Area to the Left of Z-Score Calculator

Enter a Z-score to calculate the area under the normal distribution curve to the left of the Z-score.

Z-Score:

Area to the Left of Z-Score (P(Z ≤ z)):

Understanding the Z-Score and Area Under the Normal Curve

The Z-Score, also known as the standard score, measures the number of standard deviations a data point is from the mean in a standard normal distribution (mean = 0, standard deviation = 1). This score helps in comparing individual data points within the context of the entire distribution.

In statistics, the area to the left of a Z-score represents the cumulative probability that a random variable is less than or equal to a specified Z-score value. This area is often used to find probabilities and percentiles, making it a valuable tool for decision-making in fields like psychology, finance, and research.

Key Components of the Z-Score

Z-Score: Indicates how many standard deviations a particular value is from the mean. Positive Z-scores are above the mean, while negative Z-scores are below the mean.
Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1, used as the basis for calculating Z-scores and their cumulative areas.

Formula for the Standard Normal Distribution

The probability density function (PDF) for the standard normal distribution is:

\[ f(z) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{z^2}{2}\right) \]

Programmatically Calculating the Area to the Left of a Z-Score

To calculate the area to the left of a Z-score, we can use popular libraries in JavaScript, Python, and R. Here’s how to perform these calculations:

1. Using JavaScript

In JavaScript, you can use the jStat library to calculate the cumulative probability for a given Z-score.

// Calculate cumulative probability for a Z-score
const z = 1.5;  // Example Z-score

// CDF: P(Z ≤ z)
const cumulativeProbability = jStat.normal.cdf(z, 0, 1);

console.log('Cumulative Probability (P(Z ≤ 1.5)):', cumulativeProbability);

Note: Ensure you have included the jStat library in your project to perform these calculations.

2. Using Python

In Python, the SciPy library offers a function for cumulative distribution calculations.

from scipy.stats import norm

# Define the Z-score
z = 1.5  # Example Z-score

# CDF: P(Z ≤ z)
cumulative_probability = norm.cdf(z)

print("Cumulative Probability (P(Z ≤ 1.5)):", cumulative_probability)

Note: Install SciPy by running pip install scipy if it’s not already installed.

3. Using R

In R, the stats package includes functions for cumulative probability calculations for the normal distribution.

# Define the Z-score
z <- 1.5  # Example Z-score

# CDF: P(Z ≤ z)
cumulative_probability <- pnorm(z)

cat("Cumulative Probability (P(Z ≤ 1.5)):", cumulative_probability, "\n")

Note: The pnorm function in R calculates the cumulative probability for a specified Z-score.

Example Calculation

Let's assume we have a Z-score of \( z = 1.5 \). The area to the left of this Z-score, or \( P(Z \leq 1.5) \), represents the probability that a randomly chosen value from the distribution is less than or equal to 1.5 standard deviations above the mean.