1. What is the Normal Probability Calculator?

The Normal Probability Calculator computes the probability that a normally distributed random variable X is less than a given value x, using the mean (μ) and standard deviation (σ) of the distribution.

2. How Does the Calculator Work?

The calculator uses the standard normal distribution formula:

Where:

• \( x \) — The value you want to evaluate
• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation of the distribution
• \( \Phi \) — Cumulative Distribution Function (CDF) of the standard normal distribution

Explanation: The formula converts your value to a z-score and then uses the standard normal CDF to find the probability.

3. Importance of Normal Distribution

Details: The normal distribution is fundamental in statistics, describing many natural phenomena. Calculating probabilities under this curve is essential for statistical inference, quality control, and risk assessment.

4. Using the Calculator

Tips: Enter your value (x), the mean (μ) of your distribution, and the standard deviation (σ). Standard deviation must be positive. The calculator will return P(X < x).

5. Frequently Asked Questions (FAQ)

Q1: What does P(X < x) mean?
A: It represents the probability that a random variable from this normal distribution will be less than the value x.

Q2: How accurate is this calculator?
A: It uses a precise approximation of the normal CDF with accuracy to at least 7 decimal places.

Q3: Can I calculate P(X > x) with this?
A: Yes, simply subtract the result from 1 (P(X > x) = 1 - P(X < x)).

Q4: What if my data isn't normally distributed?
A: This calculator assumes normal distribution. For non-normal data, consider transformations or non-parametric methods.

Q5: What's the range of possible probabilities?
A: Probabilities range from 0 to 1, where 0 means impossible and 1 means certain.