Bell Curve Calculator

The Bell Curve Calculator allows users to input mean, standard deviation, and an x value to compute probability density, z-score, cumulative probability, and the percentages of data within one, two, and three standard deviations.

Use Our Bell Curve Calculator

How to Use the Bell Curve Calculator

Overview

This Bell Curve Calculator helps in computing statistical values such as probability density, z-score, cumulative probability, and percentages within standard deviations based on the normal distribution. Follow the guide below to understand how to use the calculator efficiently.

Step 1: Input Values

Start by entering the required values into their respective fields:

  • Mean (μ): Enter the mean value of the dataset. This field is required and must be a number between -1,000,000 and 1,000,000. The input is accepted in increments of 0.01.
  • Standard Deviation (σ): Enter the standard deviation value. Ensure the value is greater than 0.0001 and less than or equal to 1,000,000. The step value for input is 0.01, and this field is required.
  • X Value: This is the specific data point for which you want to calculate the probability. The value should range between -1,000,000 and 1,000,000 with permissible increments of 0.01. This is also a mandatory field.

Step 2: Calculate the Results

After entering the values, the calculator will automatically compute the following statistical measures:

  • Probability Density: This is computed as ( (1 / (σ times sqrt{2 times π})) times exp[-(x – μ)^2 / (2 times σ^2)] ). The result is displayed as a number up to six decimal places.
  • Z-Score: Calculated using the formula ( (x – μ) / σ ), the Z-Score represents how many standard deviations away your X Value is from the mean. The output has three decimal places.
  • Cumulative Probability: Use the formula ( 0.5 times (1 + text{erf}((x – μ) / (σ times sqrt{2}))) ) to find the cumulative distribution function value, represented as a percentage with two decimal places.
  • Within ±1 Standard Deviation: This fixed value of 68.27% indicates the probability of values within one standard deviation of the mean.
  • Within ±2 Standard Deviations: This fixed percentage of 95.45% shows the probability of values falling within two standard deviations.
  • Within ±3 Standard Deviations: The fixed result of 99.73% suggests the likelihood of values within three standard deviations from the mean.

By following these steps, you will efficiently utilize the Bell Curve Calculator to analyze statistical data based on your inputs.