Standard Normal Distribution Calculation

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Standard Normal Distribution Calculator

Calculation Results:

Z-Score:

P(X < ): (Left-tailed)

P(X > ): (Right-tailed)

P(μ < X < ): (Center to X)

Please ensure Standard Deviation is greater than 0 and all fields are filled.
function calculateNormalDist() { var x = parseFloat(document.getElementById("inputX").value); var mean = parseFloat(document.getElementById("inputMean").value); var sd = parseFloat(document.getElementById("inputSD").value); var resultsDiv = document.getElementById("sd-results"); var errorDiv = document.getElementById("sd-error"); if (isNaN(x) || isNaN(mean) || isNaN(sd) || sd <= 0) { resultsDiv.style.display = "none"; errorDiv.style.display = "block"; return; } errorDiv.style.display = "none"; var z = (x – mean) / sd; var p = getZPercent(z); document.getElementById("zScoreRes").innerText = z.toFixed(4); document.getElementById("valXSpan").innerText = x; document.getElementById("valXSpan2").innerText = x; document.getElementById("valXSpan3").innerText = x; document.getElementById("probLower").innerText = p.toFixed(5); document.getElementById("probUpper").innerText = (1 – p).toFixed(5); document.getElementById("probMean").innerText = Math.abs(p – 0.5).toFixed(5); resultsDiv.style.display = "block"; } function getZPercent(z) { if (z 6.5) return 1.0; var factK = 1; var sum = 0; var term = 1; var k = 0; var loopStop = Math.exp(-0.5 * z * z) / Math.sqrt(2 * Math.PI); while (Math.abs(term) > 0.0000000001) { term = (Math.pow(z, k) / (doubleFactorial(k))); // Using the Taylor series approximation for the CDF // However, for efficiency in JS, we use the error function approximation break; } // Standard Abramowitz and Stegun approximation var t = 1 / (1 + 0.2316419 * Math.abs(z)); var d = 0.3989423 * Math.exp(-z * z / 2); var prob = d * t * (0.3193815 + t * (-0.3565638 + t * (1.781478 + t * (-1.821256 + t * 1.330274)))); if (z > 0) return 1 – prob; return prob; } function doubleFactorial(n) { if (n <= 1) return 1; return n * doubleFactorial(n – 2); }

Understanding the Standard Normal Distribution

The Standard Normal Distribution, often referred to as the Z-distribution, is a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. This mathematical model is the cornerstone of inferential statistics and probability theory.

What is a Z-Score?

A Z-score represents the number of standard deviations a specific data point (X) is away from the mean. It allows researchers to compare scores from different datasets that might have different scales or units.

Z = (X – μ) / σ

How to Use This Calculator

  1. Value (X): Enter the specific observed value you want to analyze.
  2. Mean (μ): Enter the average value of your entire population or sample.
  3. Standard Deviation (σ): Enter the measure of spread in your data.
  4. Results: The calculator provides the Z-score and the cumulative probability (p-value) for both the lower and upper tails.

Real-World Example: IQ Scores

IQ scores are typically normally distributed with a mean of 100 and a standard deviation of 15. If an individual has an IQ of 130, we can calculate their standing:

  • X: 130
  • Mean: 100
  • SD: 15
  • Z-Score Calculation: (130 – 100) / 15 = 2.0

A Z-score of 2.0 means this individual is 2 standard deviations above the mean. Looking at the standard normal distribution table, a Z-score of 2.0 corresponds to a percentile of approximately 97.7%, meaning they scored higher than 97.7% of the population.

The Empirical Rule (68-95-99.7)

In a normal distribution:

  • 68.2% of data falls within +/- 1 standard deviation of the mean.
  • 95.4% of data falls within +/- 2 standard deviations of the mean.
  • 99.7% of data falls within +/- 3 standard deviations of the mean.

Our calculator helps you find the exact probabilities for values that fall anywhere along this curve, providing more precision than the general empirical rule.

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