Standard Normal Distribution Calculator

Standard Normal Distribution:

\[ P = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-\frac{t^2}{2}} dt \]

Cumulative Probability (P):

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1. What is Standard Normal Distribution?

The standard normal distribution is a special case of the normal distribution with a mean of 0 and standard deviation of 1. It's used to find probabilities and percentiles for normal distributions.

2. How Does the Calculator Work?

The calculator uses the standard normal distribution formula:

\[ P = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-\frac{t^2}{2}} dt \]

Where:

\( P \) — Cumulative probability (dimensionless)
\( z \) — Standard score (dimensionless)

Explanation: The equation calculates the area under the standard normal curve to the left of a given z-score.

3. Importance of Z-Scores

Details: Z-scores allow comparison of values from different normal distributions and are fundamental in statistical hypothesis testing.

4. Using the Calculator

Tips: Enter the z-score (positive or negative) to calculate the cumulative probability. The result represents the probability that a random variable from the standard normal distribution is less than or equal to the given z-score.

5. Frequently Asked Questions (FAQ)

Q1: What does a z-score represent?
A: A z-score measures how many standard deviations an element is from the mean.

Q2: What are common z-score values?
A: ±1.96 corresponds to 95% confidence, ±2.58 to 99% confidence in two-tailed tests.

Q3: How to interpret the probability result?
A: A result of 0.95 means there's a 95% chance a random value will be below your z-score.

Q4: Can I calculate z-score from probability?
A: Yes, this would be the inverse calculation (quantile function).

Q5: When is this distribution used?
A: In statistical quality control, finance, social sciences, and whenever data follows a normal distribution.