Find The Shaded Area Under The Standard Normal Curve Calculator

Calculate Area Under Normal Curve

Find the area (probability) under the standard normal distribution curve (Z ~ N(0,1)).

What is the Shaded Area Under Standard Normal Curve Calculator?

The shaded area under standard normal curve calculator is a tool used to find the probability associated with a range of values for a standard normal variable (Z). The standard normal distribution is a special normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. The total area under this curve is equal to 1 (or 100%), and the area under a portion of the curve represents the probability of a random variable falling within that specific range.

This calculator is essential for anyone working with statistics, particularly in fields like data science, quality control, finance, and research. It helps determine probabilities (p-values) for z-scores, which are fundamental in hypothesis testing and confidence interval estimation.

Common misconceptions include thinking the z-score itself is a probability or that the curve represents any normal distribution directly without standardization.

Shaded Area Under Standard Normal Curve Formula and Mathematical Explanation

The area under the standard normal curve between two z-scores, z1 and z2, is found by calculating the difference between the cumulative distribution function (CDF) values at these points:

Area = P(z1 < Z < z2) = Φ(z2) – Φ(z1)

Where Φ(z) is the standard normal CDF, which gives the area under the curve to the left of z:

Φ(z) = P(Z ≤ z) = ∫_-∞^z (1/√(2π)) * e^(-t²/2) dt

The shaded area under standard normal curve calculator uses numerical methods to approximate Φ(z), often based on the error function (erf), as there is no simple closed-form solution for the integral.

Φ(z) = 0.5 * (1 + erf(z / √2))

For different scenarios:

• Area to the left of z: Area = Φ(z)
• Area to the right of z: Area = 1 – Φ(z)
• Area between z1 and z2: Area = Φ(z2) – Φ(z1) (assuming z1 < z2)
• Area outside z1 and z2: Area = Φ(z1) + (1 – Φ(z2)) (assuming z1 < z2)

Variables Table

Variable	Meaning	Unit	Typical Range
Z	Standard Normal Variable	None (standard deviations)	-∞ to +∞
z, z1, z2	Z-scores (points on the x-axis)	None (standard deviations)	-4 to 4 (most common)
Φ(z)	Cumulative Distribution Function at z	Probability (0 to 1)	0 to 1
Area	Probability associated with the z-scores	Probability (0 to 1)	0 to 1

Practical Examples (Real-World Use Cases)

Example 1: Test Scores

Suppose test scores in a large class are normally distributed with a mean of 70 and a standard deviation of 10. We want to find the proportion of students who scored between 60 and 85.

First, convert scores to z-scores:

z1 = (60 – 70) / 10 = -1

z2 = (85 – 70) / 10 = 1.5

Using the shaded area under standard normal curve calculator with z1 = -1 and z2 = 1.5 (between):

Φ(-1) ≈ 0.1587

Φ(1.5) ≈ 0.9332

Area = 0.9332 – 0.1587 = 0.7745

So, about 77.45% of students scored between 60 and 85.

Example 2: Manufacturing Quality Control

A machine fills bags with 500g of sugar, normally distributed with a standard deviation of 5g. Bags outside the 490g to 510g range are rejected. What proportion of bags are rejected?

z1 = (490 – 500) / 5 = -2

z2 = (510 – 500) / 5 = 2

We want the area outside z1=-2 and z2=2. Using the shaded area under standard normal curve calculator:

Φ(-2) ≈ 0.0228

Φ(2) ≈ 0.9772

Area outside = Φ(-2) + (1 – Φ(2)) = 0.0228 + (1 – 0.9772) = 0.0228 + 0.0228 = 0.0456

About 4.56% of bags are rejected.

How to Use This Shaded Area Under Standard Normal Curve Calculator

1. Select Area Type: Choose whether you want the area to the left of z, right of z, between z1 and z2, or outside z1 and z2 using the radio buttons.
2. Enter Z-scores:
   • If "left" or "right" is selected, enter the z-score in the "Z-score (z)" field.
   • If "between" or "outside" is selected, enter the z-scores in the "Lower Z-score (z1)" and "Upper Z-score (z2)" fields. Ensure z1 < z2 for "outside".
3. Calculate: Click the "Calculate Area" button or simply change the input values. The results will update automatically.
4. Read Results: The "Primary Result" shows the calculated area (probability). Intermediate values Φ(z) or Φ(z1) and Φ(z2) are also displayed.
5. View Chart: The chart below visually represents the standard normal curve and the shaded area corresponding to your input(s).
6. Reset: Click "Reset" to return to default values.
7. Copy: Click "Copy Results" to copy the area and intermediate values to your clipboard.

The calculated area is the probability of a standard normal random variable falling within the specified z-score range(s). For example, if you find the area to the left of z=1.96 is 0.975, it means there's a 97.5% chance Z is less than or equal to 1.96.

Key Factors That Affect Shaded Area Results

• Z-score(s): The values of z, z1, and z2 directly determine the boundaries of the area being calculated. Larger |z| values generally correspond to areas closer to 0 or 1 in the tails.
• Type of Area: Whether you calculate left-tail, right-tail, between, or outside dramatically changes the result based on the same z-scores.
• Mean and Standard Deviation (Implied): This is a standard normal curve calculator, so the mean is fixed at 0 and standard deviation at 1. If you start with a non-standard normal distribution, you must first convert your X values to z-scores using z = (X – μ) / σ.
• Precision of Calculation: The underlying numerical integration or approximation method for Φ(z) affects the precision of the area. Our calculator uses a standard approximation for the error function.
• Continuity Correction: When approximating discrete distributions (like binomial) with the normal distribution, a continuity correction (adding or subtracting 0.5 from X before converting to z) can significantly impact the z-scores and thus the area.
• Symmetry of the Curve: The normal distribution is symmetric around the mean (0). This means Φ(-z) = 1 – Φ(z), which is used in calculations, especially for "right tail" and "outside" areas.

Understanding these factors is crucial for accurately interpreting the results from any shaded area under standard normal curve calculator.

Frequently Asked Questions (FAQ)

What does the standard normal distribution represent?

It's a normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be standardized by converting its values to z-scores.

What is a z-score?

A z-score measures how many standard deviations an element is from the mean. A positive z-score is above the mean, and a negative z-score is below the mean.

Why is the total area under the normal curve equal to 1?

The total area represents the total probability of all possible outcomes, which must sum to 1 (or 100%).

How do I find the area for a non-standard normal distribution?

First, convert your X values to z-scores using the formula z = (X – μ) / σ, where μ is the mean and σ is the standard deviation of your distribution. Then use the shaded area under standard normal curve calculator with the calculated z-scores.

Can I find the z-score given an area?

Yes, that's called finding the inverse CDF or quantile function. This calculator finds the area given z; you'd need an inverse normal distribution calculator for the reverse. Check our z-score calculator for related functions.

What does it mean if the area is very small?

A very small area indicates that the range of z-scores is far from the mean, in the tails of the distribution, and the probability of observing values in that range is very low.

What is the area between z=-1 and z=1?

Approximately 0.6827, or 68.27% (the empirical rule).

What is the area between z=-2 and z=2?

Approximately 0.9545, or 95.45% (the empirical rule).

Related Tools and Internal Resources

• Z-Score Calculator: Calculate z-scores from raw data, mean, and standard deviation.
• Probability Calculator: Explore various probability calculations and distributions.
• Statistics Calculators: A collection of calculators for various statistical measures.
• Normal Distribution Guide: Learn more about the properties and applications of the normal distribution.
• Central Limit Theorem Explained: Understand how the normal distribution arises in many contexts.
• Hypothesis Testing Calculator: Use z-scores and areas in hypothesis testing.