5 Number Summary Calculator with Steps and Box Plot

Creator: Abdullah ArshadReviewer: Yulduz Zaripova

Type your numbers into the calculator above and it sorts them, splits the data into halves, and shows exactly how each value is found, plus the IQR and the box plot fences.

The step by step method, worked examples and box plot instructions follow.

What is a five number summary?

A five number summary is a set of five values that describe how a dataset is spread out: the smallest value, the largest value, and three quartiles that split the data into four equal parts. It tells you the range, the center, and where the middle half of your data sits, all without reading every number.

The five values, in order, are the minimum, Q1, the median (Q2), Q3, and the maximum. It is the basis for the box plot, and it flags outliers faster than reading down a table of raw numbers.

What are the five numbers?

Q1, Q2, and Q3 are also called the lower, middle, and upper quartiles or percentiles. Q1 is the 25th percentile, Q3 is the 75th, and the median is the 50th.

How to find the 5 number summary step by step

Here is the method the calculator uses. It works for any list of numbers.

1. Sort the data from smallest to largest.
2. Read off the minimum (first value) and maximum (last value).
3. Find the median of the whole dataset. This is Q2.
4. Split the data into a lower half and an upper half at the median.
5. Q1 is the median of the lower half.
6. Q3 is the median of the upper half.

One detail trips people up. When the dataset has an odd number of values, the median sits on a single number, so you leave that number out of both halves. When the count is even, the median falls between two numbers, so both halves keep all their values. This calculator uses that standard method. If you are not sure which method your course expects, check with your instructor.

Five number summary example (odd number of values)

Take the dataset 2, 4, 6, 8, 10, 12, 14. It has 7 values, an odd count.

• Sorted, it is already in order.
• Minimum = 2, Maximum = 14.
• The middle value (position 4) is the median: 8.
• Lower half is 2, 4, 6, so Q1 = 4. The median (8) is left out.
• Upper half is 10, 12, 14, so Q3 = 12.

Five number summary: Min = 2, Q1 = 4, Median = 8, Q3 = 12, Max = 14. IQR = 12 − 4 = 8. That IQR is the input for the 1.5 × IQR rule, and using the IQR to spot outliers shows how the fences are drawn from these five numbers.

Five number summary example (even number of values)

Now take 2, 4, 6, 8, 10, 12. It has 6 values, an even count, so no number gets left out.

• Minimum = 2, Maximum = 12.
• The median is the average of the two middle values (6 and 8): (6 + 8) ÷ 2 = 7.
• Lower half is 2, 4, 6, so Q1 = 4.
• Upper half is 8, 10, 12, so Q3 = 10.

Five number summary: Min = 2, Q1 = 4, Median = 7, Q3 = 10, Max = 12. IQR = 10 − 4 = 6.

Worked example: the bowling strikes dataset

A common homework question gives the number of strikes each employee bowled: 1, 3, 6, 8, 10, 12, 13, 13, 16, 18, 19. There are 11 values.

• Minimum = 1, Maximum = 19.
• The middle value (position 6) is the median: 12.
• Lower half is 1, 3, 6, 8, 10. Its median is 6, so Q1 = 6.
• Upper half is 13, 13, 16, 18, 19. Its median is 16, so Q3 = 16.

Summary: Min = 1, Q1 = 6, Median = 12, Q3 = 16, Max = 19. On a box plot, the left whisker runs 1 to 6, the box runs 6 to 16 with a line at 12, and the right whisker runs 16 to 19.

Worked example: the camping days dataset

Another textbook set: the number of days 12 friends went camping: 6, 2, 0, 10, 3, 6, 6, 4, 12, 0, 6, 2. There are 12 values, an even count.

• Sorted: 0, 0, 2, 2, 3, 4, 6, 6, 6, 6, 10, 12.
• Minimum = 0, Maximum = 12.
• Median is the average of positions 6 and 7 (4 and 6): (4 + 6) ÷ 2 = 5.
• Lower half is 0, 0, 2, 2, 3, 4. Its median is (2 + 2) ÷ 2, so Q1 = 2.
• Upper half is 6, 6, 6, 6, 10, 12. Its median is (6 + 6) ÷ 2, so Q3 = 6.

Summary: Min = 0, Q1 = 2, Median = 5, Q3 = 6, Max = 12.

Note: some textbooks use an interpolation method that gives Q1 = 1.5 for this set. This calculator uses the median-of-halves method, which is the version most stats courses teach. If your class uses a different method, check which one your instructor expects.

How do you turn a five number summary into a box plot?

A box plot (also called a box and whisker plot) is just a drawing of the five number summary. Once you have the five values, the plot builds itself:

1. Draw a number line that covers your minimum and maximum.
2. Draw a box from Q1 to Q3.
3. Draw a vertical line inside the box at the median.
4. Draw a whisker from the left of the box out to the minimum.
5. Draw a whisker from the right of the box out to the maximum.

The box holds the middle 50% of your data, which is the interquartile range. The whiskers show how far the smallest and largest values stretch. A box that sits to one side with a long whisker on the other tells you the data is skewed.

Which box plot matches a dataset?

Multiple choice questions often show four box plots and ask which one fits the data. The trick is to compute the five number summary first, then match. For the set 2, 4, 6, 8, 10, 12, the summary is Min = 2, Q1 = 4, Median = 7, Q3 = 10, Max = 12. The correct box plot has its box from 4 to 10, a median line at 7, a left whisker to 2, and a right whisker to 12. Any option with a box or median in a different spot is wrong. Run your numbers through the calculator, then read the five values off the answer.

How do you read a five number summary?

Each value answers a different question about your data.

• The gap between minimum and maximum is the full range.
• The gap between Q1 and Q3 is the interquartile range (IQR), where the middle half of the data lives.
• A small IQR means the data is tightly packed. A large IQR means it is spread out.
• If the median sits closer to Q1 than to Q3, the data leans low. Closer to Q3, it leans high.

The IQR also drives outlier detection. Anything below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is flagged as an outlier. To see which points in your data qualify, run the same numbers through the outlier calculator.

What is the IQR in a five number summary?

The interquartile range is Q3 minus Q1. It measures the spread of the middle 50% of your data and ignores the extreme values at each end, which makes it more stable than the full range when a dataset has outliers. For the bowling set above, IQR = 16 − 6 = 10. For a fuller picture of spread, pair the IQR with the variance, which measures how far every value sits from the mean.

Common mistakes to avoid

• Skipping the sort. Quartiles are meaningless until the data is in order.
• Including the median in a half. With an odd count, the median belongs to neither half.
• Mixing up median and mean. The median is the middle value, not the average.
• Using the wrong quartile method. Different methods give slightly different Q1 and Q3. Match the one your course uses.

Frequently Asked Questions (FAQs)

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