Calculator Inputs
Example Data Table
These values help verify common standard normal probabilities.
| Z score | Left tail Φ(z) | Right tail | PDF f(z) |
|---|---|---|---|
| -2.00 | 0.022750 | 0.977250 | 0.053991 |
| -1.50 | 0.066807 | 0.933193 | 0.129518 |
| -1.00 | 0.158655 | 0.841345 | 0.241971 |
| 0.00 | 0.500000 | 0.500000 | 0.398942 |
| 1.00 | 0.841345 | 0.158655 | 0.241971 |
| 1.50 | 0.933193 | 0.066807 | 0.129518 |
| 2.00 | 0.977250 | 0.022750 | 0.053991 |
Formula Used
Z score: z = (x - μ) / σ
Standard normal density: f(z) = (1 / √(2π)) × e^(-z² / 2)
Cumulative probability: Φ(z) = P(Z ≤ z)
Right tail: P(Z ≥ z) = 1 - Φ(z)
Between two values: P(a ≤ Z ≤ b) = Φ(b) - Φ(a)
Outside two values: P(Z ≤ a or Z ≥ b) = Φ(a) + 1 - Φ(b)
Two tail: P(|Z| ≥ |a|) = 2 × [1 - Φ(|a|)]
How to Use This Calculator
- Select whether your input is already a z score or a raw value.
- Choose the probability type, such as left tail or between.
- Enter the primary value. Add an upper value when needed.
- For raw values, enter the mean and standard deviation.
- Select decimal places for your result.
- Press the calculate button.
- Review the result panel above the form.
- Download the CSV or PDF report when needed.
Standard Normal Distribution Guide
What It Measures
A standard normal distribution uses a mean of zero and a standard deviation of one. It changes any normal value into a z score. That z score tells how far a value sits from the mean. Positive z scores are above the mean. Negative z scores are below it.
How Probability Is Found
Probability is found from the area under the bell curve. The whole area is one. A left tail value gives the chance of being less than a selected z score. A right tail value gives the chance of being greater. A between result measures the area between two z scores. An outside result adds both extreme tail areas.
Useful Input Options
This calculator supports z score inputs and raw value inputs. Raw values are converted using the supplied mean and standard deviation. This makes the tool useful for exam scores, quality checks, measurements, risk models, and research data. It also shows density. Density is not a direct probability at one exact point. It is the curve height at that z score.
Reading the Graph
The graph helps users see the selected region. The shaded area represents the reported probability. The results panel shows decimal probability, percent probability, cumulative values, and density values. These details help users compare tails, central regions, and extreme outcomes.
Choosing the Right Tail Type
Use left tail probability when you need a percentile. Use right tail probability for exceedance risk. Use between probability for acceptance bands or confidence style ranges. Use outside probability when both low and high extremes matter. Use two tail probability when distance from the center matters more than direction.
Accuracy Tips
Always check the standard deviation before using raw values. It must be greater than zero. Also make sure the input values use the same unit as the mean. The calculator gives numerical support, not a final statistical judgment. For formal studies, pair the output with sampling design, assumptions, and subject knowledge.
Exporting Results
Keep rounded answers for reporting. Use more decimals when comparing close cases. Save the CSV file when you need a spreadsheet record. Save the PDF when sharing a quick summary. The example table offers common reference points. It can help verify that your inputs and selected tail type match your intended probability question clearly.
FAQs
What is a standard normal distribution?
It is a normal curve with mean zero and standard deviation one. It lets you compare different normal values after converting them into z scores.
What does a z score mean?
A z score shows how many standard deviations a value is from the mean. Positive values are above the mean. Negative values are below it.
Is density the same as probability?
No. Density is the curve height at a point. Probability is an area under the curve across a range or tail.
When should I use left tail probability?
Use it when you need the chance of being less than or equal to a value. It is also useful for percentiles.
When should I use right tail probability?
Use it when you need the chance of exceeding a value. It is common in risk, rejection regions, and high score analysis.
What does between probability show?
It shows the chance that a standard normal value falls between two selected z scores. It is a central area calculation.
Can I enter raw values?
Yes. Select raw value mode. Then enter the raw value, mean, and standard deviation. The calculator converts them to z scores.
Why does standard deviation need to be positive?
Standard deviation measures spread. A zero or negative value cannot describe normal variation, so z score conversion would be invalid.