Z Score Probability Calculator

Z-Score Probability Calculator

Calculation Results

Z-Score:

P(Z < z) – Left Tail: (Probability of score less than X)

P(Z > z) – Right Tail: (Probability of score greater than X)

P(-|z| < Z < |z|) – Central: (Probability between -z and +z)

Percentile:

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Understanding Z-Scores and Normal Distribution

A Z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. It is a fundamental concept in statistics used to compare observations from different normal distributions.

The Z-Score Formula

z = (X – μ) / σ

• X: The specific value you are testing (Raw Score).
• μ (Mu): The average value of the population (Mean).
• σ (Sigma): The measure of spread in the population (Standard Deviation).

What do the probabilities mean?

When you calculate the probability associated with a Z-score, you are finding the "Area Under the Curve" of the standard normal distribution:

• Left Tail P(Z < z): This tells you the percentage of the population that falls below your raw score. This is commonly referred to as the percentile rank.
• Right Tail P(Z > z): This tells you the percentage of the population that scores higher than your raw score.
• Two-Tailed Probability: Often used in hypothesis testing to determine if a value is "extreme" in either direction (significantly higher or significantly lower than the mean).

Realistic Example: SAT Scores

Imagine the SAT Math section has a mean score (μ) of 520 and a standard deviation (σ) of 115. If a student scores 750 (X), what is their percentile?

1. Calculate Z: (750 – 520) / 115 = 2.00
2. Interpret: A Z-score of 2.00 means the student scored 2 standard deviations above the average.
3. Probability: Looking at a standard normal table, a Z of 2.00 corresponds to a left-tail probability of approximately 0.9772.
4. Result: The student is in the 97.7th percentile, scoring higher than 97.7% of test-takers.

Common Z-Score Benchmarks

Z-Score	Percentile (Approx)	Interpretation
0	50%	Exactly Average
1.0	84.1%	Above Average
1.96	97.5%	Statistically Significant (95% CI)
-1.0	15.9%	Below Average

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.abs(z) / Math.sqrt(2 * Math.PI); // Error function approximation var t = 1 / (1 + 0.2316419 * Math.abs(z)); var d = 0.39894228 * Math.exp(-z * z / 2); var p = d * t * (0.31938153 + t * (-0.356563782 + t * (1.781477937 + t * (-1.821255978 + t * 1.330274429)))); if (z > 0) return 1 – p; return p; } function calculateZProbability() { var rawX = parseFloat(document.getElementById('rawScore').value); var mean = parseFloat(document.getElementById('popMean').value); var sd = parseFloat(document.getElementById('stdDev').value); if (isNaN(rawX) || isNaN(mean) || isNaN(sd)) { alert("Please enter valid numbers for all fields."); return; } if (sd 0 ? (1 – pLeft) : pLeft)); if (pCentral < 0) pCentral = 0; // Display Results document.getElementById('resZScore').innerText = z.toFixed(4); document.getElementById('resLeftTail').innerText = pLeft.toFixed(5); document.getElementById('resRightTail').innerText = pRight.toFixed(5); document.getElementById('resCentral').innerText = pCentral.toFixed(5); document.getElementById('resPercentile').innerText = (pLeft * 100).toFixed(2) + "th percentile"; document.getElementById('zResultsArea').style.display = 'block'; // Scroll smoothly to results document.getElementById('zResultsArea').scrollIntoView({ behavior: 'smooth', block: 'nearest' }); }