What the five-number summary is
The five-number summary is a compact description of a dataset built from five ordered values: the minimum, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum. Reading left to right, these five numbers split the sorted data into four equal-count groups: about 25% of the values fall below Q1, 25% between Q1 and the median, 25% between the median and Q3, and 25% above Q3. Because the summary is built from ranks and positions rather than sums, it is resistant — a single extreme outlier barely moves the median or the quartiles, unlike the mean or standard deviation.
The five-number summary is the exact set of values needed to draw a box-and-whisker plot: the box spans Q1 to Q3, a line inside marks the median, and the whiskers reach out toward the minimum and maximum. That is why it is a staple of exploratory data analysis, introductory statistics courses, and any quick "shape of the data" check.
How this calculator computes the quartiles
Every method starts the same way: sort the values from smallest to largest and take the median (Q2). The median is the middle value when there is an odd number of data points, or the average of the two middle values when there is an even number.
To find the quartiles, this tool uses the median-of-halves method (also called the Tukey method or the Moore & McCabe method, the version taught in most U.S. AP Statistics and introductory college courses):
- Q1 is the median of the lower half of the sorted data.
- Q3 is the median of the upper half of the sorted data.
- When the dataset has an odd number of values, the overall median is excluded from both halves before finding Q1 and Q3.
- When the dataset has an even number of values, the data splits cleanly into two equal halves with no value left out.
The interquartile range (IQR = Q3 − Q1) measures the spread of the middle 50% of the data, and the range (max − min) measures the total spread. This calculator reports all five summary numbers plus the IQR and range, and echoes the sorted data so you can check the ordering by eye.
A worked example
Take the seven values 2, 4, 4, 5, 7, 9, 10 (already sorted). There are 7 values, so the median is the 4th value: 5. Excluding that middle value, the lower half is 2, 4, 4 with median Q1 = 4, and the upper half is 7, 9, 10 with median Q3 = 9. So the five-number summary is min = 2, Q1 = 4, median = 5, Q3 = 9, max = 10, giving IQR = 9 − 4 = 5 and range = 10 − 2 = 8.
Why methods can disagree
There is no single universally agreed definition of a quartile, so different software can return slightly different values for Q1 and Q3. Excel's QUARTILE.INC and the default numpy.percentile, for example, use linear interpolation between ranks and can produce quartiles that fall between two data points where the median-of-halves method would not. All methods agree exactly on the minimum, maximum, and median — the disagreements are confined to Q1 and Q3, and they shrink as the dataset grows. If you must match a specific textbook or grading system, confirm which quartile convention it expects.