Calculator Inputs
Example Data Table
This sample shows how common z scores map to tail areas and central probabilities in a standard normal distribution.
| Z Score | Left Tail | Right Tail | Two Tailed Probability | Central Probability |
|---|---|---|---|---|
| 0.50 | 0.691462 | 0.308538 | 0.617076 | 0.382924 |
| 1.00 | 0.841345 | 0.158655 | 0.317310 | 0.682690 |
| 1.64 | 0.949497 | 0.050503 | 0.101006 | 0.898994 |
| 1.96 | 0.975002 | 0.024998 | 0.049996 | 0.950004 |
| 2.58 | 0.995060 | 0.004940 | 0.009880 | 0.990120 |
Formula Used
The calculator first standardizes the observed value when needed:
z = (x - μ) / σ
It then estimates the cumulative normal probability using the error function:
Φ(z) = 0.5 × [1 + erf(z / √2)]
The two tailed probability is computed from the more extreme side:
Two Tailed Probability = 2 × min(Φ(z), 1 - Φ(z))
This result is the same p value used in many two sided z tests.
How to Use This Calculator
- Select whether you want to enter an observed value or a direct z score.
- If using an observed value, enter the value, mean, and standard deviation.
- Choose your significance level to compare against the final p value.
- Set the number of decimal places for more precise reporting.
- Press the calculate button to show the result section above the form.
- Review tail probabilities, central area, and the significance interpretation.
- Use the CSV or PDF buttons to export the calculated output.
FAQs
1. What does two tailed probability mean?
It measures the combined probability of values at least as extreme as the observed statistic on both sides of the distribution.
2. When should I use a two tailed test?
Use it when deviations in either direction matter, such as checking whether a process differs from a target without assuming direction.
3. Is this calculator only for normal distributions?
Yes. This version uses the standard normal distribution after standardizing your value into a z score.
4. What is the difference between p value and alpha?
The p value comes from your data. Alpha is your decision threshold. Compare them to judge statistical significance.
5. Why is the result doubled?
A two tailed calculation includes both tails. It doubles the smaller one tail probability to reflect extremes in either direction.
6. What happens when z equals zero?
The observed value sits exactly at the mean. The two tailed probability becomes 1.000000, showing no extremeness.
7. Can I export the calculated results?
Yes. After calculation, use the CSV button for spreadsheet output or the PDF button for a print-friendly report.