This tool will calculate the probability of a value occurring within a specified range based on the cumulative distribution function.

## How to Use the Cumulative Distribution Function (CDF) Calculator

The CDF calculator allows you to compute the cumulative probability up to a specific x value in a normal distribution, given the mean and standard deviation.

### Steps to Use:

- Enter the mean (μ) value of your dataset.
- Enter the standard deviation (σ) value of your dataset.
- Enter the x value for which you want to find the cumulative probability.
- Click “Calculate” to see the result in the “Result” field.

### Explanation of the Calculation:

The calculator first converts the given x value to a z-score using the formula: z = (x – μ) / σ. The z-score is then used to find the cumulative probability using the standard normal distribution.

### Limitations:

This calculator uses a numerical approximation to compute the CDF, which may introduce small errors for extreme values. For most practical purposes, the approximation should be sufficiently accurate.

## Use Cases for This Calculator

### Calculate Probability for a Specific Value

Enter the value of interest along with the appropriate parameters such as mean and standard deviation to find the probability of obtaining that value or a lower value in a distribution. The calculator will provide you with the cumulative probability up to that specific value.

### Estimate Probability Range

Input a range of values along with the distribution parameters to determine the cumulative probability of landing within that range. This use case will help you assess the likelihood of the random variable falling within the specified interval.

### Analyze Probability Distribution Shape

Experiment with different mean and standard deviation values using the calculator to observe how the shape of the cumulative distribution function changes. This feature can help you visualize the impact of altering the distribution parameters.

### Calculate Probability of a Range

Specify a lower and upper limit of a range to calculate the probability of the random variable falling within that specific interval. The calculator will provide you with the cumulative probability corresponding to the defined range.

### Explore Probability Density

Adjust the input parameters to explore how the probability density function translates into cumulative probabilities for different values. This use case allows for a deeper understanding of the relationship between probability density and cumulative distribution.

### Compare Cumulative Probabilities

Input multiple values to compare the cumulative probabilities associated with each value in the distribution. This feature enables you to assess the relative likelihood of different outcomes based on their cumulative probabilities.

### Evaluate Probability of Outliers

Input extreme values far from the mean to evaluate the probability of such outliers occurring in the distribution. This use case can help you assess the likelihood of rare events based on their cumulative probabilities.

### Understand Probability Scaling

Manipulate the input values to understand how scaling the distribution affects the cumulative probabilities. This feature allows you to grasp how changes in the distribution parameters impact the overall probability distribution.

### Calculate Cumulative Probability at Percentiles

Enter percentile values to calculate the cumulative probability corresponding to those percentiles in the distribution. This use case helps you determine the probability associated with specific percentiles of the cumulative distribution function.

### Validate Probability Assessments

Compare the calculated cumulative probabilities with theoretical expectations to validate the accuracy of probability assessments. This feature enables you to confirm the correctness of your probability calculations using the cumulative distribution function.