Z Calculated Formula

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Z Calculated Formula Calculator

Calculate the Z-statistic (Z-score) for hypothesis testing based on sample data and population parameters.

Calculated Z-Statistic: 0.00

Understanding the Z Calculated Formula

In statistics, the Z calculated formula (often called the Z-test statistic) is used to determine how far a sample mean deviates from the population mean, measured in units of standard error. It is a critical component of hypothesis testing, specifically when the population variance is known or the sample size is large (n > 30).

The Formula

Z = (x̄ – μ) / (σ / √n)
  • x̄ (Sample Mean): The average value obtained from your sample data.
  • μ (Population Mean): The hypothesized or known mean of the entire population.
  • σ (Standard Deviation): The measure of dispersion in the population.
  • n (Sample Size): The total number of observations in your sample.

How to Interpret the Result

The Z-score represents how many standard deviations the sample mean is away from the population mean. In hypothesis testing:

  • Positive Z: The sample mean is higher than the population mean.
  • Negative Z: The sample mean is lower than the population mean.
  • Magnitude: Typically, a Z-score greater than 1.96 or less than -1.96 indicates that the result is statistically significant at a 95% confidence level (α = 0.05).

Practical Example

Imagine a factory claims their lightbulbs last for 1,000 hours (μ = 1000) with a standard deviation of 50 hours (σ = 50). You test 100 bulbs (n = 100) and find they actually last 990 hours (x̄ = 990).

  1. Difference: 990 – 1000 = -10
  2. Standard Error: 50 / √100 = 5
  3. Z Calculated: -10 / 5 = -2.00

Since -2.00 is less than -1.96, you would likely reject the manufacturer's claim at a 5% significance level.

function calculateZScore() { var sampleMean = parseFloat(document.getElementById('sampleMean').value); var popMean = parseFloat(document.getElementById('popMean').value); var stdDev = parseFloat(document.getElementById('stdDev').value); var sampleSize = parseFloat(document.getElementById('sampleSize').value); var resultDiv = document.getElementById('zResultWrapper'); var output = document.getElementById('zOutput'); var interpretation = document.getElementById('zInterpretation'); // Validation if (isNaN(sampleMean) || isNaN(popMean) || isNaN(stdDev) || isNaN(sampleSize)) { alert("Please fill in all fields with valid numbers."); return; } if (sampleSize <= 0) { alert("Sample size must be greater than zero."); return; } if (stdDev 1.96) { msg += " This is generally considered statistically significant at the 0.05 (95%) confidence level."; } else { msg += " This is generally not considered statistically significant at the 0.05 (95%) confidence level."; } interpretation.innerHTML = msg; }

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