Calculate the minimum proportion of data within K standard deviations
Must be greater than 1
Alternative: Calculate K based on Specific Values
Value to determine probability of being outside the range |X – μ|
Results
Understanding Chebyshev's Theorem
Chebyshev's Theorem (or Inequality) is a fundamental principle in statistics that describes the minimum proportion of values that fall within a certain number of standard deviations from the mean for any data distribution, regardless of its shape.
The Formula
P(|X – μ| < kσ) ≥ 1 – (1 / k²)
Where:
k is the number of standard deviations (must be > 1).
μ is the population mean.
σ is the standard deviation.
Key Benchmarks
✅ k = 2: At least 75% of data falls within 2 standard deviations.
✅ k = 3: At least 88.89% of data falls within 3 standard deviations.
✅ k = 4.47: At least 95% of data falls within 4.47 standard deviations.
Practical Example
Suppose a company's daily production mean is 500 units with a standard deviation of 50 units. If we want to know what percentage of days the production will be between 400 and 600 units:
Calculate k: (600 – 500) / 50 = 2.
Apply formula: 1 – (1 / 2²) = 1 – 0.25 = 0.75.
Result: At least 75% of the time, production will fall in this range.
function calculateChebyshevByK() {
var k = parseFloat(document.getElementById('k_value').value);
var resultArea = document.getElementById('chebyshev-result-area');
var output = document.getElementById('chebyshev-output');
if (isNaN(k) || k <= 1) {
alert("Please enter a K value greater than 1.");
return;
}
var proportion = 1 – (1 / Math.pow(k, 2));
var percentage = (proportion * 100).toFixed(2);
var maxOutside = ((1 / Math.pow(k, 2)) * 100).toFixed(2);
resultArea.style.display = 'block';
output.innerHTML = 'For k = ' + k + ':' +
'Minimum percentage of data within ' + k + ' standard deviations: ' + percentage + '%' +
'Maximum percentage of data outside this range: ' + maxOutside + '%';
}
function calculateChebyshevByValues() {
var mean = parseFloat(document.getElementById('mean').value);
var sd = parseFloat(document.getElementById('std_dev').value);
var target = parseFloat(document.getElementById('target_val').value);
var resultArea = document.getElementById('chebyshev-result-area');
var output = document.getElementById('chebyshev-output');
if (isNaN(mean) || isNaN(sd) || isNaN(target)) {
alert("Please enter valid numbers for Mean, Std. Deviation, and Target Value.");
return;
}
if (sd <= 0) {
alert("Standard deviation must be greater than 0.");
return;
}
var diff = Math.abs(target – mean);
var k = diff / sd;
if (k <= 1) {
resultArea.style.display = 'block';
output.innerHTML = 'Note: The calculated K value is ' + k.toFixed(4) + '. Chebyshev\'s Inequality only provides useful bounds when K > 1. At K ≤ 1, the theorem states that 0% or more of the data is within the range, which is always true for any probability.';
return;
}
var proportion = 1 – (1 / Math.pow(k, 2));
var percentage = (proportion * 100).toFixed(2);
var maxOutside = ((1 / Math.pow(k, 2)) * 100).toFixed(2);
var lowerBound = mean – (k * sd);
var upperBound = mean + (k * sd);
resultArea.style.display = 'block';
output.innerHTML = 'Analysis for X = ' + target + ':' +
'Calculated K: ' + k.toFixed(4) + " +
'Data Range (|X – μ|): ' + lowerBound.toFixed(2) + ' to ' + upperBound.toFixed(2) + " +
'Minimum percentage of data within this range: ' + percentage + '%' +
'Maximum percentage of data outside this range: ' + maxOutside + '%';
}