The Interquartile Range (IQR) is a vital statistical measure that describes the spread of the middle 50% of your data. It's a robust indicator of variability because it's less sensitive to extreme outliers compared to the simple range (the difference between the maximum and minimum values).
What is the IQR?
The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1):
IQR = Q3 - Q1
Key Terms:
Quartiles: Quartiles divide a sorted dataset into four equal parts.
Q1 (First Quartile): The value below which 25% of the data falls. It's the median of the lower half of the dataset.
Q2 (Second Quartile): This is the median of the entire dataset. 50% of the data falls below Q2.
Q3 (Third Quartile): The value below which 75% of the data falls. It's the median of the upper half of the dataset.
Median: The middle value in a sorted dataset. If the dataset has an even number of points, the median is the average of the two middle values.
How to Calculate the IQR Manually:
Sort the Data: Arrange all your data points in ascending order.
Find the Median (Q2): Determine the median of the entire dataset.
Divide the Dataset:
The lower half consists of all data points strictly below the median. If the dataset has an odd number of points, do NOT include the median in either half.
The upper half consists of all data points strictly above the median. If the dataset has an odd number of points, do NOT include the median in either half.
Find Q1: Calculate the median of the lower half. This is your first quartile (Q1).
Find Q3: Calculate the median of the upper half. This is your third quartile (Q3).
Identifying Outliers: The IQR is often used to detect potential outliers. A common rule of thumb is that data points falling below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR are considered potential outliers.
Measuring Spread in Skewed Data: When data is skewed (not symmetrically distributed), the IQR provides a more reliable measure of spread than the standard deviation.
Understanding Data Distribution: It gives insight into the variability within the central portion of the data.
Example Calculation:
Let's calculate the IQR for the dataset: 2, 5, 7, 8, 10, 12, 15, 18, 20, 22
Median (Q2): There are 10 data points. The median is the average of the 5th and 6th values: (10 + 12) / 2 = 11.
Divide the Dataset:
Lower Half: 2, 5, 7, 8, 10
Upper Half: 12, 15, 18, 20, 22
Q1: The median of the lower half (5 points) is the 3rd value: 7. So, Q1 = 7.
Q3: The median of the upper half (5 points) is the 3rd value: 18. So, Q3 = 18.
IQR: IQR = Q3 – Q1 = 18 – 7 = 11.
The Interquartile Range for this dataset is 11.
function calculateIQR() {
var dataInput = document.getElementById("dataPoints").value;
var resultDiv = document.getElementById("result");
var resultValueElement = document.getElementById("resultValue");
// Clear previous results
resultDiv.style.display = 'none';
resultValueElement.textContent = ";
// Input validation and parsing
var dataPoints = dataInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(item) {
return !isNaN(item); // Keep only valid numbers
});
if (dataPoints.length < 4) { // Need at least 4 points to meaningfully calculate quartiles and IQR
resultValueElement.textContent = "Error: Please enter at least 4 valid numbers.";
resultDiv.style.display = 'block';
resultDiv.style.backgroundColor = '#dc3545'; // Red for error
return;
}
// Sort the data points
dataPoints.sort(function(a, b) {
return a – b;
});
var n = dataPoints.length;
var q1, q3;
// Function to calculate median
function calculateMedian(arr) {
var len = arr.length;
var mid = Math.floor(len / 2);
if (len % 2 === 0) {
// Even number of elements
return (arr[mid – 1] + arr[mid]) / 2;
} else {
// Odd number of elements
return arr[mid];
}
}
// Determine Q1 and Q3 based on the method (inclusive/exclusive median)
// Using the method where the median is NOT included in halves for odd N
var midIndex = Math.floor(n / 2);
var lowerHalf, upperHalf;
if (n % 2 === 0) {
lowerHalf = dataPoints.slice(0, midIndex);
upperHalf = dataPoints.slice(midIndex);
} else {
// For odd N, exclude the median itself
lowerHalf = dataPoints.slice(0, midIndex);
upperHalf = dataPoints.slice(midIndex + 1);
}
q1 = calculateMedian(lowerHalf);
q3 = calculateMedian(upperHalf);
var iqr = q3 – q1;
resultValueElement.textContent = "Q1: " + q1 + ", Q3: " + q3 + ", IQR: " + iqr;
resultDiv.style.display = 'block';
resultDiv.style.backgroundColor = 'var(–success-green)'; // Reset to green
}