Casio fx-991EX Normal Distribution: PD, CD and Inverse

Choose the operation that matches the question

On the fx-991EX, the Distribution mode offers Normal PD, Normal CD and Inverse Normal. These are different operations. PD returns a density at a specified x. CD calculates probability between lower and upper bounds. Inverse Normal starts from an area and finds a corresponding boundary; the manual specifies a left-tail area for that function. Select Distribution from the main menu, then choose the named operation and enter its requested variables. Do not copy a procedure for a different Casio model without checking that its menus and field definitions match.

Enter standard deviation rather than variance

A normal distribution is described here by mean μ and positive standard deviation σ. If a problem states variance 225, its standard deviation is √225=15. Entering 225 in the σ field would create a radically wider distribution. For the example X with mean 100 and standard deviation 15, a value of 115 is one standard deviation above the mean because z=(115−100)/15=1. This standardized value helps check the setup. The mean and standard deviation must use the same measurement unit as the x value and any interval boundaries.

Calculate an interval with Normal CD

For the example distribution, the probability that X lies between 85 and 115 is approximately 0.682689, or 68.2689%. In Normal CD, use Lower=85, Upper=115, σ=15 and μ=100. The two boundaries are one standard deviation below and above the mean. This is an interval probability, so it should be between zero and one. Reversing the lower and upper fields does not describe the same requested interval. If the result is unexpectedly tiny, inspect the standard deviation, units and boundary order before attributing the difference to rounding.

Distinguish a density from a probability

At x=100, the normal density is 1/(15√(2π)), approximately 0.0265962 per measurement unit. That is not a 2.65962% chance of observing exactly 100. A continuous normal model assigns zero probability to any single exact point; probabilities are areas over intervals. Density can also exceed one for sufficiently small standard deviations without violating probability rules. The distinction is essential when reading Normal PD output. If the question asks for the chance of being below a threshold or within a range, use a cumulative calculation rather than treating the density height as the answer.

Translate a right-tail requirement into a left-tail area

Suppose the question asks for the cutoff exceeded by the highest 10% of observations. The left-tail area is 90%, so use Area=0.9 with μ=100 and σ=15 in Inverse Normal. The boundary is approximately 119.2233. Entering 0.1 instead would produce the lower tenth-percentile boundary, approximately 80.7767. The two are symmetric around 100 for this distribution, which provides another check. Write the requested event in words and identify the tail before entering the probability; the calculator cannot infer which side of the boundary the question intends.

Use the browser tool for a cumulative cross-check

The site's Normal Distribution Calculator takes x, mean and standard deviation and reports a cumulative probability at that x along with related values. For the example, evaluate x=115 and x=85 separately, keeping mean 100 and standard deviation 15. Their left-tail probabilities are approximately 0.8413447 and 0.1586553. Subtracting the second from the first gives the interval probability 0.6826894. The local tool does not reproduce the fx-991EX Distribution menu or directly accept the four Normal CD fields, so compare equivalent quantities rather than expecting identical input screens.

Keep the model assumptions visible

These calculations assume the variable follows the specified normal distribution. They do not prove that a real data set is normal, that an estimated standard deviation is reliable or that an observation is independent of previous observations. A rounded calculator display can also differ slightly in its last digit from a table or another numerical method. Report the probability with sensible precision and retain μ, σ and the event boundaries. A complete worked answer identifies both the model and the event, then gives the numerical result, so another reader can reproduce the calculation without guessing which tail or function was selected.

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