Normal Distribution Calculator

Calculate z-scores, cumulative probability, tail probability, and interval probability for a normal distribution

Example Results

2 examples

Evaluate an exam score in a normal model

Calculate z-score, percentile, and tail probability for a score when the mean and standard deviation are known

{
  "result": {
    "zScore": 1,
    "cumulativeProbability": 0.8413,
    "upperTailProbability": 0.1587,
    "percentile": 84.1345
  }
}
View input parameters
{ "rawValue": "85", "meanInput": "75", "standardDeviationInput": "10", "lowerBound": "", "upperBound": "", "precision": 4 }

Compute interval probability between two values

Estimate the probability that a normally distributed value falls inside a given range

{
  "result": {
    "zScore": 0,
    "cumulativeProbability": 0.5,
    "intervalProbability": 0.6827,
    "percentile": 50
  }
}
View input parameters
{ "rawValue": "100", "meanInput": "100", "standardDeviationInput": "15", "lowerBound": "85", "upperBound": "115", "precision": 4 }

Key Facts

Category
Math, Date & Finance
Input Types
text, number
Output Type
json
Sample Coverage
2
API Ready
Yes

Overview

The Normal Distribution Calculator allows you to perform statistical analysis by calculating z-scores, cumulative probabilities, and interval probabilities based on your specific mean and standard deviation parameters.

When to Use

  • •When you need to determine the percentile rank of a specific data point within a normal distribution.
  • •When you want to calculate the probability of a value falling within a specific range or interval.
  • •When you are performing statistical hypothesis testing and need to find tail probabilities.

How It Works

  • •Enter the raw value, the population mean, and the standard deviation into the respective fields.
  • •Optionally, provide lower and upper bounds if you need to calculate the probability of a value falling within a specific interval.
  • •Set your desired decimal precision to ensure the output meets your reporting requirements.
  • •Click calculate to receive the z-score, cumulative probability, and relevant tail or interval probabilities.

Use Cases

Academic grading: Determining the percentage of students who scored above or below a certain threshold.
Quality control: Calculating the probability that a manufactured part falls within acceptable tolerance limits.
Financial analysis: Estimating the likelihood of asset returns falling within a specific range based on historical volatility.

Examples

1. Evaluate an exam score

Student
Background
An exam has a mean score of 75 with a standard deviation of 10.
Problem
Determine how a score of 85 compares to the rest of the class.
How to Use
Input 85 as the raw value, 75 as the mean, and 10 as the standard deviation.
Example Config
rawValue: 85, meanInput: 75, standardDeviationInput: 10
Outcome
The calculator returns a z-score of 1 and a cumulative probability of 0.8413, indicating the score is in the 84th percentile.

2. Compute interval probability

Quality Assurance Engineer
Background
A machine produces parts with a mean length of 100mm and a standard deviation of 15mm.
Problem
Find the probability that a part length is between 85mm and 115mm.
How to Use
Input 100 as the mean, 15 as the standard deviation, and set the lower bound to 85 and upper bound to 115.
Example Config
meanInput: 100, standardDeviationInput: 15, lowerBound: 85, upperBound: 115
Outcome
The tool calculates an interval probability of 0.6827, showing that approximately 68.27% of parts fall within this range.

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FAQ

What is a z-score?

A z-score represents the number of standard deviations a data point is from the mean of the distribution.

What does cumulative probability measure?

It measures the probability that a random variable will take a value less than or equal to the specified raw value.

Can I calculate probabilities for ranges?

Yes, by entering values into the Lower Bound and Upper Bound fields, the tool calculates the probability of a value falling within that range.

What is the default decimal precision?

The tool defaults to 4 decimal places, but you can adjust this between 0 and 10 based on your needs.

Is this tool suitable for non-normal distributions?

No, this calculator is specifically designed for data that follows a normal (Gaussian) distribution.

API Documentation

Request Endpoint

POST /en/api/tools/normal-distribution-calculator

Request Parameters

Parameter Name Type Required Description
rawValue text Yes -
meanInput text Yes -
standardDeviationInput text Yes -
lowerBound text No -
upperBound text No -
precision number No -

Response Format

{
  "key": {...},
  "metadata": {
    "key": "value"
  },
  "error": "Error message (optional)",
  "message": "Notification message (optional)"
}
JSON Data: JSON Data

AI MCP Documentation

Add this tool to your MCP server configuration: 

{
  "mcpServers": {
    "elysiatools-normal-distribution-calculator": {
      "name": "normal-distribution-calculator",
      "description": "Calculate z-scores, cumulative probability, tail probability, and interval probability for a normal distribution",
      "baseUrl": "https://elysiatools.com/mcp/sse?toolId=normal-distribution-calculator",
      "command": "",
      "args": [],
      "env": {},
      "isActive": true,
      "type": "sse"
    }
  }
}

You can chain multiple tools, e.g.: `https://elysiatools.com/mcp/sse?toolId=png-to-webp,jpg-to-webp,gif-to-webp`, max 20 tools.

If you encounter any issues, please contact us at [email protected]