Calculator Inputs
Example Data Table
| Case | Mean | Standard deviation | Value | Z score | Meaning |
|---|---|---|---|---|---|
| Center | 215 | 7 | 215 | 0 | Exactly at the mean |
| One deviation high | 215 | 7 | 222 | 1 | Upper edge of the 68% band |
| Two deviations low | 215 | 7 | 201 | -2 | Lower edge of the 95% band |
| Three deviations high | 215 | 7 | 236 | 3 | Near the upper rare zone |
Formula Used
The calculator treats N(215,7) as a normal model with mean 215 and standard deviation 7.
Z score: z = (x - μ) / σ
Normal density: f(x) = [1 / (σ√(2π))] × e^[-(x - μ)² / (2σ²)]
Cumulative probability: P(X ≤ x) = Φ((x - μ) / σ)
Interval probability: P(a ≤ X ≤ b) = Φ((b - μ) / σ) - Φ((a - μ) / σ)
Empirical rule: about 68% lies within one standard deviation, 95% within two, and 99.7% within three.
How to Use This Calculator
Enter the mean and standard deviation. The default setup uses 215 and 7.
Enter an x value to calculate its z score, density, left tail, and right tail.
Enter lower and upper bounds to calculate the probability inside an interval.
Use the sample size field when you also want the standard error.
Press the calculate button. The result appears above the form and below the header.
Use the CSV option for spreadsheet work. Use the PDF option for a saved report.
Understanding the 68-95-99.7 Rule
A normal curve is a smooth model for balanced variation. This calculator starts with N(215,7), where 215 is the mean and 7 is the standard deviation. It then maps values around the center, so the spread is easy to read. The famous 68-95-99.7 rule gives a fast view of that spread.
About the Default Distribution
For this distribution, one standard deviation covers 208 to 222. Two standard deviations cover 201 to 229. Three standard deviations cover 194 to 236. These bands help you judge whether a value is ordinary, high, low, or rare. They also give practical checkpoints before a deeper probability calculation.
Why the Graph Matters
The graph places the bell curve beside the numbers. A vertical line marks the mean. Extra lines show the sigma limits. The entered x value and interval endpoints are also drawn. This makes the result useful for reports, study notes, classroom work, and quality checks.
Using Z Scores
A z score tells how many standard deviations a value sits from the mean. A value of 222 has a z score of 1. A value of 201 has a z score of -2. The sign shows direction. The size shows distance. Larger absolute z scores mean more unusual values.
Probability and Tail Areas
The calculator estimates the left tail, right tail, central interval, and percentile. These outputs help compare one cutoff, two cutoffs, or a selected range. The density value shows the curve height at x. It is not a probability by itself. It supports the graph and the shape analysis.
Good Inputs Improve Results
Use a positive standard deviation. Keep the lower bound below the upper bound. Choose enough decimals for your purpose. More decimals help technical reports. Fewer decimals help quick learning. For N(215,7), the default settings are ready for the empirical rule.
Exporting the Work
After calculation, the page can download a CSV summary. It can also create a printable report. This helps save results, attach evidence, or compare several scenarios. Always review assumptions before using a normal model for real decisions. A clear record also prevents mistakes later. Users can revisit older numbers. Teams can share classroom examples during practice and review sessions with confidence today.
FAQs
What does N(215,7) mean here?
It means a normal distribution with mean 215 and standard deviation 7. The calculator uses this as the default model for all bands, z scores, and probabilities.
What is the 68-95-99.7 rule?
It is an empirical rule for normal curves. About 68% falls within one standard deviation, 95% within two, and 99.7% within three.
What are the default sigma bands?
For N(215,7), the one deviation band is 208 to 222. The two deviation band is 201 to 229. The three deviation band is 194 to 236.
What does the z score show?
The z score shows distance from the mean in standard deviation units. Positive values are above the mean. Negative values are below the mean.
Is density the same as probability?
No. Density is curve height at a point. Probability comes from area under the curve over an interval or tail.
Can I change the default mean?
Yes. Enter any numeric mean. The calculator will redraw the curve and rebuild every probability, band, and export result from that value.
Why include sample size?
Sample size is used to calculate standard error. It helps when comparing a sample mean, but it does not change the original distribution spread.
How do I save the results?
Press Download CSV for spreadsheet use. Press Download PDF to create a report containing the main calculated outputs.