Why is the Z score not universally agreed upon?

Question

I'm trying to compute a 95% confidence interval for a problem but it seems that multiple sources are saying different things of what the Z score should be. Some claim that it is 1.65 and some claim that it is 1.96. Which one is it really?

1.96 Table

Boston University uses 1.96 for this table

Boston University uses 1.65 for this table on the same website

1.65 Table

I'm just really confused why there is a disagreement / inconsistency here? Aren't we all talking about the same thing? How could this not be standardized / universally agreed upon?

Answer 1

A $1$-tailed test has $5\%$ significance at $p=0.95$, i.e. $z=1.65$. A $2$-tailed test has $5\%$ significance at $p=0.975$ (because this is the probability to the left of the $2.5\%$-probability right-hand tail), i.e. $z=1.96$.

Answer 2

For a random variable $Z$ with a standard normal distribution and with mean $0$ and standard deviation $1$, you can say

• $\mathbb P(Z \le 1.645) \approx 0.95$

and

• $\mathbb P(-1.96 \le Z \le 1.96) \approx 0.95$

so the critical values depend on whether you are trying to give a one-tailed or two-tailed hypothesis test or confidence interval.

By contrast $\mathbb P(-1.645 \le Z \le 1.645) \approx 0.9$ and $\mathbb P(Z \le 1.96) \approx 0.975$, so in a sense this matters.

It is worth remembering that the reason $95\%$ is such a common test is that Ronald Fisher's practical experience at Rothamsted Experimental Station had led him to the empirical view that two standard deviations was enough to justify making further investigations. He then calculated the $95\%$ as a round version of $\mathbb P(-2 \le Z \le 2) \approx 0.9545$; if he had concentrated on one-tailed tests then he might have recommended a different number for the typical confidence.