Box Plot Formula

The Formula

\text{Lower fence} = Q_1 - 1.5 \cdot \text{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\%.

Quick Example

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

Notation

Five-number summary: \{x_{\min}, Q_1, \tilde{x}, Q_3, x_{\max}\} where \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\%.

Formal View

Five-number summary \{x_{(1)}, Q_1, Q_2, Q_3, x_{(n)}\}; outlier fences at Q_1 - 1.5 \cdot \text{IQR} and Q_3 + 1.5 \cdot \text{IQR}

Worked Examples

Example 1

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

Solution

  1. 1
    Order the data (already ordered). Find quartiles: Q_1 = 8 (median of lower half \{3,7,8,10,12\}... median is 8), Q_2 = 13 (average of 12 and 14), Q_3 = 20 (median of upper half \{14,18,20,25,100\}... median is 20)
  2. 2
    Calculate IQR: IQR = Q_3 - Q_1 = 20 - 8 = 12
  3. 3
    Find fences: Lower fence = Q_1 - 1.5 \times IQR = 8 - 18 = -10; Upper fence = Q_3 + 1.5 \times IQR = 20 + 18 = 38
  4. 4
    Identify outliers: any value below -10 or above 38. The value 100 exceeds 38, so it is an outlier
  5. 5
    Draw box plot: whiskers extend to 3 (min non-outlier) and 25 (max non-outlier); box from Q_1=8 to Q_3=20; line at median Q_2=13; plot 100 as a separate dot

Answer

100 is an outlier (exceeds upper fence of 38). Whiskers: [3, 25]. Box: [8, 20]. Median: 13.
The 1.5×IQR rule identifies potential outliers by establishing fences beyond which data is unusual. Box plots provide a five-number summary (min, Q1, median, Q3, max) while flagging extreme values as separate dots.

Example 2

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

Common Mistakes

  • Assuming the whiskers always extend to the minimum and maximum — they typically stop at 1.5 \times \text{IQR} from the quartiles
  • Thinking the median line must be in the center of the box — a skewed distribution shifts it to one side
  • Interpreting the box width as the data range — the box only covers the middle 50\% (IQR)

Why This Formula Matters

Box plots excel at comparing distributions across groups and spotting skewness and outliers at a glance — especially valuable with many data sets side by side.

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\%.

What do the symbols mean in the Box Plot formula?

Five-number summary: \{x_{\min}, Q_1, \tilde{x}, Q_3, x_{\max}\} where \tilde{x} is the median

Why is the Box Plot formula important in Math?

Box plots excel at comparing distributions across groups and spotting skewness and outliers at a glance — especially valuable with many data sets side by side.

What do students get wrong about Box Plot?

Whiskers don't always go to min/max—they may stop at 1.5 \times \text{IQR} from the box.

What should I learn before the Box Plot formula?

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

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