Quartile and Box Plot Calculator: Five-Number Summary, IQR, Outliers, Standard Deviation and Coefficient of Variation

Summarise a small data set and find its outliers.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Set how many values you have and enter them.

Choose the quartile method your course uses.

The calculator sorts the data and gives the five-number summary, IQR, fences, outliers and the mean and spread.

Formulas used

  • IQR = Q_3 - Q_1
  • Q_1 - 1.5\,IQR,\quad Q_3 + 1.5\,IQR
  • s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}

Worked example

With the default numbers in the calculator above:

  1. The data in order: 7, 15, 36, 39, 40, 41, 42, 43, 47, 49 (n = 10)
  2. Five-number summary: minimum, Q1, median, Q3, maximum: 7, 36, 40.5, 43, 49
  3. Interquartile range: 7
  4. Range: 42
  5. Outlier fences (1.5 x IQR beyond the quartiles): 25.5 and 53.5
  6. Outliers: 7, 15
  7. Mean: 35.9
  8. Sample standard deviation (divide by n - 1): 13.7715
  9. Population standard deviation (divide by n): 13.0648
  10. Coefficient of variation: 38.361 %

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Quartile method Inclusive, as in Excel QUARTILE.INC.

  1. The data in order: 7, 15, 36, 39, 40, 41, 42, 43, 47, 49 (n = 10)
  2. Five-number summary: minimum, Q1, median, Q3, maximum: 7, 36.75, 40.5, 42.75, 49
  3. Interquartile range: 6
  4. Range: 42
  5. Outlier fences (1.5 x IQR beyond the quartiles): 27.75 and 51.75
  6. Outliers: 7, 15
  7. Mean: 35.9
  8. Sample standard deviation (divide by n - 1): 13.7715
  9. Population standard deviation (divide by n): 13.0648
  10. Coefficient of variation: 38.361 %

Common questions

How do I find the quartiles?

Split the ordered data in half. Q1 is the median of the lower half, Q3 the median of the upper half, and for an odd count the middle value is left out.

How do I find outliers?

A value is an outlier if it is more than 1.5 times the IQR below Q1 or above Q3.

Why do different calculators give different quartiles?

There are several accepted methods. Excel has an inclusive and an exclusive one that differ from the school method.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any school or first-year statistics text (summary statistics and box plots).

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