1. What is Normal Versus Standard Curves Comparison?

This calculator compares probabilities between normal distributions with different parameters and the standard normal distribution (μ=0, σ=1). It transforms any normal distribution to the standard normal scale using z-scores.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( x \) — Value of interest
• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation of the distribution
• \( \Phi \) — Standard normal cumulative distribution function

Explanation: The formula converts any normal distribution to the standard normal distribution by calculating the z-score, then finds the probability using the standard normal CDF.

3. Importance of Probability Calculation

Details: This calculation is fundamental in statistics for hypothesis testing, confidence intervals, and understanding how extreme a particular value is within a distribution.

4. Using the Calculator

Tips: Enter your value (x), the mean (μ) and standard deviation (σ) of your normal distribution. The calculator will show the equivalent z-score and probability.

5. Frequently Asked Questions (FAQ)

Q1: What is a z-score?
A: A z-score measures how many standard deviations an observation is from the mean. Positive z-scores are above the mean, negative are below.

Q2: Why use the standard normal distribution?
A: The standard normal (μ=0, σ=1) provides a common reference frame and its probabilities are well-tabulated, making calculations easier.

Q3: What does the probability represent?
A: The probability P(X < x) represents the area under the normal curve to the left of your x value.

Q4: Can I use this for non-normal distributions?
A: No, this calculation is specifically for normal distributions. Other distributions require different methods.

Q5: How accurate is the probability calculation?
A: The calculator uses a precise approximation of the standard normal CDF, accurate to at least 4 decimal places.