Box Plot Formula

A box plot displays the five-number summary (minimum, Q1, median, Q3, maximum) of a data set using a box and whiskers.

The Formula

Lower fence=Q1−1.5⋅IQR\text{Lower fence} = Q_1 - 1.5 \cdot \text{IQR}, Upper fence=Q3+1.5⋅IQR\text{Upper fence} = Q_3 + 1.5 \cdot \text{IQR}

When to use: A summary of spread and center in one picture. Box shows the middle 50%50\%.

Quick Example

Box from Q1 to Q3, line at median, whiskers to min/max, dots for outliers.

Notation

Five-number summary: {xmin⁡,Q1,x~,Q3,xmax⁡}\{x_{\min}, Q_1, \tilde{x}, Q_3, x_{\max}\} where x~\tilde{x} is the median

What This Formula Means

A box plot displays the five-number summary (minimum, Q1, median, Q3, maximum) of a data set using a box and whiskers.

A summary of spread and center in one picture. Box shows the middle 50%50\%.

Formal View

Five-number summary {x(1),Q1,Q2,Q3,x(n)}\{x_{(1)}, Q_1, Q_2, Q_3, x_{(n)}\}; outlier fences at Q1−1.5⋅IQRQ_1 - 1.5 \cdot \text{IQR} and Q3+1.5⋅IQRQ_3 + 1.5 \cdot \text{IQR}

Worked Examples

Example 1

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For the data set {3,7,8,10,12,14,18,20,25,100}\{3, 7, 8, 10, 12, 14, 18, 20, 25, 100\}, construct a box plot and identify any outliers using the 1.5×IQR1.5 \times IQR rule.

Answer

100 is an outlier (exceeds upper fence of 38). Whiskers: [3, 25]. Box: [8, 20]. Median: 13.

First step

1
Order the data (already ordered). Find quartiles: Q1=8Q_1 = 8 (median of lower half {3,7,8,10,12}\{3,7,8,10,12\}... median is 8), Q2=13Q_2 = 13 (average of 12 and 14), Q3=20Q_3 = 20 (median of upper half {14,18,20,25,100}\{14,18,20,25,100\}... median is 20)

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Example 2

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Two box plots compare exam scores for two classes. Class A: median=72, Q1=65Q_1=65, Q3=80Q_3=80. Class B: median=78, Q1=70Q_1=70, Q3=85Q_3=85. Compare center and spread for both classes.

Example 3

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Data {2,4,5,7,8,10,12,15}\{2,4,5,7,8,10,12,15\}. Find the five-number summary.

Common Mistakes

  • Reading box width as a count of data — each section holds 25% of values; width shows spread, not frequency.
  • Forgetting outliers extend the whiskers only to non-outlier values — points beyond the 1.5×IQR1.5\times\text{IQR} fences are plotted separately.
  • Confusing the median line with the mean — the box plot shows the median, which the mean need not match in skewed data.

Why This Formula Matters

The box plot is the fast comparison tool — line up several side by side and you instantly see which group has higher median, wider spread, or more outliers, all from five numbers. It is where median, quartiles, and IQR come together visually. Recognizing it by "Am I summarizing or comparing distributions using min, Q1, median, Q3, and max?" — rather than by familiar numbers — is what lets a student tell it apart from histogram and five-number summary and dot plot in a mixed problem set.

Frequently Asked Questions

What is the Box Plot formula?

A box plot displays the five-number summary (minimum, Q1, median, Q3, maximum) of a data set using a box and whiskers.

How do you use the Box Plot formula?

A summary of spread and center in one picture. Box shows the middle 50%50\%.

What do the symbols mean in the Box Plot formula?

Five-number summary: {xmin⁡,Q1,x~,Q3,xmax⁡}\{x_{\min}, Q_1, \tilde{x}, Q_3, x_{\max}\} where x~\tilde{x} is the median

Why is the Box Plot formula important in Math?

The box plot is the fast comparison tool — line up several side by side and you instantly see which group has higher median, wider spread, or more outliers, all from five numbers. It is where median, quartiles, and IQR come together visually. Recognizing it by "Am I summarizing or comparing distributions using min, Q1, median, Q3, and max?" — rather than by familiar numbers — is what lets a student tell it apart from histogram and five-number summary and dot plot in a mixed problem set.

What do students get wrong about Box Plot?

The procedure for box plot is the easy part; the trap is reading box width as a count of data. Asking "Am I summarizing or comparing distributions using min, Q1, median, Q3, and max?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Box Plot formula?

Before studying the Box Plot formula, you should understand: median, quartiles.

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