Drag a bound on the bell curve and read the shaded probability.
Normal Distribution Visualizer simulator
Drag the bound
This simulation needs a modern browser with canvas support. The measurements and formula below still describe the physics.
Live measurements
Controls
Formula
z = (x - μ) / σ
- z z-score
- —
Observations
What's happening
Real-world examples
One sigma
Mean 0 and sigma 1 with bound 1 give z = 1 and a left area near 0.84.
Half the curve
Bound 0 on that same curve shades a left area of 0.5, exactly half.
Two sigma
Bound 2 with mean 0 and sigma 1 selects empirical band 2, the 95 percent middle.
Center drag
A drag at the middle of the canvas sets the bound to 0, because the axis runs from -4 to 4.
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Uses of normal distribution calculator
- Checking a statistics homework z-score against the shaded area
- Showing a class why 68 percent is the middle band and not the left tail
- Comparing one bound on a normal curve with a list of sample numbers
- Estimating a left-tail probability before a longer statistics course
normal distribution calculator pitfalls
- Reading a z-score of 1 as if it were the shaded probability
- Treating the 68 percent band as the left tail area
normal distribution calculator questions (7)
What is a normal distribution?
A normal distribution is a bell curve fixed by a mean and a standard deviation. Most of the area sits near the mean, and both tails thin out. Sigma sets the width while the mean slides the peak left or right. With mean 0 and sigma 1, this is the standard normal curve.
What is the difference between a z-score and the shaded area?
The z-score counts standard deviations from the mean to the bound. The shaded area is the probability of landing at or below that bound. For example, bound 1 on the standard normal gives z = 1. The left area there is near 0.84, not 1.
Why does a zero standard deviation get refused?
A standard deviation of zero would divide the z-score by zero. The sigma slider therefore stops at 0.2, which keeps a real width. That floor is how a zero width was refused. Raise sigma and the bell widens while z shrinks.
How do the 68, 95, and 99.7 bands work?
These bands are memory aids for area near the mean, not a second shaded tail. About 68 percent of the area lies within one sigma of the mean. About 95 percent lies within two sigma, and about 99.7 percent lies within three. A bound of 2 with mean 0 and sigma 1 selects band 2.
Where does this bell curve get used?
Checking a statistics homework z-score against the shaded area takes one drag of the bound. Showing a class why 68 percent is the middle band and not the left tail fits here. Comparing one bound on a normal curve with a list of sample numbers belongs beside the statistics visualizer. Estimating a left-tail probability before a longer statistics course is the edge of this page.
What does a drag on the canvas change?
A horizontal drag moves only the bound, while mean and sigma stay on the sliders. The middle of the canvas is bound 0, because the axis runs from -4 to 4. On the standard normal, bound 0 shades a left area of 0.5.
What numbers show up at the default setting?
The page opens on mean 0 and sigma 1 with the bound at 1, so z = 1. The shaded left area is near 0.84 and the right area is near 0.16. Those two areas add to 1 on every curve.