Normal Distribution Area Calculator

Calculator Inputs

Mean (μ)

Standard Deviation (σ)

Calculation Mode

Lower Bound / Cutoff

Upper Bound

Decimal Places

Use both lower and upper values for interval calculations.

Example Data Table

Scenario	Mean	Standard Deviation	Mode	Value(s)	Expected Area
Standard score below 1.25	0	1	Less than	x = 1.25	0.894350
Central interval inside one deviation	0	1	Between	-1 to 1	0.682689
Upper tail beyond 1.645	0	1	Greater than	x = 1.645	0.049985
Exam scores between 60 and 80	70	10	Between	60 to 80	0.682689

Formula Used

The calculator converts raw values into z-scores, then uses the standard normal cumulative distribution function to measure shaded probability.

1) Standardization z = (x - μ) / σ

2) Cumulative distribution Φ(z) = 0.5 × [1 + erf(z / √2)]

3) Left-tail probability P(X ≤ x) = Φ((x - μ) / σ)

4) Right-tail probability P(X ≥ x) = 1 - Φ((x - μ) / σ)

5) Interval probability P(a ≤ X ≤ b) = Φ((b - μ) / σ) - Φ((a - μ) / σ)

6) Outside interval probability P(X < a or X > b) = 1 - P(a ≤ X ≤ b)

How to Use This Calculator

Enter the distribution mean and standard deviation.
Select whether you need a left tail, right tail, inside interval, or outside interval area.
Enter the cutoff value for one-sided mode, or both bounds for interval mode.
Choose the number of decimal places for output precision.
Press Calculate Area to show the probability summary above the form.
Review the graph, z-scores, complement probability, and interpretation.
Use the export buttons to save the current result as CSV or PDF.

Frequently Asked Questions

1) What does the shaded area represent?

The shaded area is the probability that values from the chosen normal distribution fall inside the selected tail or interval.

2) Can I use any mean and standard deviation?

Yes. The calculator works with standard normal and custom normal distributions, as long as the standard deviation is greater than zero.

3) What is the difference between between and outside mode?

Between mode measures probability inside two bounds. Outside mode combines both tails and measures probability beyond the chosen interval.

4) Why are z-scores shown in the results?

Z-scores standardize your values. They show how far each point sits from the mean in standard deviation units.

5) What happens if my standard deviation is zero?

A normal distribution cannot exist with zero spread. The calculator stops and asks for a positive standard deviation value.

6) Are inclusive and exclusive bounds different here?

For continuous distributions, inclusive and exclusive endpoints have the same probability area. The difference is effectively zero.

7) Can this help with confidence range analysis?

Yes. You can measure central coverage around a mean by testing ranges such as one, two, or three standard deviations.

8) Why does my probability look almost zero or one?

That usually means the cutoff is far into a tail. Large absolute z-scores push the probability close to 0 or 1.