Standard normal distribution

The standard normal distribution is the single normal curve with mean 0 and standard deviation 1, written $Z \sim N(0, 1)$. Any normal variable maps onto it through the z-score $z = \frac{x - \mu}{\sigma}$, where $\mu$ (mu) and $\sigma$ (sigma) are the mean and standard deviation of the original variable. Subtracting and dividing is a linear change of scale, so it moves no area: the proportion of $N(\mu, \sigma)$ below $x$ equals the proportion of $N(0, 1)$ below $z$. That equality is the reason one table can serve every normal distribution.

Suppose scores are $N(500, 100)$ and you want the share below 650. Standardize: $z = \frac{650 - 500}{100} = 1.5$. The area to the left of $z = 1.5$ is 0.9332, so about 93.3 percent of scores fall below 650 and 6.7 percent above. Symmetry hands you the mirror image for free, since the area to the left of $z = -1.5$ is that same 0.0668.

The sentence to unlearn is "I converted to z-scores, so the distribution is normal now." Standardizing relocates the center to 0 and rescales the spread to 1. It changes nothing about shape. A right-skewed set of values becomes a right-skewed set of z-scores with exactly the same skew, and reading a standard normal table on it returns an answer that is simply wrong. The table is valid because the distribution was normal to begin with, not because you standardized it.

Two smaller habits cost points. A table entry is the area to the left of $z$, so a "greater than" question needs 1 minus the entry. And a negative $z$ is not a mistake; it only means the value sits below the mean.

The z-table on this site runs from $z = -3.49$ to $z = 3.49$, matching the AP Statistics Table A layout. Past the ends there is very little left to account for: the area below $z = -3.49$ is about 0.00024.