IQR Calculator

Introduction

The IQR Calculator finds the interquartile range of any data set you enter. The IQR is the difference between the third quartile (Q3) and the first quartile (Q1) of your data. It tells you how spread out the middle 50% of your values are. This is one of the best ways to measure spread because it ignores extreme values, also called outliers. Scientists, students, and analysts use the IQR every day to better understand their data. Simply enter your numbers into the calculator above, and it will give you Q1, Q3, and the IQR right away.

How to use our IQR Calculator

Enter a set of numbers and this calculator will find the interquartile range (IQR), quartiles (Q1, Q2, Q3), five-number summary, outliers, and a box plot for your data.

Enter Your Data: Type or paste your numbers into the text box. You can separate values with commas, spaces, semicolons, or newlines. Fractions like 1/2 or 3/4 also work. You need at least 4 numbers for the calculator to find the quartiles.

Quartile Method: Pick how you want the quartiles calculated. "Inclusive (Type-7)" is the standard method used in most statistics classes. "Exclusive" matches Excel's QUARTILE.EXC function. "Tukey's Hinges" splits the data at the median and finds the median of each half.

Decimal Places: Choose how many decimal places to show in your results, from 0 to 6. The default is 2.

Outlier Detection: Select which outlier fences to use. "1.5×IQR" finds mild outliers, "3×IQR" finds extreme outliers, and "Show Both" displays both types at the same time.

Box Plot: Choose "Show" to display a box-and-whisker plot of your data, or "Hide" to turn it off.

Click Calculate IQR to see your results, including the five-number summary, IQR value, outlier fences, detected outliers, a box plot, sorted data with color-coded highlights, and a step-by-step breakdown of how each value was calculated. Click Reset to return all settings and data to their defaults.

What Is the Interquartile Range (IQR)?

The interquartile range (IQR) is a measure of how spread out the middle 50% of a data set is. You find it by subtracting the first quartile (Q1) from the third quartile (Q3). In simple terms, it tells you the range of values where the central half of your data falls. Because the IQR ignores the highest and lowest extremes, it gives a more reliable picture of spread than the full range, especially when your data contains outliers.

How to Calculate the IQR

To calculate the interquartile range by hand, follow these steps:

1. Sort your data from smallest to largest.
2. Find Q1 (the 25th percentile). This is the median of the lower half of your data.
3. Find Q3 (the 75th percentile). This is the median of the upper half of your data.
4. Subtract: IQR = Q3 − Q1.

For example, with the data set 7, 15, 36, 39, 40, 41, 42, 43, 47, 49, 55, 70, 92, Q1 is 39 and Q3 is 49. So the IQR is 49 − 39 = 10. This means the middle half of the values spans a width of 10 units.

Quartile Methods

There is more than one way to calculate quartiles, and different methods can give slightly different results. The three most common methods are:

• Inclusive (Type-7): The standard method used in most statistics textbooks. It uses the formula h = 1 + (n − 1) × p and interpolates between data points when needed.
• Exclusive (QUARTILE.EXC): The method used by Excel's QUARTILE.EXC function. It uses h = (n + 1) × p, which positions the quartiles slightly differently.
• Tukey's Hinges: This method splits the sorted data at the median, then finds the median of each half. It is commonly used when drawing box-and-whisker plots.

All three methods produce the same IQR for many data sets, but they can differ when the number of data points is small or when values don't divide evenly.

Using the IQR to Detect Outliers

One of the most useful things about the IQR is that it helps you find outliers—values that are unusually far from the rest of your data. You do this by calculating fences:

• Mild outlier fences (1.5 × IQR): Lower fence = Q1 − 1.5 × IQR, Upper fence = Q3 + 1.5 × IQR. Any value outside these fences is considered a mild outlier.
• Extreme outlier fences (3 × IQR): Lower fence = Q1 − 3 × IQR, Upper fence = Q3 + 3 × IQR. Values beyond these fences are considered extreme outliers.

This fence method, sometimes called Tukey's rule, is the standard approach for outlier detection in box plots and many real-world data analyses.

Why the IQR Matters

The IQR is widely used in statistics because it is resistant to outliers. Unlike the standard deviation or full range, a single extreme value does not heavily change the IQR. This makes it especially helpful when working with skewed data or data sets that contain errors. You will see the IQR used in box plots, data cleaning, quality control, and research across fields like science, business, and education. When analyzing your results, you may also want to look at related measures such as percent change to understand how values shift over time, or use a percent error calculator to compare observed values against expected ones. For a quick check on basic proportions in your data, our percentage calculator is another handy tool, while the rate of change calculator can help you examine trends across data points.

Frequently Asked Questions

What is the IQR formula?

The IQR formula is IQR = Q3 − Q1. Q3 is the third quartile (75th percentile) and Q1 is the first quartile (25th percentile). You subtract Q1 from Q3 to get the interquartile range.

How many numbers do I need to calculate the IQR?

You need at least 4 numbers to calculate the IQR. With fewer than 4 data points, the calculator cannot split your data into quartiles properly.

What is the difference between the IQR and the range?

The range is the difference between the largest and smallest values in your data set. The IQR is the difference between Q3 and Q1, which only covers the middle 50% of the data. The IQR is more useful because it is not affected by extreme values or outliers.

Which quartile method should I use?

Use Inclusive (Type-7) if you are in a statistics class, as it is the most common textbook method. Use Exclusive if you need to match Excel's QUARTILE.EXC function. Use Tukey's Hinges if you are drawing a box plot by hand. When in doubt, Inclusive (Type-7) is a safe default.

What is a mild outlier vs. an extreme outlier?

A mild outlier is a data point that falls outside the 1.5×IQR fences but inside the 3×IQR fences. An extreme outlier is a data point that falls outside the 3×IQR fences. Extreme outliers are much farther from the rest of the data than mild outliers.

Can I enter fractions into the calculator?

Yes. You can type fractions like 1/2, 3/4, or 7/8 directly into the input box. The calculator will convert them to decimal numbers before doing the math.

What does the box plot show?

The box plot shows a visual summary of your data. The box spans from Q1 to Q3, and the line inside the box marks the median. The whiskers extend to the smallest and largest values that are not outliers. Outlier points are shown as colored dots outside the whiskers.

What are outlier fences?

Outlier fences are boundary values used to decide if a data point is an outlier. The lower fence is Q1 minus a multiple of the IQR, and the upper fence is Q3 plus that same multiple. For mild outliers the multiple is 1.5, and for extreme outliers the multiple is 3.

Can the IQR be zero?

Yes. The IQR is zero when Q1 and Q3 are the same value. This happens when more than half of your data points are the same number. An IQR of zero means there is no spread in the middle 50% of your data.

Can the IQR be negative?

No. Since Q3 is always greater than or equal to Q1, the IQR is always zero or positive. It can never be a negative number.

Why do different tools give me different Q1 and Q3 values?

Different tools use different quartile methods. For example, Excel's QUARTILE.INC uses one formula while QUARTILE.EXC uses another, and many textbooks use Tukey's Hinges. All are correct—they just follow different rules. This calculator lets you pick the method so you can match whatever tool or textbook you are comparing against.

What is the five-number summary?

The five-number summary includes the minimum, Q1, median (Q2), Q3, and maximum of your data set. These five values give you a quick overview of the spread and center of your data. The calculator shows all five in the results.

How is the IQR different from standard deviation?

Both measure spread, but in different ways. The standard deviation uses every data point and is affected by outliers. The IQR only looks at the middle 50% and ignores extreme values. The IQR is a better choice when your data is skewed or has outliers.

What does the sorted data display with colors mean?

The sorted data section highlights key values with colors. Blue marks Q1, yellow marks the median, green marks Q3, orange marks mild outliers, and red marks extreme outliers. This helps you quickly see where important values fall in your data.

Can I paste data from a spreadsheet?

Yes. Copy a column or row from your spreadsheet and paste it into the input box. The calculator accepts numbers separated by spaces, commas, semicolons, or newlines, so most spreadsheet formats will work automatically.