Enter Distribution Values
Example Data Table
This table uses mean 0 and standard deviation 1.
| X | Z Score | Density f(x) | Left Area | Right Area |
|---|---|---|---|---|
| -2 | -2 | 0.05399097 | 0.02275006 | 0.97724994 |
| -1 | -1 | 0.24197072 | 0.15865526 | 0.84134474 |
| 0 | 0 | 0.39894228 | 0.5 | 0.5 |
| 1 | 1 | 0.24197072 | 0.84134474 | 0.15865526 |
| 2 | 2 | 0.05399097 | 0.97724994 | 0.02275006 |
Formula Used
The normal probability density function is:
f(x) = 1 / (σ√(2π)) × e^(-((x - μ)² / (2σ²)))
Here, x is the observed value, μ is the mean, and σ is the standard deviation.
The z score is:
z = (x - μ) / σ
The cumulative probability uses the error function approximation:
P(X ≤ x) = 0.5 × [1 + erf((x - μ) / (σ√2))]
How to Use This Calculator
- Enter the x value where you need the density.
- Enter the mean and standard deviation.
- Add two range limits to calculate between-area probability.
- Enter a percentile to find its matching x value.
- Set table start, end, and step values.
- Press Calculate to show the result above the form.
- Use CSV or PDF buttons to export the results.
Understanding the Normal Density Curve
What the Calculator Measures
A normal probability density function describes a smooth bell-shaped curve. It is widely used in statistics, research, quality control, finance, testing, and measurement analysis. This calculator finds the density at a selected x value. It also returns the z score, left cumulative area, right tail area, range probability, and percentile point. These outputs help you understand where a value sits inside a normal distribution.
Why Density Is Different From Probability
The density value is not a direct probability for one exact point. In continuous distributions, probability comes from area under the curve. A single exact value has almost zero probability. A range has useful probability. That is why this tool includes lower and upper limits. The range result shows the area between two selected x values.
Role of Mean and Standard Deviation
The mean controls the center of the curve. The standard deviation controls the spread. A small standard deviation creates a taller and narrower curve. A large standard deviation creates a wider and flatter curve. The calculator uses both values to standardize x into a z score. The z score shows how many standard deviations the value is from the mean.
Using Results in Statistical Work
The left area is useful for percentile and threshold analysis. The right tail helps with exceedance questions. The between-area result supports interval probability checks. The generated table helps compare many x values at once. You can export the output for reports, records, lessons, or repeated analysis. Always use a positive standard deviation. Also check that your inputs match the scale of your real data.
Frequently Asked Questions
1. What does the normal density value mean?
It shows the curve height at a selected x value. It is not the probability of that exact value. Probability is found from area under the curve across an interval.
2. Why must standard deviation be positive?
Standard deviation measures spread. A zero or negative spread is not valid for a normal distribution. The formula also divides by standard deviation.
3. What is a z score?
A z score shows how far x is from the mean in standard deviation units. Positive values are above the mean. Negative values are below it.
4. What is left cumulative area?
Left cumulative area is the probability that a random value is less than or equal to x. It is also used for percentile interpretation.
5. What is right tail area?
Right tail area is the probability that a random value is greater than x. It is useful for exceedance, risk, and upper threshold questions.
6. Can I calculate probability between two values?
Yes. Enter range start and range end values. The calculator subtracts their cumulative areas to estimate the probability between those limits.
7. What does the percentile result show?
It returns the x value below which the selected percent of observations falls. For example, the 95th percentile marks a high cutoff.
8. Can I export the calculation?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable summary of the main calculation results.