2419707245 0.8413447461
1.1 0.2178521770 0.8643339391
1.2 0.1941860550 0.8849303298
1.3 0.1713685920 0.9031995154
1.4 0.1497274656 0.9192433408
1.5 0.1295175957 0.9331927987
1.6 0.1109208347 0.9452007083
1.7 0.0940490774 0.9554345372
1.8 0.0789501583 0.9640696809
1.9 0.0656158148 0.9712834402
2 0.0539909665 0.9772498681
2.1 0.0439835960 0.9821355794
2.2 0.0354745928 0.9860965525
2.3 0.0283270377 0.9892758900
2.4 0.0223945303 0.9918024641
2.5 0.0175283005 0.9937903347
2.6 0.0135829692 0.9953388120
2.7 0.0104209348 0.9965330262
2.8 0.0079154516 0.9974448697
2.9 0.0059525324 0.9981341867
3 0.0044318484 0.9986501020
3.1 0.0032668191 0.9990323968
3.2 0.0023840882 0.9993128621
3.3 0.0017225689 0.9995165759
3.4 0.0012322192 0.9996630707
3.5 0.0008726827 0.9997673709
Jacob Bernoulli (~1700) - John Arbuthnot (1710) - Nicolaus Bernoulli (1710s) - Abraham de Moivre (1718)
The case of Arbuthnot1 see explanation in note below, can also be read about in de Moivre's Doctrine of Chance (1718) from page 251-254 who extends this line of thinking further.
De Moivre makes two steps/advancements:
The normal approximation of a Bernoulli distribution, which helps to easily calculate probabilities for results being within or out a certain range. In the section before the example about Arbuthnot's case, de Moivre writes about his approximation (now called the Gaussian/normal distribution) for the Bernoulli distribution. This approximation allows to easily calculate a p-value (which Arbuthnot could not do).
Generalization of Arbuthnot's argument. He mentions that "this method of reasoning may also be usefully applied in some other very interesting inquiries". (which may give partial credit to de Moivre for seeing the general applicability of the argument)
According to de Moivre, Jacob Bernoulli wrote about this problem in his Ars Conjectandi. De Moivre names this in English 'Assigning the limits within which, by the repetition of experiments, the probability of an event may approach indefinitely to a probability given', but the original text by Bernouilli is in Latin. I do not know sufficient Latin to be able to figure out if Bernoulli was writing about a concept like the p-value or more like the law of large numbers. Interesting to note is that Bernouilli claims to have had these ideas for 20 years (and also the work 1713 was published after his death 1705 so it seems to precede the date 1710 mentioned in the comments by @Glen_b for Arbuthnot).
One source of inspiration for de Moivre was Nicolaus Bernouilli, who in 1712/1713 made the calculations for the probability of the number of boys being born is not less than 7037 and not bigger than 7363, when 14000 is the number of total born kids and the probability for a boy is 18/35.
(The numbers for this problem were based on 80 years of statistics for London. He wrote about this in letters to Pierre Raymond de Montmort published in the second edition (1713) of Montmort's Essay d'analyse sur les jeux de hazard.)
The calculations, which I did not quite follow, turned out a probability of 43.58 to 1. (Using a computer summing all terms probability of a binomial from 7037 up to 7363 I get 175:1 so I may have misinterpreted his work/calculation.)
1: John Arbuthnot wrote about this case in An argument for divine providence, taken from the constant regularity observed in the births of both sexes (1710).
Explanation of Arbuthnot's argument: the boy:girl birth ratio is remarkably different from the middle. He does not calculate exactly the p-value (which is not his goal), but uses the probability to get boys>girls 82 times in a row $$\frac{1}{2}^{82} \sim \frac{1}{4 \,8360\,0000\,0000\,0000\,0000\,0000}$$ arguing that this number would be even more small when you would consider that one could take a smaller range and that it happened more than in just London and 82 years, he ends up at the conclusion that it is very unlikely and that this must be some (divine) providence to counter the greater mortality among men to finally end up with equal men and women.
Arbuthnot: then A’s Chance will be near an infinitely small Quantity, at least less than any assignable Fraction. From whence it follows that it is Art, not Chance that governs.
Best Answer
Laplace was the first to recognize the need for tabulation, coming up with the approximation:
$$\begin{align}G(x)&=\int_x^\infty e^{-t^2}dt\\[2ex]&=\small \frac1 x- \frac{1}{2x^3}+\frac{1\cdot3}{4x^5} -\frac{1\cdot 3\cdot5}{8x^7}+\frac{1\cdot 3\cdot 5\cdot 7}{16x^9}+\cdots\tag{1} \end{align}$$
The first modern table of the normal distribution was later built by the French astronomer Christian Kramp in Analyse des Réfractions Astronomiques et Terrestres (Par le citoyen Kramp, Professeur de Chymie et de Physique expérimentale à l'école centrale du Département de la Roer, 1799). From Tables Related to the Normal Distribution: A Short History Author(s): Herbert A. David Source: The American Statistician, Vol. 59, No. 4 (Nov., 2005), pp. 309-311:
$$\vdots$$
But... how accurate could he be? OK, let's take $2.97$ as an example:
Amazing!
Let's move on to the modern (normalized) expression of the Gaussian pdf:
The pdf of $\mathscr N(0,1)$ is:
$$f_X(X=x)=\large \frac{1}{\sqrt{2\pi}}\,e^{-\frac {x^2}{2}}= \frac{1}{\sqrt{2\pi}}\,e^{-\left(\frac {x}{\sqrt{2}}\right)^2}= \frac{1}{\sqrt{2\pi}}\,e^{-\left(z\right)^2}$$
where $z = \frac{x}{\sqrt{2}}$. And hence, $x = z \times \sqrt{2}$.
So let's go to R, and look up the $P_Z(Z>z=2.97)$... OK, not so fast. First we have to remember that when there is a constant multiplying the exponent in an exponential function $e^{ax}$, the integral will be divided by that exponent: $1/a$. Since we are aiming at replicating the results in the old tables, we are actually multiplying the value of $x$ by $\sqrt{2}$, which will have to appear in the denominator.
Further, Christian Kramp did not normalize, so we have to correct the results given by R accordingly, multiplying by $\sqrt{2\pi}$. The final correction will look like this:
$$\frac{\sqrt{2\pi}}{\sqrt{2}}\,\mathbb P(X>x)=\sqrt{\pi}\,\,\mathbb P(X>x)$$
In the case above, $z=2.97$ and $x=z\times \sqrt{2}=4.200214$. Now let's go to R:
Fantastic!
Let's go to the top of the table for fun, say $0.06$...
What says Kramp? $0.82629882$.
So close...
The thing is... how close, exactly? After all the up-votes received, I couldn't leave the actual answer hanging. The problem was that all the optical character recognition (OCR) applications I tried were incredibly off - not surprising if you have taken a look at the original. So, I learned to appreciate Christian Kramp for the tenacity of his work as I personally typed each digit in the first column of his Table Première.
After some valuable help from @Glen_b, now it may very well be accurate, and it's ready to copy and paste on the R console in this GitHub link.
Here is an analysis of the accuracy of his calculations. Brace yourself...
$0.000001200764$ - in the course of $301$ calculations, he managed to accumulate an error of approximately $1$ millionth!
mean(abs(difference))
withdifference = R - kramp
:$0.000000003989249$ - he managed to make an outrageously ridiculous $3$ one-billionth error on average!
On the entry in which his calculations were most divergent as compared to [R] the first different decimal place value was in the eighth position (hundred millionth). On average (median) his first "mistake" was in the tenth decimal digit (tenth billionth!). And, although he didn't fully agree with with [R] in any instances, the closest entry doesn't diverge until the thirteen digital entry.
mean(abs(R - kramp)) / mean(R)
(same asall.equal(R[,2], kramp[,2], tolerance = 0)
):$0.00000002380406$
sqrt(mean(difference^2))
:$0.000000007283493$
If you find a picture or portrait of Chistian Kramp, please edit this post and place it here.