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Normal Probability Distribution Graph Interactive

You can explore the concept of the standard normal curve and the numbers in the z-Table using the following applet.


The (colored) graph can have any mean, and any standard deviation.

The gray curve on the left side is the standard normal curve, which always has mean = 0 and standard deviation = 1.

We work out the probability of an event by first working out the z-scores (which refer to the distance from the mean in the standard normal curve) using the formulas shown.


Drag any of the colored dots left or right to change the values of:

  • The mean, = μ (red dot)
  • The standard deviation = σ (red dot, minimum value 0.2 for this graph), and
  • the starting and end points of the region of interest (x1 and x2, the green dots).

The green shaded area represents the probability of an event with mean μ, standard deviation σ occuring between x1 and x2, while the gray shaded area is the normalized case, where `mu=0` and `sigma = 1.`


Probability = P(z1) + P(z2) =

NOTE: The values given in the probability calculations come from the z-table.

Things to do:

  1. Drag the `mu` and `sigma` sliders to change the mean and standard deviation, and to see the effect on the bell curve.
  2. Drag the `x_1` and `x_2` sliders to change the portion of the curve for which you need to find the probability.
  3. Now click on "Show standard normal curve" to see the equivalent shaded area when the blue curve is translated to the standard form.
  4. Now click on "Show `z`-score calculations" to see how this is done.
  5. Now see the probability calculation for your situation.
  6. Observe the proportion of the area under the curve from `mu` to `1` standard deviation from `mu` (around 34%, and observe on the gray curve it's `sigma=1`), `mu` to `2` standard deviations (around 47.7%, or `sigma=2` on the standard normal curve) and from `mu` to 3 standard deviations (almost the whole right hand side, around 49.9%, and `{:sigma=3)`
  7. Try setting `mu=0` and `sigma=1` for the green curve. In this case, you have done what the `z`-score does for us - that is, used the standard normal curve.

How does it find the probabilities?

This javascript applet is using numerical integration to find the shaded area under the curve between the vertical lines.

It is actually using Simpson's Rule, which you will learn about later in the calculus section.

These are the scores we (humans) need to get from the z-table.

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