Find The Area Under The Normal Curve Calculator

Use this calculator to find the area under the normal curve between two values, to the left, or to the right of a value, given the mean and standard deviation.

Point	X Value	Z-Score	Area to the Left (Φ(z))
Lower (X1)	-1	-1.0000	0.1587
Upper (X2)	1	1.0000	0.8413

Table of X values, Z-scores, and Cumulative Probabilities

What is an Area Under the Normal Curve Calculator?

An area under the normal curve calculator is a statistical tool used to determine the probability or proportion of data falling within a specific range of values in a normal distribution (also known as a Gaussian distribution or bell curve). It calculates the area under the curve between two given points (X1 and X2), to the left of a point (X), or to the right of a point (X), based on the mean (μ) and standard deviation (σ) of the distribution.

This area represents the probability that a random variable following the normal distribution will take a value within that range. For example, if you find the area between X1 and X2 is 0.68, it means there's a 68% chance that a randomly selected value from this distribution will lie between X1 and X2.

Who Should Use It?

This calculator is beneficial for:

• Students and Educators: For learning and teaching statistics, probability, and the normal distribution.
• Researchers and Analysts: To analyze data, determine probabilities, and make inferences based on normally distributed data.
• Engineers and Quality Control Professionals: For process control and quality assurance, where many measurements follow a normal distribution.
• Finance Professionals: To model asset returns and assess risks, as many financial models assume normal distribution.
• Anyone working with data that is or can be approximated by a normal distribution.

Common Misconceptions

• All data is normally distributed: While the normal distribution is common, not all datasets follow it. It's important to check the distribution of your data first.
• The area is the same as the height of the curve: The area represents cumulative probability over a range, while the height (probability density) is the likelihood at a single point (though probability at a single point is zero for continuous distributions).
• Mean and Median are always the same: In a perfectly normal distribution, the mean, median, and mode are identical. However, real-world data might only approximate a normal distribution.

Area Under the Normal Curve Formula and Mathematical Explanation

The normal distribution's probability density function (PDF) is given by:

f(x; μ, σ) = (1 / (σ * √(2π))) * e^{-0.5 * ((x – μ) / σ)2}

Where:

• x is the value on the x-axis
• μ is the mean of the distribution
• σ is the standard deviation of the distribution
• e is Euler's number (approx. 2.71828)
• π is Pi (approx. 3.14159)

To find the area under this curve between two points, X1 and X2, we need to integrate the PDF from X1 to X2. However, this integral doesn't have a simple closed-form solution. So, we first convert the X values to standard normal scores (z-scores) using:

z = (x – μ) / σ

This transforms the distribution to a standard normal distribution with a mean of 0 and a standard deviation of 1. The PDF of the standard normal distribution is:

φ(z) = (1 / √(2π)) * e^{-0.5 * z2}

The area to the left of a z-score is given by the cumulative distribution function (CDF), Φ(z):

Φ(z) = ∫_-∞^z φ(t) dt

The area between z1 and z2 (corresponding to X1 and X2) is then:

Area = Φ(z2) – Φ(z1)

The area to the left of z is Φ(z), and the area to the right of z is 1 – Φ(z).

The calculator uses a numerical approximation for the error function (erf) to compute Φ(z), as Φ(z) = 0.5 * (1 + erf(z / √2)).

Variables Table

Variable	Meaning	Unit	Typical Range
μ (Mean)	The average value or center of the distribution.	Same as data	Any real number
σ (Standard Deviation)	A measure of the spread or dispersion of the data around the mean.	Same as data	Positive real number (>0)
X1, X2, X	Specific values from the distribution for which we want to find the area/probability.	Same as data	Any real number
z (z-score)	The number of standard deviations a value X is from the mean μ.	Dimensionless	Typically -4 to +4, but can be any real number
Φ(z) (CDF)	The cumulative probability from -∞ up to z; the area to the left of z under the standard normal curve.	Probability	0 to 1

Practical Examples (Real-World Use Cases)

Example 1: Exam Scores

Suppose exam scores in a large class are normally distributed with a mean (μ) of 75 and a standard deviation (σ) of 10. We want to find the percentage of students who scored between 65 and 85.

• Mean (μ) = 75
• Standard Deviation (σ) = 10
• X1 = 65
• X2 = 85

Using the area under the normal curve calculator with these inputs:

• z1 = (65 – 75) / 10 = -1
• z2 = (85 – 75) / 10 = 1
• Area = Φ(1) – Φ(-1) ≈ 0.8413 – 0.1587 = 0.6826

So, about 68.26% of students scored between 65 and 85.

Example 2: Manufacturing Tolerances

A machine produces bolts with a mean diameter (μ) of 10 mm and a standard deviation (σ) of 0.05 mm. The diameters are normally distributed. What percentage of bolts will have a diameter between 9.9 mm and 10.1 mm?

• Mean (μ) = 10
• Standard Deviation (σ) = 0.05
• X1 = 9.9
• X2 = 10.1

Using the area under the normal curve calculator:

• z1 = (9.9 – 10) / 0.05 = -2
• z2 = (10.1 – 10) / 0.05 = 2
• Area = Φ(2) – Φ(-2) ≈ 0.9772 – 0.0228 = 0.9544

About 95.44% of the bolts will have a diameter within the desired tolerance.

How to Use This Area Under the Normal Curve Calculator

1. Enter the Mean (μ): Input the average value of your normally distributed dataset.
2. Enter the Standard Deviation (σ): Input the standard deviation, ensuring it's a positive number.
3. Select Calculation Type: Choose whether you want to find the area "Between X1 and X2", "To the Left of X", or "To the Right of X".
4. Enter X Values:
   • If "Between X1 and X2", enter the lower bound (X1) and upper bound (X2).
   • If "To the Left of X" or "To the Right of X", enter the single X value.
5. View Results: The calculator automatically updates the "Area", z-scores, and cumulative probabilities (Φ(z)) as you enter the values.
6. Interpret the Area: The "Area" result is the probability or proportion of the distribution within the specified range.
7. Examine Chart and Table: The chart visually represents the area, and the table provides the z-scores and cumulative probabilities for your X values.
8. Reset: Click "Reset" to return to default values.
9. Copy Results: Click "Copy Results" to copy the main area, intermediate values, and input parameters to your clipboard.

Key Factors That Affect Area Under the Normal Curve Results

1. Mean (μ): The mean shifts the entire curve left or right along the x-axis. Changing the mean changes the center of the distribution, thus affecting the X values relative to the center and the resulting area for a fixed X range not centered at the mean.
2. Standard Deviation (σ): The standard deviation determines the spread of the curve. A smaller σ makes the curve taller and narrower (more values close to the mean), while a larger σ makes it shorter and wider (values more spread out). This directly impacts the area within a given range of X values.
3. X Values (X1, X2, X): The specific points on the x-axis define the boundaries for the area calculation. The further these values are from the mean (relative to σ), the more extreme the z-scores, and the areas will change accordingly.
4. Choice of Area Type: Whether you calculate the area between two points, to the left, or to the right fundamentally changes which portion of the curve is being measured.
5. Symmetry of the Normal Curve: The normal curve is symmetric around the mean. This means the area to the left of μ – kσ is the same as the area to the right of μ + kσ.
6. Total Area: The total area under any normal curve is always 1 (or 100%), representing the total probability.

Frequently Asked Questions (FAQ)

Q: What is a z-score? A: A z-score (or standard score) measures how many standard deviations an element is from the mean. A z-score of 0 means the element is exactly at the mean, a z-score of 1 is 1 standard deviation above the mean, and -1 is 1 standard deviation below.

Q: Can I use this calculator for any type of distribution? A: No, this area under the normal curve calculator is specifically for data that follows a normal distribution or can be reasonably approximated by one.

Q: What if my standard deviation is zero? A: A standard deviation of zero is theoretically impossible for a distribution of data points (it would mean all data points are identical). The calculator requires a positive standard deviation.

Q: How is the area related to probability? A: The area under the curve between two points represents the probability that a random variable from that normal distribution will fall between those two points.

Q: What does an area of 0.5 mean? A: An area of 0.5 to the left of a point means that point is the median (and also the mean and mode in a normal distribution). 50% of the values fall below it.

Q: Can X1 be greater than X2 when calculating the area between them? A: Typically, X1 is the lower bound and X2 is the upper bound. If you enter X1 > X2, the calculator will likely show a result based on Φ(X2) – Φ(X1), which would be negative or zero if X1 is truly larger. It's best to ensure X1 ≤ X2. Our calculator will handle it, but it's good practice.

Q: What is the empirical rule (68-95-99.7 rule)? A: For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean (μ ± σ), 95% within two (μ ± 2σ), and 99.7% within three (μ ± 3σ). You can verify this using the area under the normal curve calculator.

Q: How do I know if my data is normally distributed? A: You can use methods like histograms, Q-Q plots, or statistical tests (like the Shapiro-Wilk test or Kolmogorov-Smirnov test) to assess normality.

Related Tools and Internal Resources

• Z-Score Calculator: Calculate the z-score for a given value, mean, and standard deviation.
• Standard Deviation Calculator: Compute the standard deviation and variance for a dataset.
• Probability Calculator: Explore various probability calculations and concepts.
• Statistics Basics: Learn fundamental concepts in statistics.
• Data Analysis Tools: Discover other tools for analyzing data.
• Normal Distribution Explained: A detailed guide to understanding the normal distribution.