Five Number Summary Calculator

What Does a Five Number Summary Calculator Do?

A five number summary calculator sorts your data and reports five landmarks: the minimum, the first quartile (Q1), the median, the third quartile (Q3) and the maximum. Together they describe where a data set starts, where the middle half sits, where the center is and where it ends, without needing a mean or a bell curve.

This one adds the pieces people usually go looking for next. You get the interquartile range (IQR), the full range, the outlier fences built from the 1.5 × IQR rule, a list of any values that fall outside them, and a box plot drawn from your numbers. A table underneath shows what all four common quartile methods would give for the same data, so you can see at once why your textbook, your TI-84 and Excel sometimes disagree. To see every value as a dot instead of a box, try the strip chart maker.

Every step is written out. If you are checking homework, you can follow the sorting, the halving and the median-finding line by line instead of trusting one final answer. For the mean, median and mode of the same data, the mean, median and mode calculator does it in one step.

What Are the Five Numbers in a Five-Number Summary?

The five numbers are the minimum, Q1, the median, Q3 and the maximum, listed from smallest to largest. They split a sorted data set into four chunks that each hold about a quarter of the values.

The statistician John Tukey popularized the idea in his 1977 book Exploratory Data Analysis, where it was the starting point for the box-and-whisker plot. Because it relies on ordering rather than adding, a single extreme value cannot drag it around the way it drags a mean. That is the main reason the summary is worth reporting next to, or instead of, an average.

How Do You Use This Five Number Summary Calculator?

Paste your numbers and the answer appears immediately. There is nothing to submit.

1. Enter the data. Separate values with commas, spaces, tabs, semicolons or new lines. A column copied out of Excel or Google Sheets pastes in as is. Negatives and decimals work, and up to 500 values are accepted.
2. Pick a quartile method. The default, median of halves with the median left out, matches the TI-84 and most statistics textbooks. Choose the other options to match Tukey’s hinges (R’s fivenum) or Excel’s QUARTILE.INC and QUARTILE.EXC. The comparison table shows all four together.
3. Set the decimal places from 2 to 8. Rounding only affects the display; the arithmetic runs at full precision.
4. Read the results. The five numbers are at the top, the IQR and outlier fences next, then the working, a labelled list of every sorted value, the software formulas and the box plot.

The quick buttons load ready-made examples, including an odd-sized list, an even-sized list and a data set with an outlier. Press Clear to return to the starting example.

How Do You Find a Five-Number Summary by Hand?

Sort the data, read off the smallest and largest values, find the median, then find the median of each half. That gives all five numbers in four moves.

1. Sort the values from smallest to largest.
2. Minimum and maximum: the first and last values in the sorted list.
3. Median: the middle value if n is odd, or the average of the two middle values if n is even.
4. Split the list at the median into a lower half and an upper half. With an even n the split is obvious. With an odd n you have to decide what to do with the median itself (more on that below).
5. Q1 is the median of the lower half and Q3 is the median of the upper half.

Two checks catch most slips. The five numbers must never decrease from left to right, and Q1 and Q3 should each sit between the median and the nearer end of the data.

Example 1: What Is the Five-Number Summary of an Even-Sized Data Set?

A bakery recorded eight delivery times in minutes: 12, 15, 18, 21, 25, 28, 33 and 40. The list is already sorted and n = 8.

Half of the deliveries took between 16.5 and 30.5 minutes. The median is 6.5 above Q1 and 7.5 below Q3, so the box is nearly balanced, but the top whisker (9.5 minutes long) is about twice as long as the bottom one (4.5), which hints at a few slow runs stretching the high end.

The outlier fences are 16.5 − 1.5 × 14 = −4.5 and 30.5 + 1.5 × 14 = 51.5. Nothing is outside them, so the plot has no outlier dots. For spread measured around the mean rather than the quartiles, see the standard deviation calculator and the variance calculator.

Example 2: What Do You Do With the Median When n Is Odd?

This is where textbooks and software split. Take nine values, the number of items a market stall sold each day: 2, 4, 4, 5, 7, 9, 10, 12 and 15. The median is the fifth value, 7.

Median left out of both halves (the TI-84 and most classroom method). The lower half is 2, 4, 4, 5 and the upper half is 9, 10, 12, 15. Q1 = (4 + 4) / 2 = 4 and Q3 = (10 + 12) / 2 = 11, so the summary is 2, 4, 7, 11, 15 with IQR 7.

Median kept in both halves (Tukey’s hinges, R’s fivenum). The lower half is 2, 4, 4, 5, 7 and the upper half is 7, 9, 10, 12, 15. Q1 is still 4 but Q3 = 10, so the summary is 2, 4, 7, 10, 15 with IQR 6.

Same data, different Q3, different IQR, and even different outlier fences (21.5 against 19). Neither answer is wrong. They follow different definitions, and the only mistake is mixing them up. If your instructor gave a rule, use it. If not, say which method you used when you report the result. The quartile calculator works out the quartile positions behind each method step by step.

Example 3: How Does an Outlier Show Up in a Five-Number Summary?

Ten patients waited this many minutes at a clinic: 12, 13, 14, 15, 16, 17, 18, 19, 20 and 95. The median is (16 + 17) / 2 = 16.5. With the median left out of the halves, the lower half is 12 to 16, so Q1 = 14, and the upper half is 17, 18, 19, 20, 95, so Q3 = 19.

The 95 is far above the upper fence of 26.5, so it is flagged as an outlier and drawn as a separate dot. The upper whisker stops at 20, the largest value that is still inside the fence.

Notice what the outlier did to different measures. The mean is 23.9 minutes, higher than 9 of the 10 patients actually waited. The range is 83. The median (16.5) and the IQR (5) barely noticed it: drop the 95 and the median becomes 16 and the IQR 5. That contrast is the whole argument for the five-number summary. When one stray value can distort the average, the quartiles still describe the typical patient.

Why Do Different Calculators Give Different Quartiles?

Because there is no single definition of a quartile for a small sample. At least four conventions are in everyday use, and they agree only when the data happen to line up neatly. Hyndman and Fan counted nine quantile definitions across statistical packages in a 1996 paper; these four cover what students and analysts meet most.

Here is the delivery-time data from Example 1 (12, 15, 18, 21, 25, 28, 33, 40) under each method:

With n = 8, the inclusive method puts Q1 at position 1 + 0.25 × 7 = 2.75, three quarters of the way from 15 to 18, which is 17.25. The exclusive method puts it at 0.25 × 9 = 2.25, a quarter of the way from 15 to 18, which is 15.75. The gap between the two IQRs (12 and 16) is large for a list of only eight numbers. With hundreds of values the methods converge, and the difference stops mattering.

How Do You Read a Box Plot From the Five Numbers?

A box plot draws the five-number summary as a picture, and the box plot maker will draw one for up to three data sets. The box runs from Q1 to Q3, a line inside it marks the median, and whiskers extend to the smallest and largest values that are not outliers. Any outlier is drawn as its own dot beyond the whisker.

Three things are worth reading off it:

• Spread. A long box means the middle half of the data is spread out. A short box means it is tightly packed.
• Symmetry. If the median line sits near the middle of the box and the whiskers are about equal, the data are roughly symmetric. A median pushed toward Q1 with a long upper whisker points to right skew, which is common with times, prices and incomes.
• Outliers. Dots past the whiskers deserve a second look. They may be typing errors, a different group mixed in, or real and interesting extremes.

The box plot on this page is drawn from the same numbers shown in the result cards, using the quartile method you selected, so the picture and the table never disagree.

What Are the IQR, the Range and the 1.5 × IQR Rule?

The interquartile range is Q3 minus Q1, the width of the middle 50% of the data. The range is the maximum minus the minimum. The IQR ignores the top and bottom quarters, so it resists outliers; the range is set entirely by the two most extreme values and does not.

Tukey’s 1.5 × IQR rule turns the IQR into a yardstick for unusual values:

A value outside the fences is called a suspected outlier. The factor 1.5 is a convention, not a law of nature. For data from a normal distribution the rule flags only about 0.7% of values, which is why a flagged point in a bell-shaped sample is worth a look. In skewed data, a longer tail will produce more flagged points without anything being wrong.

The fences depend on the quartile method too. In Example 2 the fences were 21.5 or 19 on the high side depending on how the median was treated. When a value sits close to a fence, check which method your course expects before calling it an outlier. For a different way to measure spread, the mean absolute deviation calculator gives the average distance from the center instead of the width of the middle half.

How Do You Get a Five-Number Summary in Excel, R, Python and a TI-84?

Each tool has a one-line route, but they do not all use the same quartile method. Match the tool to the method you need.

Excel (inclusive)   =MIN(A2:A9)  =QUARTILE.INC(A2:A9,1)  =MEDIAN(A2:A9)  =QUARTILE.INC(A2:A9,3)  =MAX(A2:A9)
Excel (exclusive)   =QUARTILE.EXC(A2:A9,1)  =QUARTILE.EXC(A2:A9,3)   (needs enough values; returns #NUM! otherwise)
TI-84               STAT > CALC > 1-Var Stats > scroll to minX, Q1, Med, Q3, maxX
R (hinges)          fivenum(x)
R (inclusive)       quantile(x, c(0, .25, .5, .75, 1), type = 7)
R (exclusive)       quantile(x, c(0, .25, .5, .75, 1), type = 6)
Python (inclusive)  np.percentile(x, [0, 25, 50, 75, 100])
Python (exclusive)  np.percentile(x, [0, 25, 50, 75, 100], method="weibull")

On the eight delivery times, QUARTILE.INC returns 17.25 and 29.25, QUARTILE.EXC returns 15.75 and 31.75, while the TI-84 and fivenum both return 16.5 and 30.5. Those match the table in the section on quartile methods. Older Excel files use plain QUARTILE, which is the same as QUARTILE.INC. The method= argument in NumPy needs version 1.22 or later.

The calculator above prints the exact formulas for your data and your chosen method in the “Same calculation in other tools” box.

What Mistakes Cause a Wrong Five-Number Summary?

• Forgetting to sort. The median of an unsorted list is just whatever value happens to be in the middle. Always sort first.
• Using the mean instead of the median as the middle number. The summary is built from order statistics only.
• Mixing quartile methods. Taking Q1 from a TI-84 and Q3 from Excel gives a pair that belongs to no definition.
• Splitting an odd-sized list wrongly. Decide whether the median goes into both halves, one half or neither, and apply that rule to both sides.
• Reading Q1 as the first quarter of the values. Q1 is a cut point, not a group. It is the value with about 25% of the data at or below it.
• Dropping an outlier before checking it. Flagged does not mean wrong. Look at where the value came from first.
• Comparing the IQR across methods. As the delivery-time table showed, the same list can have an IQR of 12, 14 or 16.

When Should You Use a Five-Number Summary Instead of the Mean?

Use the five-number summary when the data are skewed, contain outliers, or you simply want a quick, honest picture of the whole distribution. It is the default way to describe incomes, house prices, waiting times, response times and test scores with a few very low or very high marks.

The mean and standard deviation are better when the data are roughly symmetric and you plan to go on to further analysis built on them. If some observations carry more importance than others, the weighted mean calculator handles that case. For data that really are bell-shaped, the normal curve calculator turns a mean and standard deviation into percentiles and probabilities, and the cumulative area between two bounds is available in the normal CDF calculator.

For counts of successes in repeated trials, a quartile summary is a poor fit, and a model such as the one in the binomial distribution calculator describes the data better. A sensible habit for any numeric data set is to report the median and IQR, glance at the box plot, and only then decide whether a mean is a fair summary.

Frequently Asked Questions

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, box plot (section 1.3.3.7); Tukey, J. W. (1977), Exploratory Data Analysis; Hyndman, R. J. and Fan, Y. (1996), “Sample quantiles in statistical packages,” The American Statistician 50(4), 361–365; NumPy documentation for numpy.percentile; R documentation for fivenum and quantile; Microsoft documentation for QUARTILE.INC and QUARTILE.EXC.

Related Calculators

Looking at spread or center from another angle? Try these, or browse the full list of free statistics calculators:

Mean Absolute Deviation CalculatorAverage distance from the mean or median, with a step table. Weighted Mean CalculatorFind the center when some values count more than others. Normal Curve CalculatorZ-scores, percentiles and bell-curve probabilities. All StatsSolve CalculatorsBrowse the full directory of free statistics tools.