How to Graph Box Plot?

TL;DR: Want to summarize a whole data set in one quick picture? A box plot does it with just five numbers: minimum, Q1, median, Q3, and maximum. The box itself stretches from Q1 to Q3 with a line at the median, giving you the middle 50 percent of your data at a glance. Whiskers reach out to the smallest and largest non-outlier values, and any true outliers float off as separate dots. Read one and you instantly know the spread, the center, and the weird points.

Key takeaways:

Box plots show the five-number summary: min, Q1, median, Q3, max.
The box spans Q1 to Q3, so its width is the IQR.
A line inside the box marks the median (Q2), splitting the data in half.
Whiskers extend to the smallest and largest values within \(1.5\,\text{IQR}\) of the box.
Points outside the whiskers (the \(1.5\,\text{IQR}\) fences) are plotted as outliers.

Tutor-style math help

Graph Box Plot: what to notice and how to work it

Statistics skill

Statistics is about describing data honestly. A calculation is useful only when it helps explain center, spread, shape, or association.

What to notice first

Ask what the display is trying to show. Histograms show shape, box plots show spread by quartiles, and scatter plots show association.

Common student mistake

Do not report a number without context. A mean, median, IQR, or regression line should answer a question about the data.

Key formulas and cues

\(\text{mean}=\frac{\text{sum}}{\text{number of values}}\)

\(IQR=Q_3-Q_1\)

\(z=\frac{x-\mu}{\sigma}\)

\(\text{slope of best-fit line}=\frac{\text{change in prediction}}{\text{change in }x}\)

A reliable path

Identify the questionDecide whether you need center, spread, shape, or association.
Use the right displayChoose a histogram, box plot, scatter plot, or summary statistic.
Write the meaningExplain what the statistic says about the data set.

Worked examples

Find IQR

Example: \(Q_1=8\), median \(=12\), \(Q_3=20\)

IQR measures the middle 50%.
Subtract Q1 from Q3.
20 – 8 = 12.

Answer: \(IQR=12\)

Read association

Example: A scatter plot rises from left to right.

As x increases, y tends to increase.
That is a positive association.
A best-fit line should have positive slope.

Answer: Positive association

Open pop-up practice

Try one before moving on

Try: Find the mean of 6, 8, and 10.

Answer: \(8\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

A box diagram is a special type of diagram that shows the quartiles in a box and the line that extends from the lowest to the highest value.

A step-by-step guide to the box plot

A method for summarizing a set of data that is measured using an interval scale is called a box-and-whisker plot. These are mostly used for data analysis. We use these types of charts or graphs to know:

Distribution shape
A central value of it
Variability of it

A box plot is a graph that shows data from a five-number summary containing one of the measures of central tendency. It does not show the distribution in particular as much as a stem and leaf plot or a histogram does. But it is primarily used to indicate whether a distribution is skewed or not, and if there are potential unusual observations (also called outliers) present in the data set.

Parts of box plots

Minimum: The minimum value in the given dataset.
First Quartile \((Q1)\): The first quarter is the middle of the lower half of the data set.
Median: The median is the middle value of the dataset that divides the given dataset into two equal parts. The median is considered the second quartile.
Third Quartile \((Q3)\): The third quartile is the middle of the upper half of the data.
Maximum: The maximum value in the given dataset.

Apart from these five terms, the other terms used in the box diagram are:

Interquartile Range \((IQR)\): The difference between the third quartile and first quartile is known as the interquartile range. \(IQR = Q3-Q1\)
Outlier: The data that falls on the far left or right side of the sorted data is tested to be the outlier. In general, the outliers fall more than the specified distance from the first and third quartiles. Outliers are greater than \(Q3+(1.5\times IQR)\) or less than \(Q1-(1.5\times IQR)\).

Box plot distribution

The box diagram distribution explains how the data is grouped, how the data is skewed, and also the symmetry of the data.

Positively Skewed: If the distance from the middle to the maximum is greater than the distance from the middle to the minimum, the box diagram has a positive skew.
Negatively Skewed: If the distance from the middle to the minimum is greater than the distance from the middle to the maximum, the box diagram will be negatively skewed.
Symmetric: If the median is equidistant from the maximum and minimum values, the box plot is symmetric.

Box plot chart

In a box and whisker plot:

The end of the box is the top and bottom quadrants so that the box crosses the interquartile range.
A vertical line inside the box marks the median.
The two lines outside the box are the whiskers drawn to the highest and lowest observations.

Box Plot , Example 1:

Find the maximum, minimum, median, first quartile, and third quartile for the given data set: \(24, 46, 12, 10, 15, 14, 8\).

Solution:

Given: \(24, 46, 12, 10, 15, 14, 8\).

Arrange the given dataset in ascending order.

\(8, 10, 12, 14, 15, 24, 46\)

Therefore,

Minimum \(= 8\)

Maximum \(= 46\)

Median \(= 14\)

First quartile \(= 10 \left(Middle\:value\:of\:\:8,\:10,\:12\:is\:10\right)\:\)

Third quartile \(= 24 \left(Middle\:value\:of\:15,\:24,\:46\:is\:24\right)\).

Exercises for Box Plot

Draw a box and whisker plot for each data set.

\(\color{blue}{22, 24, 25, 26, 21, 22, 28, 29, 23}\)
\(\color{blue}{51, 53, 52, 56, 52, 55, 55, 56, 52}\)

Recommended EffortlessMath Books

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Frequently Asked Questions

What is a box plot graph used for?

A box plot shows how a data set is spread out using five numbers: the minimum, Q1, median, Q3 and maximum. It is especially handy for comparing groups, such as test scores from two classes, because you can see at a glance which group has the higher median and which has the wider spread. It also makes outliers easy to spot.

How do you draw a box plot graph?

Sort the data and find the five-number summary. For 2, 4, 5, 7, 8, 10, 12, the minimum is 2, Q1 is 4, the median is 7, Q3 is 10 and the maximum is 12. Draw a number line, make a box from 4 to 10 with a vertical line at 7, then draw whiskers out to 2 and 12.

How do you graph a box plot with outliers?

Find IQR = Q3 − Q1, then the fences Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. Plot any value outside the fences as a separate dot, and stop each whisker at the most extreme value still inside. For example, with Q1 = 10 and Q3 = 18, the IQR is 8, so the fences are 10 − 12 = −2 and 18 + 12 = 30. A value of 35 would be an outlier.

How do you graph a box plot on a TI-84?

Enter the data in L1 using STAT, then Edit. Press 2nd, Y= to open STAT PLOT, turn Plot1 on and pick a box plot type; the modified box plot shows outliers as separate points. Set Xlist to L1 and Freq to 1, then press ZOOM and choose ZoomStat to fit the window. Press TRACE to read the five-number summary.

How do you read a box plot graph?

The line inside the box is the median, the box edges are Q1 and Q3, and the whisker ends are the smallest and largest non-outlier values. Each section holds about 25% of the data. So if a box plot of quiz scores has Q1 = 70 and Q3 = 88, the middle half of the students scored between 70 and 88, and the IQR is 18 points.

What is the difference between a bar graph and a box plot?

A bar graph compares counts or amounts across categories, like the number of students who chose each favorite sport. A box plot summarizes one set of numerical data, like heights or test scores, by showing its median, quartiles and range. Use a bar graph to answer how many are in each group, and a box plot to see how spread out the values are.

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