The Normal Distribution and z-Scores
In a normal distribution the z-score says how many standard deviations a value is from the mean: $z = \dfrac{x - \mu}{\sigma}$. The probability of an interval is the area under the curve over it. On the horizontal axis the z-scores are on top and the raw values for your $\mu$ and $\sigma$ are below (in green).
What to notice
- The empirical rule: about 68% lies within $z = \pm 1$, 95% within $\pm 2$ and 99.7% within $\pm 3$. Set the bounds to these values and check.
- Changing $\mu$ and $\sigma$ does not change the area; only the raw values on the axis change. This is why every normal distribution can be handled with one standard normal table.
- The "two tails" option is the picture of a two-sided p-value in a hypothesis test: the area outside $z = \pm 1.96$ is 5%.