Normal Curve Calculator

This Normal Curve Calculator allows users to compute key statistics such as Z-Score, Probability Density, Cumulative Probability, and the area above a given X value for a normal distribution, given the population mean, standard deviation, and an X value.

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How to Use the Normal Curve Calculator

The Normal Curve Calculator is designed to compute various statistics related to the normal distribution. Follow these steps to get started:

Step 1: Provide Input Values

  1. Enter the Population Mean (μ): Locate the input field labeled “Population Mean (μ)”. Enter your data’s mean value. Ensure the value is between -999999 and 999999 and can have up to two decimal points.

  2. Enter the Standard Deviation (σ): Find the input field labeled “Standard Deviation (σ)”. Input the standard deviation of the population. It must be greater than 0.0001 and can be as large as 999999 with precision up to two decimal places.

  3. Enter the X Value: Use the input field labeled “X Value” to enter the specific point x on the distribution. This value needs to be within the range of -999999 to 999999 and may also include decimal precision up to two decimal places.

Step 2: Review Result Fields

Once you’ve entered the input data, the calculator provides several output fields based on standard statistical formulas:

  • Z-Score: This field calculates and displays the z-score, using the formula ( z = frac{x – mu}{sigma} ). The value is formatted to four decimal places.

  • Probability Density: Shows the probability density function (PDF) at the specific X value. It uses the formula ( frac{1}{sigma sqrt{2pi}} expleft(-0.5 left(frac{x – mu}{sigma}right)^2right) ) and displays the result to six decimal places.

  • Cumulative Probability P(X ≤ x): This field determines the cumulative probability up to the x value using the formula ( 0.5 left(1 + text{erf}left(frac{x – mu}{sigma sqrt{2}}right)right) ), formatted as a percentage with four decimal places.

  • Area Above P(X > x): Calculates and displays the area to the right of x, computed as ( 1 – (0.5 left(1 + text{erf}left(frac{x – mu}{sigma sqrt{2}}right)right)) ). The result is shown as a percentage with four decimal place precision.

Step 3: Understand Standard Deviation Probabilities

The calculator also estimates the probabilities of values falling within certain standard deviations from the mean:

  • Probability Within ±1σ: Based on the error function, this shows the probability of an observed value lying within one standard deviation from the mean. The percentage is shown with two decimal places.

  • Probability Within ±2σ: Displays the likelihood of data being within two standard deviations, again derived from the error function and formatted as a percentage with two decimal places.

  • Probability Within ±3σ: Shows the probability that a value lies within three standard deviations of the mean, also displayed as a percentage with two decimal digit accuracy.

By following the above steps, you can efficiently utilize the Normal Curve Calculator for standard normal distribution analysis.