Quartile Calculator - Q1, Q2 & Q3 With Both Methods

About the Quartile Calculator

Quartiles divide sorted data into four equal parts. The first quartile, Q1, has a quarter of the values below it; the second quartile, Q2, is the median; and the third quartile, Q3, has three quarters below it. Quartiles summarise where the bulk of the data lies and how it is spread, and they are the backbone of box plots, the interquartile range and the five-number summary.

This quartile calculator finds Q1, Q2 and Q3 for any list of numbers, by either the median-of-halves method used in most textbooks or the inclusive method used by Excel and many statistics packages. It shows how the quartiles were found, how many values fall in each quarter, and the interquartile range with the minimum and maximum.

How to Use the Quartile Calculator

Enter your numbers, separated by commas, spaces or new lines. At least four values are needed.

Choose the quartile method that matches your course or software.

The three quartiles appear with the working.

The Two Methods

  Median of halves (textbook):
    Q2 = median of all the data
    Q1 = median of the lower half,  Q3 = median of the upper half
    (with an odd count, the median is left out of both halves)

  Inclusive (Excel QUARTILE.INC):
    Q1 at position (n − 1) × 0.25,  Q3 at position (n − 1) × 0.75
    counting the first sorted value as position 0, interpolating between values

Step-by-Step Example: Median of Halves

The values 1, 3, 4, 7, 8, 10, 12, 15, 18.

  Q2:          9 values, the 5th is 8
  Lower half:  1, 3, 4, 7        → Q1 = (3 + 4) ÷ 2 = 3.5
  Upper half:  10, 12, 15, 18    → Q3 = (12 + 15) ÷ 2 = 13.5

The quartiles are Q1 = 3.5, Q2 = 8 and Q3 = 13.5. The middle value, 8, belongs to neither half.

Step-by-Step Example: Inclusive Method

The values 1, 2, 3, 4, 5, 6, 7, 8.

  Q1 position:  (8 − 1) × 0.25 = 1.75  → between 2 and 3 → 2.75
  Q2 position:  (8 − 1) × 0.5  = 3.5   → between 4 and 5 → 4.5
  Q3 position:  (8 − 1) × 0.75 = 5.25  → between 6 and 7 → 6.25

The quartiles are 2.75, 4.5 and 6.25. By the median-of-halves method the same data gives 2.5, 4.5 and 6.5 — the median agrees, but Q1 and Q3 differ slightly.

Why Methods Differ

Quartiles for small data sets usually fall between two values, and there is more than one reasonable way to decide exactly where. The Hyndman–Fan survey of statistical software identified nine methods in common use. They converge as the data grows, so for hundreds of values the choice hardly matters, but for a class of 12 students it can move a quartile by half a mark or more. If your answer must match a textbook, exam or spreadsheet, use the same method it does.

Reading Quartiles

Once found, quartiles are easy to interpret and hard to distort, which is why they are so widely used in reports.

Quartiles describe a distribution without being distorted by extremes. If the gap from Q1 to Q2 is much smaller than the gap from Q2 to Q3, the upper half of the middle data is more spread out — a sign of right skew, common in incomes and prices. Equal gaps suggest a roughly symmetric distribution. Together with the minimum and maximum, quartiles form the five-number summary that a box plot draws.

Quartiles in Practice

Schools report pupils' results by quartile to show where a pupil stands relative to peers. Economists describe income distribution by quartile or quintile. Hospitals and businesses track the 25th and 75th percentiles of waiting or delivery times to see how typical cases vary. In each case, quartiles give a more informative picture than a single average, especially when the data is uneven.

Quartiles, Percentiles and Deciles

Quartiles are special percentiles: Q1 is the 25th percentile, Q2 the 50th and Q3 the 75th. Deciles divide data into ten parts and percentiles into a hundred. The same calculation methods apply to all of them, so the inclusive quartile method is the same as Excel's PERCENTILE.INC at 25, 50 and 75.

A Worked Comparison

Suppose a teacher has ten test scores: 42, 55, 61, 64, 68, 70, 73, 79, 85 and 94. The median is 69, halfway between 68 and 70. By the median-of-halves method, Q1 is the median of the lower five, 61, and Q3 the median of the upper five, 79, so the interquartile range is 18. The quartiles tell the teacher that the middle half of the class scored between 61 and 79, a far more useful summary than the full range of 52, which is stretched by the single low score of 42 and the single high score of 94.

Understanding Your Result

The headline gives Q1, Q2 and Q3.

The values in each quarter line counts how many data points fall in each part.

The how they were found line shows the halves used, or the interpolation positions.

The IQR and extremes line gives the interquartile range with the minimum and maximum.

When Should You Use This Calculator?

Use it for homework and exams on quartiles and box plots.

Use it to check spreadsheet results against the textbook method.

Use it to summarise scores, incomes, prices or times.

Use it as a first step toward the IQR and outlier checks.

Common Mistakes

Forgetting to sort. Quartiles depend on the sorted order, not on the order you typed the values in.

Including the median in the halves for odd counts. The textbook method leaves it out.

Assuming all software agrees. Check which method a tool uses.

Confusing quartiles with quarters of the range. Quartiles depend on the data, not on equal divisions between the minimum and maximum.

Rounding positions instead of interpolating. The inclusive method interpolates between the two neighbouring values.

Reporting quartiles without the method. Say which method you used, so others can reproduce the figures.