Please enter at least 4 numbers to calculate quartiles.
Accepts any positive or negative numbers — minimum 4 values required
Five Number Summary
Sorted Data (Q1 Q2 Q3 highlighted)
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Sources & Methodology
All calculations use standard statistical methods verified against NIST and Khan Academy references. The quartile method used is Tukey’s inclusive (median-of-halves) approach, the most widely taught in introductory statistics courses.
Khan Academy — Quartile FAQ & Box Plot Methods
Defines Q1, Q2, Q3 calculation methods and the relationship between quartiles and box-and-whisker plots used in K–12 and undergraduate courses
NIST/SEMATECH e-Handbook of Statistical Methods — Quartiles & IQR
National Institute of Standards and Technology reference for quartile computation, IQR, and Tukey’s outlier fence method used in engineering and scientific research
Introductory Statistics — OpenStax (Illowsky & Dean, 2023)
Peer-reviewed open textbook used as curriculum reference for five number summary, box plot construction, and quartile percentile interpretation
Methodology: Data is sorted in ascending order. Q2 is the median of the full set. For odd-count sets, Q2 is the exact middle value and is excluded from both halves. Q1 = median of lower half. Q3 = median of upper half. IQR = Q3 − Q1. Outlier lower fence = Q1 − 1.5 × IQR. Outlier upper fence = Q3 + 1.5 × IQR (Tukey’s fence method).
⏱ Last reviewed: March 2026
How to Calculate Quartiles Step by Step
Quartiles are three values that divide a sorted data set into four equal parts. Each part contains 25% of the observations. Q1 marks the 25th percentile, Q2 marks the 50th percentile (the median), and Q3 marks the 75th percentile. Together with the minimum and maximum, they form the five number summary — the foundation of every box-and-whisker plot.
Unlike the mean and standard deviation, quartiles are resistant to outliers. They describe the shape and spread of real-world data sets without being distorted by extreme values, which is why they are used in finance, medicine, education, and social science research.
Step-by-Step Worked Example
Data set: {14, 3, 9, 7, 18, 2, 11, 5}
Step 1 — Sort ascending: {2, 3, 5, 7, 9, 11, 14, 18} (8 values, even count)
Q2 (Median) = (7 + 9) ÷ 2 = 8
Step 2: Average of 4th and 5th values. For odd counts, Q2 is the exact middle value.
Q1 = median of {2, 3, 5, 7} = (3 + 5) ÷ 2 = 4
Step 3: Lower half = first 4 values. Q1 = average of 2nd and 3rd values in lower half.
Q3 = median of {9, 11, 14, 18} = (11 + 14) ÷ 2 = 12.5
Step 4: Upper half = last 4 values. Q3 = average of 2nd and 3rd values in upper half.
IQR = Q3 − Q1 = 12.5 − 4 = 8.5
Step 5: Outlier fences: lower = 4 − (1.5 × 8.5) = −8.75 upper = 12.5 + (1.5 × 8.5) = 25.25
Quartile Reference Table
| Term | Percentile | Meaning | Box Plot Position |
|---|---|---|---|
| Minimum | 0th | Smallest non-outlier value | Left whisker end |
| Q1 — First Quartile | 25th | 25% of data falls below this | Left edge of box |
| Q2 — Median | 50th | Middle value, splits data equally | Line inside box |
| Q3 — Third Quartile | 75th | 75% of data falls below this | Right edge of box |
| Maximum | 100th | Largest non-outlier value | Right whisker end |
| IQR | — | Spread of middle 50% (Q3 − Q1) | Width of box |
IQR vs Standard Deviation — Which Should You Use?
Use IQR when your data is skewed or contains outliers. Because IQR only measures the spread of the central 50%, extreme values do not affect it. This makes it ideal for income, home prices, medical measurements, and test scores — any real-world data set that is not perfectly symmetric.
Use standard deviation when data is approximately normally distributed and every data point should influence the spread measurement equally. Standard deviation is the right choice for manufacturing tolerances, measurement error, and naturally symmetric phenomena like heights or IQ scores.
💡 Pro tip: In a perfectly symmetric data set, Q2 sits exactly halfway between Q1 and Q3. When Q2 is closer to Q1, the data is right-skewed (long upper tail). When Q2 is closer to Q3, the data is left-skewed. The calculator shows this skewness interpretation automatically in your results.
Frequently Asked Questions
Sort your data from smallest to largest. Q2 (median) is the middle number — for even counts it is the average of the two middle values. Q1 is the median of all values below Q2. Q3 is the median of all values above Q2. For odd-count data sets, do not include Q2 itself in either half when finding Q1 and Q3.
IQR = Q3 − Q1. It is the difference between the third and first quartiles. For example, if Q1 = 15 and Q3 = 40, then IQR = 25. This means the middle 50% of your data covers a range of 25 units. The IQR is resistant to outliers because it ignores the top and bottom 25% of values entirely.
Use Tukey’s fence method. Lower fence = Q1 − 1.5 × IQR. Upper fence = Q3 + 1.5 × IQR. Any value below the lower fence or above the upper fence is flagged as an outlier. This method is preferred over mean ± 2 standard deviations because it is not influenced by the very extreme values it is trying to detect.
There are three common quartile calculation methods. Excel QUARTILE.INC uses a percentile interpolation method. This calculator uses the Tukey median-of-halves method, the most widely taught in introductory statistics. The results are identical or nearly identical for large data sets. Differences appear only with small samples under around 20 values. All three methods are mathematically valid.
A box-and-whisker plot visualises the five number summary. The box spans Q1 to Q3, representing the IQR. A line inside the box marks Q2 (median). Whiskers extend from Q1 and Q3 to the nearest non-outlier values within 1.5 × IQR. Individual data points beyond the whiskers are plotted separately as outliers. Box plots are the standard tool for comparing distributions across groups.
Yes. For an odd count, Q2 is the exact middle value. Q1 is the median of values strictly below Q2, and Q3 is the median of values strictly above Q2 — the median value itself is excluded from both halves. For example, with 9 values, Q2 is the 5th value, Q1 is the median of values 1–4, and Q3 is the median of values 6–9.
Use IQR when your data is skewed or contains outliers — income, house prices, test scores, wait times. IQR measures only the spread of the central 50% and is unaffected by extreme values. Use standard deviation when data is approximately normal and you want all values to influence the spread measurement equally.
You need at least 4 data points to compute meaningful quartiles. With fewer than 4, Q1 and Q3 cannot be reliably derived from distinct halves. For stable, reliable results aim for at least 8 to 12 values. Quartiles become increasingly meaningful and interpretable as sample size grows beyond 20 observations.
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