“A $z$-score of 1.4,” said the student, reaching for his tables.

“0.92,” said the Mathematical Ninja, without skipping a beat.

“0.9192,” said the student, with a hint of annoyance. “How on earth…”

“Oh, it’s terribly simple,” said the Mathematical Ninja. “It turns out, for smallish values of $z$, the normal cumulative distribution function is roughly quadratic.”

“I understood about every third word of that.”

“OK - well then, I did some mathematical trickery.”

“Can you give an explanation that’s somewhere in between the two?”

The Mathematical Ninja looked baffled for a moment. “Um… ok. There’s a formula: $\Phi(x) \simeq \frac{1}{10} x(4.4-|x|) + 0.5$.” They wrote it on the board.

The student rubbed his eyes; he wasn’t sure he’d ever seen the Mathematical Ninja write a decimal number down before. “So… you took 1.4 away from 4.4 to get 3… and multiplied that by 1.4 to get 4.2. You divided by 10 to get 0.42, and added it on to a half. That… doesn’t seem so bad.”

The Mathematical Ninja bowed. “It’s good to two decimal places between $z=-2.2$ and $z=2.2$.”

“What if you had, say 2.5?”

“Good question! It’s about 0.99. 0.993 at a guess.”

The student studied his table. “0.994, actually. I see… between 2.2 and 2.6, you get something that rounds to 0.99, and above that it rounds to 1.00!”

A slight nod. “May it serve you well.”

  • Edited 2021-01-03 to give the Mathematical Ninja the correct gender.