Is it acceptable to make a linear approximation of z scores between values in normal tables?

Question

If I want to find the probability for a $z$ score of say $1.405$ and my printed normal tables tables only give me $z_{(1.40)}=0.9192$ and $z_{(1.41)}=0.9207$, is it acceptable to do: $z_{1.405}\approx \frac{0.9207+0.9192}{2}=0.91995$?

In this example the three numbers all round to $0.92(2sf)$ but this is not always the case.

In some books the $1.405$ is first rounded up to $1.41$ and hence $z_{1.41}=0.9207$ is taken.

But we know than $z_{1.405}$ is neither of and lies between $0.9192$ and $0.9207$. And although we know that $z_{x}$ does not change linearly with $x$ is it reasonable to make this approximation when no other means are available?

Of course one could check online for more detailed tables or tackle the integration of the probability function for the desired result but if one does not have such options is the method above reasonable?

Answer 1

Yes, linear interpolation in printed normal tables works well. It can be used in "both directions":

(1) In your case you are trying to find the CDF of standard normal evaluated at $z = 1.405,$ formally $\Phi(1.405),$ but most tables have only two-place accuracy of $z.$ By interpolation you got $0.91995.$ Of course you can use statistical software or a statistical calculator to get results that may be a little more accurate. In R, I got $\Phi(1.405) = 0.9199894,$ where all decimal places should be correct:

 pnorm(1.405)
 ## 0.9199894

(2) Sometimes you need to use the tables in reverse. For example, if you want the value $c$ such that $\Phi(c) = 0.95,$ my printed normal CDF table has $\Phi(1.64) = 0.9495$ and $\Phi(1.65) = 0.9595,$ so interpolation gives $c = \Phi^{-1}(.9500) = 1.645.$ That is, very nearly 5% of the area under a standard normal curve lies above $1.645.$ More precisely, $\Phi^{-1}(0.95) = 1.644854:$

 qnorm(.95)
 ## 1.644854

Sometimes the inverse CDF function is called the 'quantile' function.

Notes:

(a) If you have not already done so, you may soon use printed tables of 'Student's t distribution.' The bottom row of that table, usually marked Inf or $\infty,$ shows a few quantiles of the standard normal distribution. In my book, the value $\Phi^{-1}(0.95) = 1.645$ is given on that row.

(b) While it is OK to use linear interpolation in tables of the normal CDF, interpolation in tables for other distributions does not always work well.

(c) Many kinds of statistical software give standard normal CDF and quantile values (including R, Minitab, SPSS, SAS, and Excel). With the increasing availability of software, the use of printed tables is slowly decreasing over time.