Normal Distribution & Z-Score Explorer (AP Statistics)

How to use the simulator

The controls:

Mean (μ): −4 to 4 in steps of 0.1, starting at 0.
Standard deviation (σ): 0.3 to 3 in steps of 0.05, starting at 1.
Between, Left tail and Right tail buttons pick the region: $P(a < X < b)$ , $P(X < x)$ or $P(X > x)$ .
Lower bound (a) and Upper bound (b) sliders set the two edges in Between mode. In the tail modes only one slider shows, relabelled Cutoff (x).

The bound sliders always run from

\mu - 4\sigma

to

\mu + 4\sigma

in 160 steps, so each step is

\sigma/20

and moves the z-score by 0.05. You can land on z = 1.95 or 2.00, but not on 1.96.

Under each bound slider is its z-score to three decimal places. The plot shows the shaded area, a dashed line at μ, and tick marks at μ ± 1σ, 2σ and 3σ labelled with their x-values. The Live probability box gives the area to four decimals, for example P(−1.00 < X < 1.00) = 0.6827, with the z-scores underneath.

In Between mode the bounds can cross; the simulator always shades from the smaller to the larger and labels the result that way. Changing μ or σ redraws the curve but keeps each bound at the same x-value, so its z-score and the shaded area change with it.

Because μ only goes up to 4 and σ up to 3, you cannot type in a real-world scale such as a mean of 64.5 inches. Standardize first: set μ = 0 and σ = 1 and put the cutoff at your z-score. The area is the same, because standardizing only relabels the axis.

Empirical-rule check at μ = 0, σ = 1, Between mode: bounds ±1 give 0.6827, ±2 give 0.9545, ±3 give 0.9973.

The formulas

To standardize a value

x

from a normal distribution with mean

\mu

and standard deviation

\sigma

:

z = \frac{x - \mu}{\sigma}

To go back from a z-score to a value:

x = \mu + z\sigma

.

Z-scores also compare values from different distributions: 85 on a test with mean 70 and SD 10 (

z = 1.5

) is more unusual than 90 on a test with mean 80 and SD 8 (

z = 1.25

).

The z-score counts how many standard deviations $x$ is above (positive) or below (negative) the mean.
The area under the curve over an interval is the probability that $X$ falls in that interval. The total area is 1.
For a continuous distribution $P(X = x) = 0$ , so $P(X < x)$ and $P(X \le x)$ are equal.
Right-tail areas use the complement: $P(X > x) = 1 - P(X < x)$ .
Empirical rule: about 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.

Worked example

Problem: Heights of adult women are approximately normal with mean 64.5 inches and standard deviation 2.5 inches. (a) What proportion are between 62 and 68 inches tall? (b) What proportion are taller than 68 inches? (c) How tall is a woman at the 90th percentile?

Step 1: Standardize the cutoffs.

z = \dfrac{62 - 64.5}{2.5} = -1.00

and

z = \dfrac{68 - 64.5}{2.5} = 1.40

.

Step 2: Area between.

P(62 < X < 68) = P(-1.00 < Z < 1.40) \approx 0.7606

. About 76% of women are between 62 and 68 inches.

Step 3: Right tail.

P(X > 68) = P(Z > 1.40) = 1 - 0.9192 \approx 0.0808

.

Step 4: Work backwards for a percentile. The z-score with area 0.90 to its left is about 1.2816 (invNorm(0.90)). Then

x = 64.5 + 1.2816(2.5) \approx 67.7

inches.

Check it in the simulator: set μ = 0 and σ = 1. In Between mode, drag a to −1.00 and b to 1.40: the box reads 0.7606. Switch to Right tail with the cutoff at 1.40 for 0.0808. For part (c), use Left tail and slide the cutoff: z = 1.25 gives 0.8944 and z = 1.30 gives 0.9032, so the 90th percentile lies between them, as the exact 1.2816 does.

Common mistakes on the AP exam

Flipping the subtraction. $z = (x - \mu)/\sigma$ . A value below the mean must have a negative z-score.
Using the variance. Divide by $\sigma$ , not $\sigma^2$ . The AP exam writes $N(\mu, \sigma)$ with the standard deviation second; the simulator's legend uses the $N(\mu, \sigma^2)$ form, so check which one a source means.
Forgetting the complement. A table or normalcdf with no upper limit gives the area to the left. For "more than," subtract from 1.
Mixing up area and value. In a percentile question you know the area and need $x$ , so work backwards with invNorm, then $x = \mu + z\sigma$ .
Assuming normality. Normal calculations only apply when the distribution is stated or shown to be approximately normal.
Unlabelled calculator work. Write the distribution, the parameters and the boundary, such as $N(64.5, 2.5)$ , $P(X > 68)$ , or show the z-score, not just normalcdf(68, 1E99, 64.5, 2.5).

When the AP exam uses this

The normal distribution is topic 2.11, and z-scores return throughout inference as test statistics and critical values. Expect area and percentile questions in both multiple choice and free response, where you should show the boundary, the parameters and the region.

Normal Distribution & Z-Score Explorer

How to use the simulator

The formulas

Worked example

Common mistakes on the AP exam

When the AP exam uses this

More AP Statistics simulators