Why use Table for Normal Distribution I've recently learned about the Normal Distribution. I had seen this Probability distribution before in Calculus, where we just used a calculator for numerical integration. But now, in the Probability course I was told the actual way of solving the probability is using the cdf of the Standardization and using a table. Is there any reason why this is the "correct" way? Why use a Table when you can just use calculator to get the value of the Integral?
 A: This generation of statistics students and texts may be the last to deal with printed tables. Because printed normal tables are for the standard normal distribution one must standardize in order to use them and then lose some accuracy due to rounding errors. The body of most printed tables gives probabilities rounded to four places, which usually introduces no substantial error; but z-values on the margins are given only to two-place accuracy, which may lead to noticeably inaccurate answers.
Example: Scores on the ABC college admissions test are distributed according to $\mathsf{Norm}(\mu = 200, \sigma =25).$ State U wants to admit only students scoring in the top 15% on the ABC. What
minimum acceptable score should they publish?
Using R to solve this problem, which is similar to using a statistical calculator, gives the answer $226$ easily and directly. [In R, qbinom is the inverse CDF or 'quantile' function.]
qnorm(.85, 200, 25)
[1] 225.9108

Using tables, we need to standardize:
$$ 0.85 = P(X \le c) = 
P\left(\frac{X-\mu}{\sigma} \le \frac{c-200}{25}\right) =
P\left(Z \le \frac{c-200}{25}\right),$$ where $Z$ is standard normal.
Then we search the body of a printed table to find the number closest to 0.85 (0.8508, or a number equivalent, depending on the style of the table) and the corresponding number in the margins
is $1.04$  Thus $(c-200)/25 = 1.04$ and $c = 226.$
To be sure, there situations in which standardization is worthwhile for reasons other than using printed tables, but
one wonders what truly usefult topics might have been covered
while students puzzle over the use of printed normal tables.
Some "old-fashioned" instructors may regret the richly deserved
obsolescence of printed normal tables, but they will retire in due course.
[I finished high school before even basic calculators were available and was forced to learn how to extract the square root by longhand, a process that makes long division by hand seem a pleasure. I recall the hand wringing whether math would survive
without the character-building regimen of doing square roots by
longhand. Math education may be in trouble, but I think not specifically because of that.]
Moving one step beyond exact normal computations, I will end
with an example using the normal approximation to binomial.
Example: I roll a fair die 60 times, what is the probability
I will see fewer than 9 sixes?
That is $X \sim \mathsf{Binom}(60, 1/6),$ find $P(X < 9).$
In R the exact answer is easily obtained, as it is on many statistical calculators:
pbinom(8, 60, 1/6)
[1] 0.3120469

By normal approximation $\mu = np = 60(1/6) = 10, \sigma =
\sqrt{10*5/6} =  2.886751,$ With continuity correction this
is $P(X^\prime < 8.5)$ for $X^\prime \sim \mathsf{Norm}(\mu=10, \sigma=2.8869).$
pnorm(8.5, 10, 2.8868)
[1] 0.3016689

Standardization and use of printed normal tables will give
roughly the same answer.
Although the usual rules of thumb
for using a normal approximation apply here, we do not get
even the two-place accuracy one ordinarily expects. The reason
is that the so-called best-fitting normal distribution
does not match the binomial distribution as closely
as one might like.
It is often useful to know about the
normal approximation to binomial distributions, but it would
be a pity to use standardization and printed normal tables, only
to get a pretty good approximation of normal curve of mediochre fit.
k = 0:30;  pdf = dbinom(k, 60, 1/6)
hdr = "BINOM(60, 1/6) with Normal Approx"
plot(k, pdf, type="h", xlab="x", main=hdr)
 curve(dnorm(x, 10, 2.8868), add=T, col="blue") 
 abline(h=0, col="green2")


