Graphical calculators for continuous distributions

This tutorial covers how to use the StatCrunch calculators for continuous distributions. StatCrunch offers such calculators for the beta, Cauchy, chi-square, exponential, f, gamma, normal, t and Weibull distributions. These calculators allow for the calculation of a probability given reference value(s) or the calculation of a reference value associated with a specific probability. A graphic with the appropriate region shaded is always provided. In this example, focus will be placed on using the normal calculator, but the same techniques can be applied to use the calculators for other continuous distributions. The normal calculator acts as a more accurate and more interactive version of a z-table (standard normal table) found in the back of many textbooks. To begin, load an Empty Data Table.

Creating the calculator

All StatCrunch calculators are found under the Stat > Calculators menu. To create a normal calculator, choose the Stat > Calculators > Normal menu option. This will load the calculator shown below for a normal distribution with a mean of 0 and a standard deviation of 1, which is otherwise known as a standard normal distribution. The parameters defining the distribution can be changed by editing the inputs in the first row below the graph. For the normal calculator below, these inputs are labeled as Mean and Std. dev. (standard deviation). The second row of inputs is used to find probabilities associated with the distribution. By default, the normal calculator shows the area below the mean of 0 in red and indicates the probability of a standard normal variable being at or below 0 to be 0.5 (50% of the total area).

Finding the associated probability above or below a reference value

A probability associated with another reference value can be easily computed. For example, type -2.1 for the reference value, in the box to the right of the ≤ sign as shown below, and click Compute (or press the *Enter* key). The probability of being less than or equal to -2.1 is then computed to be 0.01786442. Another way of thinking about this result is that 1.786442% of the area under this distribution is below -2.1. The distributional picture changes to reflect the new reference value with the resulting probability shown in red. To find the probability above this reference value, change the inequality sign from ≤ to ≥.

Finding the value with an associated probability above or below it

The StatCrunch calculators for continuous distributions can also be used in reverse to find a reference value, given a probability to be above or below it. This is often done when finding the critical value for a hypothesis test or a multiplier in a confidence interval. In this case for the standard normal distribution, to find the value where there is a probability of 0.05 of being above it (or 5% of the total area above it), switch the inequality from ≤ to ≥ as shown below. Enter 0.05 for the probability input to the right of the equals sign. This input accepts values between 0 and 1 (noninclusive). Click Compute or hit the *Enter* key to update the calculator. The resulting distributional picture shown below is updated with 5% of the area colored in red. The reference value (at the left edge of the red area) is also computed to be 1.6448536.

Computing the probability between two reference values

To find the probability of falling between two reference values, click the Between button to the upper left of the calculator window. For this example, this will produce a new graph with the area between the default reference values of -1 and 1 colored in red as shown below. The probability of a value falling between these two reference points is also shown to be 0.68268949.

Change the first reference value to -0.5 and the second to 2 as shown below. The first reference value must always be less than the second one. Press Compute (or press *Enter*) to calculate and graph the probability between these two reference values as shown below. The area between -0.5 and 2 is shown in red and corresponds to a probability of 0.66871233. One of the two reference values or the associated probability must be left empty and the calculator will fill in the remaining input. Click the Standard button in the top left of the calculator to switch back to the original calculator with only one reference value.

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