1. A box-and-whisker plot uses a number line to represent the data visually. I.E. It shows visually the GRE - Mean, Median, Mode, and Range

2. It displays a data set along a number line using medians. Quartiles divide the data set into four equal parts. The median (second quartile) divides the data set into two halves. The median of the lower half is the first quartile. The median of the upper half is the third quartile.

3. The box plot does not show individual data but it analyzes the spread of data. A measure of spread is a single number that describes how to spread out or clustered together the data are.

4. Another measure of spread is the interquartile range (IQR). The “Interquartile Range” is the difference between the smallest value and the largest value of the middle 50% of a set of data.

The "Interquartile Range" is from Q1 to Q3:




To find the interquartile range of a set of data:

• First put the list of numbers in order;
• The list must be divided into four equal parts;
• These four equal parts (25% each, regardless of the numbers contained in each individual sector) are the quartiles;
• The interquartile range is the distance between the two middle sets of data, or in other words: the third quartile (or chunk) less the second quartile or chunk.

How to construct a box plot?





More Examples




Box plot with odd numbers in that data set

The first step in constructing a box-and-whisker plot is to first find the median (Q2), the lower quartile (Q1) and the upper quartile (Q3) of a given set of data.

Step 1: Find the median. The median is the value exactly in the middle of an ordered set of numbers.





Step 2: Consider the values to the left of the median: 18 27 34 52 54 59 61 Find the median of this set of numbers. The median is 52 and is called the lower quartile.

Step 3: Consider values to the right of the median: 78 82 85 87 91 93 100 Find the median of this set of numbers. The median 87 is therefore called the upper quartile.


Note that when there is an odd number of values, as in this example. we don't include the median in the set of numbers used to calculate the upper and lower quartiles.

We are able now to find the interquartile range (IQR). The interquartile range is the difference between the upper quartile and the lower quartile. In example 1, the IQR = Q3 — Q1 = 87 - 52 = 35.
The IQR is a very useful measurement. It is useful because it is less influenced by extreme values as it limits the range to the middle 50% of the values.


Box plot with even numbers in that data set


Step 1: Find the median. The median is the value exactly in the middle of an ordered set of numbers.




The median in this example is in-between 70 and 73, so the median is calculated by taking the mean of 70 and 73:

Median \(= \frac{70+73}{2} =71.5\)

Step 2: Consider the values to the left of the median: 42 63 64 64 70 Find the median of this set of numbers. The median is 64.

Step 3: Consider the values to the right of the median: 72 76 77 81 81


Note that when the number of values is even the median lies between the two middle values. As in this example, we include the data value just below the median in the set of numbers used to calculate the lower quartile. and the number just above the median in the set of numbers used to calculate the upper quartile.


The median is 77 and is called the upper quartile. We are able now to find the interquartile range (IQR). The interquartile range is the difference between the upper quartile and the lower quartile. The IQR = Q3 — Q1 = 77 - 64 = 13. The IQR is a very useful measurement. It is useful because it is less influenced by extreme values as it limits the range to the middle 50% of the values.


The Box Plots can be either vertical




Or horizontal