Inverse Normal Distribution Calculator
Find the value of x for a given cumulative probability in a normal distribution. Enter your mean, standard deviation, and probability to get the corresponding x value instantly.
Find x for a Given Probability
Need More? Try the Full Normal Distribution Calculator!
This tool is focused only on finding the value of x for a given probability (inverse CDF/quantile). If you want to calculate probabilities, Z-scores, areas between values, or visualize more types of normal distribution calculations, the comprehensive calculator offers all these features in one place, with step-by-step examples and advanced graphing.
What is the Inverse Normal Distribution?
The Inverse Normal Distribution (also called the Quantile Function or Probit Function) allows you to find the value of x for a given cumulative probability in a normal distribution. In other words, if you know the probability (area under the curve to the left of x), you can find the corresponding x value.
Formulas:
1. Probability Density Function (PDF)
The PDF, denoted as \(f(x)\), gives the probability density at a given point \(x\). For continuous variables, the PDF itself is not a probability, but rather the relative likelihood of the variable taking on a given value. The actual probability for a continuous variable is found by calculating the area under the PDF curve over a specific range.
Where:
- \(x\) is the value of the variable.
- \(\mu\) is the mean (location parameter).
- \(\sigma\) is the standard deviation (scale parameter).
- \(\pi \approx 3.14159\)
- \(e \approx 2.71828\) (Euler's number, the base of the natural logarithm)
2. Cumulative Distribution Function (CDF)
The CDF, denoted as \(F(x)\), gives the probability that a random variable \(X\) will take a value less than or equal to \(x\). This is represented by the area under the PDF curve from \(-\infty\) to \(x\).
3. Z-score (Standard Score)
The Z-score measures how many standard deviations an individual data point (\(x\)) is away from the mean (\(\mu\)) of the distribution. It's crucial for standardizing values from different normal distributions, allowing for direct comparison and the use of a single standard normal distribution table. A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it's below the mean.
4. Inverse CDF (Quantile Function)
The inverse CDF (also called the quantile function or probit function) finds the value \(x\) for a given cumulative probability \(p\):
Where:
- \(x\) is the value corresponding to the given probability.
- \(\mu\) is the mean of the distribution.
- \(\sigma\) is the standard deviation.
- \(Z_p\) is the Z-score for the given cumulative probability \(p\).
This is useful in statistics, quality control, finance, and many other fields where you need to determine thresholds or cut-off values for a given probability.
Frequently Asked Questions (FAQs)
1. What is the Inverse Normal Distribution?
It is a function that gives the value of x for a given cumulative probability in a normal distribution. It's also called the quantile or probit function.
2. What is a quantile?
A quantile is a value below which a given percentage of data in a distribution falls. The inverse normal finds the quantile for a specified probability.
3. What is the formula for the inverse normal?
\(x = \mu + Z_p \times \sigma\), where \(Z_p\) is the Z-score for the given probability, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
4. What is a Z-score?
A Z-score is the number of standard deviations a value is from the mean. In the inverse normal, you find the Z-score for a given probability.
5. What is the range of probability values I can use?
You can use any probability between 0 and 1 (not including exactly 0 or 1). For 0 or 1, the result is negative or positive infinity.
6. What happens if I enter a probability of 0 or 1?
The result will be negative infinity for 0 and positive infinity for 1, since the normal distribution extends infinitely in both directions.
7. Can I use this for non-normal distributions?
No, this tool is only for the normal (Gaussian) distribution. Other distributions have different inverse functions.
8. What are some real-world uses of the inverse normal?
It's used in quality control (finding specification limits), finance (VaR), setting cut-off scores in exams, and more.
9. Why is the result sometimes negative?
If the probability is less than 0.5 and the mean is 0, the result will be negative, as it falls to the left of the mean.
10. Can I use this to find percentiles?
Yes! Enter the percentile as a decimal (e.g., 0.95 for the 95th percentile) to find the corresponding value.
11. What if my standard deviation is very small or very large?
A small standard deviation means values are tightly clustered around the mean; a large one means they're more spread out. The result will reflect this spread.
12. How accurate is this calculator?
It uses a standard rational approximation for the inverse normal, which is highly accurate for practical purposes.
13. Can I use this for academic research?
For most purposes, yes. For extremely high-precision needs, use specialized statistical software (like R or Python's SciPy).
14. What is the difference between the inverse normal and the CDF?
The CDF gives the probability for a value; the inverse normal gives the value for a probability.
15. How does this relate to Z-tables?
Z-tables give the probability for a Z-score. The inverse normal finds the Z-score for a given probability, then converts it to x using your mean and standard deviation.
Explore More Engineering Insights
Continue your learning journey with our extensive resources.