Normal Model Area Explorer

1 / Set up the question

Your normal model

Mean (μ) Standard deviation (σ)

σ must be greater than 0.

What do you want to find? Shaded region

Enter cutoffs as

Cutoff (x)

Try a classroom example

Predict, then check

Your probability estimate (0 to 1)

Estimate the shaded fraction, then compare it with the answer.

2 / See the connection

Same area, different units

Drag a cutoff or the shaded region. Both scales and the calculation update together.

3 / Follow the math

Standardize, then find the area

4 / Do the same in Excel

Equivalent Excel formulas

Shown values are rounded; calculations use full precision. Excel formulas keep the original inputs.

How the four Excel functions fit together

NORM.DIST(x, μ, σ, TRUE): Original value → area to its left.
NORM.S.DIST(z, TRUE): Z-score → area to its left. S means standard normal: μ = 0, σ = 1.
NORM.INV(p, μ, σ): Area to the left → original value.
NORM.S.INV(p): Area to the left → z-score. Then x = μ + σz.

For DIST functions, TRUE gives cumulative area. FALSE gives the height of the density curve, not a probability. A continuous distribution has P(X = x) = 0.

Standardization changes the scale, not the probability. A right tail is 1 minus the left area; a between region is the upper left area minus the lower left area.

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