5 Number Summary Calculator
Enter your data in the Calculate tab first, then return here to see the full step-by-step solution.
Practice Datasets — Click to Load
Exam Scores (n=12)
Statistics class final exam results
55, 61, 69, 72, 77, 78, 80, 83, 85, 88, 91, 94
Annual Salaries (n=10)
Department salary data (USD) with a high-earner outlier
42000, 55000, 61000, 68000, 72000, 75000, 80000, 95000, 110000, 145000
Product Weights — QC (n=12)
Manufacturing quality control measurements (grams)
18.2, 18.9, 19.1, 19.4, 19.5, 19.8, 20.1, 20.5, 20.7, 21.0, 21.3, 28.4
Daily Website Visits (n=13)
Traffic data with a viral spike outlier
2, 4, 4, 7, 8, 9, 10, 12, 12, 15, 18, 22, 35
What Is a Five Number Summary?
A five number summary is a set of five descriptive statistics — Minimum, First Quartile (Q1), Median (Q2), Third Quartile (Q3), and Maximum — that together describe the full distribution of a dataset. It gives you the center of the data (median), the spread of the middle 50% (IQR = Q3 − Q1), and the overall range (Max − Min), all without assuming the data follows a normal distribution.
The five number summary was introduced by statistician John W. Tukey in his 1977 book Exploratory Data Analysis. It forms the numerical foundation for the box plot (also called a box-and-whisker plot). Together, the five values let you see whether data is symmetric, skewed left, or skewed right, and whether any values sit far outside the main cluster. The National Institute of Standards and Technology’s NIST Engineering Statistics Handbook lists the five number summary as a core technique in exploratory data analysis (EDA).
The Five Values: Definitions and Formulas
Each of the five values answers a specific question about your dataset. Taken together, they give a complete picture of the data’s shape and spread without requiring you to draw a graph.
| Value | Symbol / Formula | What It Tells You | Also Called |
|---|---|---|---|
| Minimum | Min = x(1) | The smallest value in the dataset; the starting point of the box plot whisker | Lower extreme, L |
| First Quartile | Q1 = P25 | 25% of values fall at or below Q1. It is the median of the lower half of the data. | Lower quartile, 25th percentile |
| Median | Q2 = P50 | The middle value. Half the data is below it, half above. Resistant to outliers. | Q2, 50th percentile, second quartile |
| Third Quartile | Q3 = P75 | 75% of values fall at or below Q3. It is the median of the upper half of the data. | Upper quartile, 75th percentile |
| Maximum | Max = x(n) | The largest value in the dataset; the end of the box plot whisker | Upper extreme, H |
| IQR | IQR = Q3 − Q1 | Interquartile range. Measures spread of the middle 50% of data. A sixth output computed from the five number summary. | Midspread, H-spread |
The IQR is technically a sixth output derived from the five number summary, but it appears so often in box plot construction and outlier detection that it is almost always reported alongside the five values. The IQR is resistant to extreme values, which makes it a better measure of spread than the full range whenever outliers are present. See the full guide to interquartile range on Statistics Fundamentals for a deeper treatment.
How to Calculate a Five Number Summary — Step by Step
To find the five number summary: sort your data, identify the minimum and maximum, find the median, then find Q1 as the median of the lower half and Q3 as the median of the upper half. Here is the complete method with a worked example.
Rearrange all values from smallest to largest. This is the only way to correctly identify quartile positions.
Sorted: 55, 61, 69, 72, 77, 78, 80, 83, 85, 88, 91, 94
Minimum = 55 (first value after sorting). Maximum = 94 (last value after sorting). Range = 94 − 55 = 39.
With n = 12 values (even), the median is the average of the 6th and 7th values. The 6th value is 78, the 7th is 80.
Median (Q2) = (78 + 80) / 2 = 79
Split the dataset at the median. The lower half is the first 6 values: 55, 61, 69, 72, 77, 78. Q1 is the median of this lower half — the average of the 3rd and 4th values (69 and 72).
Q1 = (69 + 72) / 2 = 70.5
The upper half is the last 6 values: 80, 83, 85, 88, 91, 94. Q3 is the median of this upper half — the average of the 3rd and 4th values (85 and 88).
Q3 = (85 + 88) / 2 = 86.5
IQR = Q3 − Q1 = 86.5 − 70.5 = 16. Lower fence = 70.5 − 1.5(16) = 46.5. Upper fence = 86.5 + 1.5(16) = 110.5. All values fall between 46.5 and 110.5, so there are no outliers in this dataset.
Complete Five Number Summary: Min = 55, Q1 = 70.5, Median = 79, Q3 = 86.5, Max = 94. IQR = 16. Range = 39. No outliers detected. You can verify this result by loading the “Exam Scores Example” in the calculator above.
Five Number Summary Formulas
The key formulas for a five number summary are straightforward once the data is sorted. The only formula that varies is the median calculation, which differs depending on whether you have an odd or even number of values.
Median — Odd n
Position = (n + 1) / 2
n = 11 → position = 6
Median = 6th sorted value
Median — Even n
Average of positions n/2
and (n/2) + 1
n = 12 → average of
6th and 7th values
IQR and Range
IQR = Q3 − Q1
Range = Max − Min
IQR measures spread of
middle 50% of data
Tukey Outlier Fences
Lower fence:
Q1 − 1.5 × IQR
Upper fence:
Q3 + 1.5 × IQR
How to Detect Outliers Using Tukey’s Rule
Any value below the lower fence (Q1 − 1.5 × IQR) or above the upper fence (Q3 + 1.5 × IQR) is classified as an outlier. This method, developed by John Tukey, is the standard technique used in box plots and exploratory data analysis. It does not assume a normal distribution, making it robust for a wide variety of datasets.
Example: Salary data with Q1 = $55,000, Q3 = $95,000. IQR = $40,000. Lower fence = $55,000 − $60,000 = −$5,000 (no lower outliers possible for salary). Upper fence = $95,000 + $60,000 = $155,000. Any salary above $155,000 would be an outlier. Note that finding outliers is the start of an investigation, not the end: you still need to determine whether an outlier is a data entry error, a legitimate extreme value, or a meaningful anomaly worth reporting.
🧠 The SQMQM Framework: Reading a Five Number Summary at a Glance
The SQMQM Framework (Spread → Quartile 1 → Midpoint → Quartile 3 → Maximum extent) is a structured way to read and interpret a five number summary from left to right. It treats each value as answering a specific question about your data’s distribution.
Reading five number summaries this way — as a narrative from left to right — is faster than memorizing formulas. A symmetric dataset will show roughly equal gaps between Min–Q1, Q1–Median, Median–Q3, and Q3–Max. Any asymmetry in those gaps tells you which direction the distribution skews. This framework connects directly to how the box plot visualizes the same five values.
Five Number Summary vs. Box Plot — What’s the Difference?
The five number summary is the numerical set {Min, Q1, Median, Q3, Max}. A box plot is the visual representation of those same five numbers drawn on a number line. They contain exactly the same information — the box plot just makes that information easier to see and compare across multiple groups.
Table: Five Number Summary vs. Box Plot — Direct Comparison
| Property | Five Number Summary | Box Plot |
|---|---|---|
| Format | Five numerical values | Visual diagram on a number line |
| The box | Q1 to Q3 span (IQR) | Rectangle from Q1 to Q3 |
| Center line | Median value | Vertical line inside the box at the median |
| Whiskers | Min and Max values | Lines extending to Min and Max (or fences) |
| Outliers | Listed separately (with Tukey fences) | Plotted as individual dots beyond the whiskers |
| Best for | Reporting, tables, calculations | Comparing distributions across groups |
| Reveals skewness? | Yes, by comparing gap sizes | Yes, visually at a glance |
In practice, the five number summary and box plot always go together. The numerical summary lets you write precise values in a report or table; the box plot lets you communicate the distribution visually to an audience. The box plot generator on Statistics Fundamentals can draw the box plot directly from any five number summary you calculate above.
How to Interpret a Five Number Summary
Reading a five number summary is about comparing the gaps between the five values, not just memorizing their definitions. The size of each gap reveals whether the data is symmetric, left-skewed, right-skewed, or contains outliers.
| What You See | What It Means | Practical Example |
|---|---|---|
| Median near the center of the box (Q1 to Q3) | Symmetric distribution — mean and median are close | Heights of adults in a population |
| Median closer to Q1 (left of center box) | Right-skewed (positively skewed) — long upper tail | Income distributions, house prices |
| Median closer to Q3 (right of center box) | Left-skewed (negatively skewed) — long lower tail | Scores on a very easy exam |
| Large IQR relative to the range | Data is spread broadly; high variability in the middle | Hospital length-of-stay data |
| Max far above the upper fence | One or more high outliers worth investigating | CEO pay in a salary dataset |
| Min far below the lower fence | One or more low outliers — possible data entry errors | Negative product weight in a QC dataset |
📊 Case Study: Five Number Summary for Salary Data
The five number summary is especially useful for salary and income data, because salaries are rarely normally distributed — they almost always have a long right tail caused by a small number of high earners.
Sorted data: $42k, $55k, $61k, $68k, $72k, $75k, $80k, $95k, $110k, $145k. With n = 10:
Median (Q2) = average of 5th and 6th values = ($72k + $75k) / 2 = $73,500
Q1 = median of bottom 5 ($42k, $55k, $61k, $68k, $72k) = $61,000
Q3 = median of top 5 ($75k, $80k, $95k, $110k, $145k) = $95,000
IQR = $95,000 − $61,000 = $34,000
Upper fence = $95,000 + 1.5 × $34,000 = $146,000
Outliers: None ($145,000 falls just inside the upper fence at $146,000)
Interpretation: The median salary ($73,500) sits well below the mean, which would be pulled up by the high earner. The large gap between Q3 ($95k) and Max ($145k) compared to the relatively smaller gap between Min ($42k) and Q1 ($61k) confirms the data is right-skewed. The five number summary captures this skewness far better than the mean alone. Load this dataset into the calculator above to verify these results.
Quartile Calculation Methods — Why Results Sometimes Differ
Different textbooks, calculators, and software programs calculate Q1 and Q3 slightly differently, which is why you may get different answers from different sources for the same dataset. The three most common methods are the Tukey (inclusive) method, the exclusive method, and the interpolation method used by Excel.
Table: Quartile Methods Compared (Dataset: 2, 4, 6, 8, 10, 12, 14)
| Method | Q1 | Q3 | IQR | Used By |
|---|---|---|---|---|
| Tukey / Inclusive (this calculator) | 4 | 12 | 8 | Most statistics textbooks, R (type 2) |
| Exclusive method | 3 | 13 | 10 | Some AP Statistics materials, R (type 1) |
| Excel QUARTILE.INC | 4 | 12 | 8 | Excel QUARTILE.INC / QUARTILE |
| Excel QUARTILE.EXC | 3 | 13 | 10 | Excel QUARTILE.EXC |
| Linear interpolation | 3.5 | 12.5 | 9 | Python NumPy default, Minitab |
The differences are usually small, and they completely disappear as sample size grows. For coursework, always use the method your textbook or course specifies. This calculator uses the inclusive (Tukey) method, which matches the approach taught in most introductory statistics courses and described by the Penn State STAT 200 course.
Five Number Summary in Excel, R, and Python
The five number summary is built into every major statistical software package. Below are the exact commands for quick reference.
# ── Python (NumPy / SciPy) ──────────────────────────
import numpy as np
data = [55, 61, 69, 72, 77, 78, 80, 83, 85, 88, 91, 94]
q1, med, q3 = np.percentile(data, [25, 50, 75])
print(f"Min={min(data)}, Q1={q1}, Median={med}, Q3={q3}, Max={max(data)}")
# ── R ────────────────────────────────────────────────
data <- c(55, 61, 69, 72, 77, 78, 80, 83, 85, 88, 91, 94)
fivenum(data) # Tukey method
summary(data) # includes mean
# ── Excel ─────────────────────────────────────────────
=MIN(A1:A12) # Minimum
=QUARTILE.INC(A1:A12,1) # Q1
=MEDIAN(A1:A12) # Median
=QUARTILE.INC(A1:A12,3) # Q3
=MAX(A1:A12) # Maximum
=QUARTILE.INC(A1:A12,3)-QUARTILE.INC(A1:A12,1) # IQR
Related Topics and Calculators on Statistics Fundamentals
The five number summary connects to core concepts across descriptive statistics and exploratory data analysis.
Frequently Asked Questions
A five number summary is a set of five descriptive statistics — Minimum, Q1, Median (Q2), Q3, and Maximum — that together describe the distribution of a dataset. It tells you the center (median), the spread of the middle 50% (IQR), and the overall range. Introduced by statistician John Tukey, it forms the numerical basis for box plots and is one of the most common tools in exploratory data analysis.
To calculate a five number summary: (1) sort the data in ascending order; (2) the minimum is the smallest value and the maximum is the largest; (3) the median is the middle value (or average of the two middle values if n is even); (4) Q1 is the median of the lower half of the data (values below the overall median); (5) Q3 is the median of the upper half (values above the overall median). Then compute IQR = Q3 − Q1 for outlier detection.
The interquartile range (IQR) equals Q3 minus Q1. It measures the spread of the middle 50% of your data. The IQR matters because it is resistant to outliers — unlike the full range (Max − Min), one extreme value cannot distort the IQR. This makes the IQR a more reliable measure of spread when data is skewed or contains outliers. The IQR is also used in Tukey’s Rule to define outlier fences: Lower fence = Q1 − 1.5 × IQR; Upper fence = Q3 + 1.5 × IQR.
The five number summary is the numerical set {Min, Q1, Median, Q3, Max}. A box plot (or box-and-whisker plot) is the visual representation of those same five numbers. The box spans Q1 to Q3, a line inside the box marks the median, and whiskers extend to the minimum and maximum (or to the outlier fences). Outliers beyond the fences are plotted as individual points. The five number summary is used in tables and reports; the box plot is used to communicate the distribution visually and to compare multiple groups side by side.
Using Tukey’s Rule: calculate IQR = Q3 − Q1. Then compute the lower fence = Q1 − 1.5 × IQR and the upper fence = Q3 + 1.5 × IQR. Any data value below the lower fence or above the upper fence is an outlier. Values more than 3 × IQR beyond Q1 or Q3 are sometimes called extreme outliers or far outliers. Always investigate outliers before removing them — they may be data entry errors, measurement mistakes, or genuinely unusual observations that deserve attention.
You can read skewness from the gaps between the five values. If the gap from Median to Q3 is larger than the gap from Q1 to Median, the data is right-skewed (positively skewed) — a long upper tail pulls the mean above the median. If the gap from Q1 to Median is larger, the data is left-skewed (negatively skewed). If the gaps are equal and the Min–Q1 and Q3–Max gaps are similar, the distribution is roughly symmetric. Income and house price data are classic examples of right skewness that the five number summary captures clearly.
There is no single universally agreed method for calculating quartiles. Excel’s QUARTILE.INC function uses linear interpolation between data points, which can give fractional quartile values even when all data values are integers. Most introductory statistics textbooks use the inclusive (Tukey) method, where Q1 is the exact median of the lower half and Q3 is the exact median of the upper half. Python’s NumPy default also uses interpolation. The differences are typically small and disappear as sample size grows. This calculator uses the inclusive textbook method.
Tukey’s five number summary refers to the five-value description {Minimum, Lower Hinge (Q1), Median, Upper Hinge (Q3), Maximum} introduced by statistician John W. Tukey in his 1977 book Exploratory Data Analysis. Tukey called Q1 the “lower hinge” and Q3 the “upper hinge.” The summary was designed as a quick, distribution-free way to describe data during exploratory analysis, and it forms the basis for the box-and-whisker plot. In R, the function fivenum() implements Tukey’s original definition.
Yes. With an even number of values, the median is the average of the two middle values. For example, with 12 sorted values, the median is the average of the 6th and 7th values. After finding the median, the lower half contains the first six values and the upper half contains the last six values, then Q1 and Q3 are found as the medians of those halves — again as averages if each half has an even number of values. The calculator above handles both odd and even datasets automatically.
Use the five number summary when data is skewed, contains outliers, or does not follow a normal distribution. The mean and standard deviation assume the data is roughly symmetric — one large outlier can shift the mean and inflate the standard deviation significantly. The five number summary (and particularly the median and IQR) are resistant to those effects. It is the standard choice for income data, house prices, clinical lab values, and any measurement where extreme values are common. The mean and standard deviation are better suited for symmetric, normally distributed data such as heights, exam scores, and repeated measurements of the same physical quantity.