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# Box and Whisker Plot|Definition & Meaning

## Definition

A **special type of figure** representing first, second and third **quartiles** against some **given data** as a **box** shaped plot with **lines** protruding from its sides spanning the **lowest** and **highest** values.

A form of **graph** called a **box and whisker plot** connects the boxes that indicate the distribution of **numerical data** with lines (also known as **whiskers).** Box and whisker plots show how a set of data might **vary.** An appropriate depiction may also be provided by a **histogram analysis,** but a box and whisker plot provides **additional information** while allowing the display of multiple data sets on the same graph. An example is shown below:

**Figure 1: Example of Box and Whisker Plot**

**Box and whisker plots** are very effective at **visually summarizing** data from various sources on a **single graph**. As such, these plots allow you to compare data from **different categories** easily, leading to efficient **decision-making.**

## Some Real World Applications

When you have many data sets from **different sources** that are connected in some manner, consider box and whisker graphs. Here are several **examples** from the real world where they may prove **helpful:**

(a) Compiling the **results** of **students** from different **institutions** or for different **courses.**

(b) Suppose you suggest a **modification** in some **industrial plant** or process. Box and whisker plots can be used to depict the effect of this **modification** on production before and after this change.

(c) Different characteristics of a **mechanical system**

(d) Data coming from **comparable devices** yielding similar results

There are many other such **applications** that can be listed.

## Statistical Information Inside a Box and Whisker Plot

The box and whisker plot shows the five summary statistics of the given numerical data.

(a) Lowest Value **(Minimum)**

(b) **Median**

(c) Highest Value **(Maximum)**

(d) **Lower Quartile**

(e) **Upper Quartile**

Consequently, the **box and whisker plot** may be constructed using the same **five statistics** listed above. A thorough understanding of all these **parameters** is a prerequisite for learning the **box and whisker plots.** Lets understand these **characteristics** one by one.

#### (a) Minimum value

The **numerically smallest value** in the given data-set or population. Its a simple **minimum function.**

#### (b) Median

If the given data is sorted in **ascending order** of **numerical magnitude,** then the median value is the number in the **center** of a set of values. It is usually the **value in the middle** in case of an odd number of samples. In the case of an even number of samples, the **middle two values** are averaged to find the median. Specifically, for an even number of samples, the **median** is the arithmetic mean of the middle two values.

#### (c) Highest Value (Maximum)

The **numerically largest value** in the given data-set or population. Its a simple **maximum function.**

#### (d) Lower Quartile

If the given data is sorted in **ascending order** of numerical magnitude, then the **lower quartile** is the numeral below which the data for the lowest 25% are included. It represents the **lowest 25%** outlier values of the data also called the lower tail.

#### (e) Upper Quartile

If the given data is sorted in **ascending order** of numerical magnitude, then the **upper quartile** is the numeral above which the data for the highest 25% are included. It represents the **highest 25%** outlier values of the data also called the higher tail.

## Construction of Box and Whisker Plot

The **construction** of box and whisker plot seems simple and **intuitive** at first sight but it might get very confusing for the students not familiar with **statistics** or the ones generally not comfortable with **graphs.** The following set of paragraphs explain how to construct a **box and whisker** plot using the given data. For the sake of **example,** we will consider some example data given below:

**Given Data = { 20, 50, 40, 30, 60, 90, 80, 70, 10 }**

**First step** is to **sort** all the **data points** in ascending order of numerical magnitude. The resulting data sequence looks as follows:

**Given Data = { 10, 20, 30, 40, 50, 60, 70, 80, 90 }**

**Second step** is to find the **Lowest Value (Minimum), Median, Highest Value (Maximum), Lower Quartile** and **Higher Quartile.** For the given data sequence above, these values are listed below:

**Lowest Value (Minimum) = 10**

**Median = 50**

**Highest Value (Maximum) = 90**

**Lower Quartile = 25**

**Upper Quartile = 75**

**Third step** is to plot the **Lowest Value (Minimum), Median, Highest Value (Maximum), Lower Quartile** and **Higher Quartile** points on a chart in the form of vertical bars (for the case of horizontal box and whisker plot) as shown in the figure below:

**Figure 2**: Marking the Lowest Value **(Minimum), Median,** Highest Value **(Maximum), Lower Quartile** and **Higher Quartile** on chart

**Fourth step** is to **construct** **box** by joining the Lower Quartile and Higher Quartile Bars as shown in the figure below:

**Figure 3:** Constructing the **Box** using **Lower Quartile** and **Higher Quartile** Bars

**Fifth and final step** is to **construct the whiskers** by joining the centers of **minimum** and **maximum** value bars with the lower and higher quartile bars respectively as shown in the figure below:

**Figure 4**: Constructing the **Whiskers**

This **five step process** is a comprehensive way of constructing or **generating a box and whisker plot.** Following is a **numerical problem** for further understanding.

## Numerical Problems Related to Box and Whisker Plot

Construct a **box and whisker plot** for the following datasets containing marks of **nine students in two different subjects:**

**Science = { 80, 50, 54, 70, 60, 82, 87, 75, 55 }**

**Mathematics = { 70, 80, 95, 80, 55, 80, 66, 88, 60 }**

### Solution

Sorting the given data sets:

**Science = { 50, 54, 55, 60, 70, 75, 80, 82, 87 }**

**Mathematics = { 55, 60, 66, 70, 80, 80, 80, 88, 95 }**

Calculating the statistical Values for Science subject data:

**Lowest Value (Minimum) = 50**

**Median = 70**

**Highest Value (Maximum) = 87**

**Lower Quartile = 54.5**

**Upper Quartile = 81**

Calculating the statistical Values for Mathematics subject data:

**Lowest Value (Minimum) = 55**

**Median = 80**

**Highest Value (Maximum) = 95**

**Lower Quartile = 63**

**Upper Quartile = 84**

Constructing the **box and whisker plot** for the given data points against results of **students** in **mathematics** and **science** subjects:

**Figure 5:** Box and Whisker Plot of **Students’** Marks in **Mathematics** and **Science** Subjects

*All mathematical drawings and images were created with GeoGebra.*