Calculate the cumulative probability (area under the curve) to the left or right of a given Z-score.
Area to the Left (P(Z < z))
Area to the Right (P(Z > z))
Area in Both Tails (Two-Tailed P)
Understanding Z-Scores and Probability
The Z-score, also known as the standard score, is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviation. A Z-score of 0 indicates that the data point's value is identical to the mean value. A positive Z-score indicates that the data point is above the mean, while a negative Z-score indicates that it is below the mean.
The Z-score is calculated using the formula:
Z = (X - μ) / σ
Where:
X is the raw score (the value you want to standardize).
μ (mu) is the population mean.
σ (sigma) is the population standard deviation.
The Z-Score and the Standard Normal Distribution
The Z-score is intrinsically linked to the Standard Normal Distribution, which is a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. The beauty of the Z-score is that it allows us to compare values from different normal distributions on a common scale.
The area under the curve of the standard normal distribution represents probability. The total area under the curve is 1 (or 100%). The Z-score allows us to determine the probability of observing a value less than, greater than, or between specific Z-scores.
How This Calculator Works
This calculator takes a Z-score as input and uses the cumulative distribution function (CDF) of the standard normal distribution to find the associated probability.
Area to the Left (P(Z < z)): This is the probability that a randomly selected value from the distribution will be less than the Z-score you provided. It represents the cumulative area under the curve from the far left up to your Z-score.
Area to the Right (P(Z > z)): This is the probability that a randomly selected value will be greater than your Z-score. It can be calculated as 1 - P(Z < z).
Area in Both Tails (Two-Tailed P): This represents the probability of observing values as extreme or more extreme than your Z-score in either the positive or negative direction. For a given Z-score z, this is typically calculated as P(Z < -|z|) + P(Z > |z|), which is equivalent to 2 * P(Z < -|z|) if the distribution is symmetric and you are interested in deviation from the mean. If you input a positive Z-score, it calculates the probability of being less than -z OR greater than z. If you input a negative Z-score, it calculates the probability of being less than z OR greater than -z.
Use Cases for Z-Scores and Probability
Z-scores and their corresponding probabilities are fundamental in statistical inference and hypothesis testing. They are used in:
Hypothesis Testing: Determining if an observed result is statistically significant enough to reject a null hypothesis.
Confidence Intervals: Estimating a range of values within which a population parameter is likely to fall.
Data Analysis: Identifying outliers or understanding the distribution of data within a population.
Quality Control: Assessing whether a product or process meets certain standards.
Standardizing Scores: Comparing test results from different exams with varying means and standard deviations.
Example Calculation
Let's say you have a Z-score of 1.96.
If you select "Area to the Left", the calculator will show that the probability P(Z < 1.96) is approximately 0.9750. This means about 97.5% of the data falls below this Z-score.
If you select "Area to the Right", the calculator will show P(Z > 1.96) is approximately 0.0250 (1 – 0.9750). This means about 2.5% of the data falls above this Z-score.
If you select "Area in Both Tails", the calculator will show the probability P(Z < -1.96) + P(Z > 1.96). Since P(Z > 1.96) is 0.0250 and by symmetry P(Z < -1.96) is also 0.0250, the total is approximately 0.0500. This is often used in hypothesis testing at a 5% significance level.
// Function to calculate the cumulative distribution function (CDF) for a standard normal distribution
// This is an approximation using the erf (error function)
// Source: https://introcs.cs.princeton.edu/java/91float/NormalDistribution.java.html
function normalCdf(z) {
// constants for the approximation
var a = [0.319382667, -0.356563782, 1.781477937, -1.821255978, 1.330274429];
var pi = Math.PI;
var norm = (1.0 / Math.sqrt(2 * pi));
var erf = function(x) {
var sign = (x >= 0) ? 1 : -1;
x = Math.abs(x);
var t = 1.0 / (1.0 + 0.5 * x);
var y = 1.0 – (((((a[4] * t + a[3]) * t) + a[2]) * t + a[1]) * t + a[0]) * t * Math.exp(-x * x);
return sign * y;
};
// The CDF is 0.5 * (1 + erf(z / sqrt(2)))
return 0.5 * (1.0 + erf(z / Math.sqrt(2)));
}
function calculateProbability() {
var zScoreInput = document.getElementById("zScore");
var tailSelect = document.getElementById("tail");
var resultDiv = document.getElementById("result");
var errorMessageDiv = document.getElementById("errorMessage");
// Clear previous error messages
errorMessageDiv.textContent = "";
resultDiv.textContent = "";
// Get values from inputs
var zScore = parseFloat(zScoreInput.value);
var tailType = tailSelect.value;
// Input validation
if (isNaN(zScore)) {
errorMessageDiv.textContent = "Please enter a valid number for the Z-Score.";
return;
}
var cdfValue = normalCdf(zScore);
var probability = 0;
if (tailType === "left") {
probability = cdfValue;
} else if (tailType === "right") {
probability = 1 – cdfValue;
} else if (tailType === "both") {
// For two-tailed test, we usually look at the area in the tails beyond +/- |z|
// If zScore is positive, we want P(Z zScore)
// If zScore is negative, we want P(Z -zScore)
// In both cases, due to symmetry, it's 2 * min(P(Z zScore))
var absZScore = Math.abs(zScore);
var leftTailProb = normalCdf(-absZScore); // Equivalent to 1 – normalCdf(absZScore)
var rightTailProb = 1 – normalCdf(absZScore);
probability = leftTailProb + rightTailProb;
}
// Format the probability to 4 decimal places
var formattedProbability = probability.toFixed(4);
// Display the result
var resultText = "";
if (tailType === "left") {
resultText = "P(Z " + zScore + ") = " + formattedProbability;
} else if (tailType === "both") {
resultText = "P(|Z| > |" + zScore + "|) = " + formattedProbability;
}
resultDiv.textContent = resultText;
}