Quartile and IQR Calculator
Solve quartile iqrcalculator problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
IQR = Q3 - Q1
The Interquartile Range equals the third quartile minus the first quartile. Outlier fences are at Q1 - 1.5*IQR (lower) and Q3 + 1.5*IQR (upper). Data points outside these fences are considered outliers.
Worked Examples
Example 1: Test Scores Analysis
Problem: Find Q1, Q2, Q3, and IQR for test scores: 55, 62, 67, 70, 73, 78, 82, 85, 88, 92, 97.
Solution: Sort data (already sorted): 55, 62, 67, 70, 73, 78, 82, 85, 88, 92, 97\nn = 11, Median (Q2) = 78\nLower half: 55, 62, 67, 70, 73 => Q1 = 67\nUpper half: 82, 85, 88, 92, 97 => Q3 = 88\nIQR = 88 - 67 = 21\nLower fence = 67 - 1.5(21) = 35.5\nUpper fence = 88 + 1.5(21) = 119.5\nNo outliers detected.
Result: Q1 = 67 | Q2 = 78 | Q3 = 88 | IQR = 21 | No outliers
Example 2: Salary Data with Outlier
Problem: Find quartiles and outliers for salaries (in thousands): 35, 42, 45, 48, 50, 52, 55, 58, 62, 150.
Solution: Sorted: 35, 42, 45, 48, 50, 52, 55, 58, 62, 150\nn = 10, Q2 = (50+52)/2 = 51\nQ1 = (42+45)/2 = 43.5\nQ3 = (58+62)/2 = 60 (using interpolation: Q1=44.25, Q3=57.25)\nIQR = Q3 - Q1 = 13\nUpper fence = 60 + 1.5(13) = 79.5\n150 > 79.5, so 150 is an outlier.
Result: Q1 = 44.25 | Q2 = 51 | Q3 = 57.25 | IQR = 13 | Outlier: 150
Frequently Asked Questions
What is the Interquartile Range (IQR) and why is it important?
The Interquartile Range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1), calculated as IQR = Q3 - Q1. It measures the spread of the middle 50% of data values and is one of the most robust measures of statistical dispersion. Unlike the range or standard deviation, the IQR is resistant to outliers because it ignores the extreme values in both tails of the distribution. This makes it particularly useful when analyzing skewed data or datasets with extreme values. The IQR is a fundamental component of box plots and is used extensively in quality control, epidemiology, and data science for understanding data variability.
How do you identify outliers using the IQR method?
The IQR method identifies outliers using fences calculated from the quartiles. The lower fence is Q1 - 1.5 times IQR, and the upper fence is Q3 + 1.5 times IQR. Any data point falling below the lower fence or above the upper fence is considered a mild outlier. For extreme outliers, the multiplier increases to 3 times the IQR, creating extreme fences at Q1 - 3 times IQR and Q3 + 3 times IQR. This method was popularized by mathematician John Tukey and is widely used because it adapts to the natural spread of the data. The 1.5 multiplier was chosen because it captures approximately 99.3% of normally distributed data within the fences.
What is the difference between IQR and standard deviation?
Both IQR and standard deviation measure data spread, but they do so in fundamentally different ways and have different strengths. Standard deviation measures the average distance of all data points from the mean, making it sensitive to every value including extreme outliers. The IQR only considers the middle 50% of data, making it robust against outliers. For normally distributed data, the IQR is approximately 1.35 times the standard deviation. Standard deviation is preferred when data follows a normal distribution and you want to use all available information. The IQR is preferred when data is skewed, contains outliers, or when you need a resistant measure of spread that wont be distorted by extreme values.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Is Quartile and IQR Calculator free to use?
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Can I use Quartile and IQR Calculator on a mobile device?
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