Understanding z-score probabilities

The standard normal distribution is the normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$. A value on this scale is written as $Z$, and a specific location on the curve is written as a z-score.

A z-score probability means an area under the standard normal curve. For example, $P(Z < z)$ is the area to the left of the z-score $z$. Because the total area under the curve is 1, these areas can be read as probabilities or as percentages.

The role of the CDF

The standard normal CDF, written $\Phi(z)$, gives the left-tail probability:

Once $\Phi(z)$ is known, other common probability forms come from complements, differences, and symmetry.

How to read the probability forms

• Left-tail probability: $P(Z < z)=\Phi(z)$ shades everything to the left of $z$.
• Right-tail probability: $P(Z > z)=1-\Phi(z)$ shades everything to the right of $z$.
• Between two z-scores: $P(z_1 < Z < z_2)=\Phi(z_2)-\Phi(z_1)$ after using the lower z-score first.
• Center probability: $P(-z < Z < z)=\Phi(z)-\Phi(-z)$ for a positive distance $z$.
• Two-tail probability: $P(Z < -z \text{ or } Z > z)=1-[\Phi(z)-\Phi(-z)]$ for the area outside the center.

Worked one-z example

Suppose $z=1.25$. The CDF value is approximately:

So $P(Z<1.25)\approx 0.8944$, and the right-tail probability is:

The area between the mean and $z=1.25$ is $0.8944-0.5=0.3944$. The symmetric center area between $-1.25$ and $1.25$ is about $0.7888$, leaving about $0.2112$ in the two outside tails.

Worked two-z example

Suppose you want $P(-1 < Z < 2)$. Use the CDF at both endpoints:

Using standard normal values, $\Phi(2)\approx 0.9772$ and $\Phi(-1)\approx 0.1587$, so:

The calculator also reports the two outside pieces: $P(Z<-1)$ and $P(Z>2)$.

How the graph shading matches the formula

Each shaded curve is a picture of the same area named by the formula. A left-tail result shades from the far left up to the z marker. A right-tail result shades from the marker to the far right. A between result shades only the interval between the two marked values. A two-tail or outside result shades both ends and leaves the middle unshaded.

Common cautions

• These probabilities use the standard normal distribution. They apply directly when $Z$ follows a standard normal model.
• For a continuous distribution, $P(Z < z)$ and $P(Z \le z)$ are effectively the same because a single point has probability 0.
• For non-normal data, z-score probabilities may only be an approximation. The z-score can still describe standardized distance, but the normal probability interpretation depends on the model.

FAQ

Why is $P(Z < 0)=0.5$?

The standard normal curve is symmetric around 0. Half of the total area lies to the left of the mean and half lies to the right, so $\Phi(0)=0.5$.

What is the difference between one-tail and two-tail probability?

A one-tail probability looks in one direction from a cutoff, such as $P(Z>z)$. A two-tail probability combines both extremes, such as $P(Z<-z \text{ or } Z>z)$.

Why does the calculator use $-z$ and $z$ for center and outside probabilities?

Those forms answer symmetric questions around the mean. The center probability asks how much area lies within the same distance of 0 on both sides, while the outside probability asks how much area lies beyond that distance in either tail.