How to find the Z score?

Question

I HAVE READ THIS ONLINE AND i WANTED TO KNOW AND UNDERSTAND ON HOW THE Z SCORE IS BEING COMPUTED? HOW DID IT END UP AS 0.675? PLEASE REFER BELOW.

"Let X be length of pregnancies.

The middle 50% of most averages corresponds to the area between the 25th percentile and the 75th percentile.

To find the values corresponding to the 25th percentile and 75th percentile, we first need to find the z-score corresponding to the 75th percentile, which is 0.675.

We then equate that z score to the standardized value of X to find X at the 75th percentile:

0.675 = (X - 263)/16

0.675 * 16 + 263 = X

X = 273.8, which is 10.8 above the mean

Since the middle 50% is symmetric about the mean, the X value for the 25th percentile is 10.8 below the mean, or 263-10.8 = 252.2

The range to find the middle 50% is 252.2, 273.8"

Answer 1

For a normal distribution, a z-score of 0.67449 results in 75% of the area under the curve:

If you have a TI-84 calculator, do "invNorm", area: 0.75, μ = 0, σ = 1, Tail: LEFT, paste and the result (0.6744897495) will be the calculation of the z-score that produces a 75% probability (75% of the area under the curve lies between -∞ and a z-score of 0.6744897495)