We have learned that the median is identified as the middle value in a data set. It essentially divides a data set into two parts.
A data set can also be divided into four parts. The scores contained in these four parts are called quartiles. As a result, a data set has three quartiles (also known as Tukey's Hinges):
The lower quartile (Q1) is the middle score of the lower group.
The middle (median; Q2) is the middle score. That middle score divides the data set into two equal groups.
The upper quartile (Q3) is the middle score of the upper group.
In terms of quartiles, we must also note the interquartile range (IQR).
To calculate the IQR, we would subtract the lower quartile (Q1) from the upper quartile (Q3).
Q3 – Q1 = IQR
The IQR range is a measure of the spread of the middle 50% of the data set.
STATS FACT:
25% of the test scores will be less than Q1 and 75% greater than Q1
50% of the test scores will be less than Q2 and 50% greater than Q2
75% of the test scores will be less than Q3 and 25% greater than Q3
In our example, we will use the Benchmark Assessment data set from two English Language Arts (Junior) classes: Period 1 and Period 10 test scores.
There are 20 students in each class. This will constitute a great number of test scores. We will find Q1, Q2, Q3 and the IQR for the test scores.
The test scores from the Period 1 class is as follows:
45, 67, 87, 64, 94, 78, 56, 47, 99, 93, 71, 96, 86, 21, 16, 82, 65, 80, 71, 68
Step 1: Reorder the values in your data set from lowest to highest.
16, 21, 45, 47, 56, 64, 65, 67, 68, 71 | 71, 78, 80, 82, 86, 87, 93, 94, 96, 99
(We have an even set of values. 71 + 71 = 142 142 ÷ 2 = 71
The median (Q2) = 71.
The middle score in the set 16, 21, 45, 47, 56, 64, 65, 67, 68, 71
56 + 64 = 120
120 / 2 = 60
Q1 = 60
The middle score in the set 71, 78, 80, 82, 86, 87, 93, 94, 96, 99
86 + 87 = 173
173 / 2 = 86.5
Q3 = 86.5
IQR = Q3 – Q1
= 86.5 – 60
= 26.5
IQR = 26.5
Figure 1. Box plot configuration
From From Figure 1, we can learn the different parts of a box plot.
A box plot will contain two whiskers that project out from the box.
The highest value in the data set is represented by the top whisker.
The lowest value in the data set is represented by the bottom whisker.
A box plot also contains the quartiles (also known as Tukey's Hinges) discussed above.
The upper quartile (Q3) is represented by the line on the top of the box.
The median (Q2 ) is represented by the line within the box.
The lower quartile (Q1 ) is represented by the line on the bottom of the box.
From viewing a box plot, we can learn about a data set’s highest and lowest score, its range of scores, median, upper and lower quartiles, and its interquartile range.
For example, in Figure 2., the box plot can reveal the highest and lowest values in the data set just by looking at the top and bottom whiskers.
The top whisker is set at 58 (highest score/value).
The bottom whisker is set at 15 (lowest score/value).
The range of scores = 58 – 15 = 43
The median (Q2 ) is set at 41.
The upper quartile (Q3) is set at 50, while the
lower quartile (Q1 ) is set at 23.
Figure 2. Box plot example
Since we have Q3 and Q1, we can calculate the interquartile range (IQR):
50 – 23 = 27
From visualizing the box plot, we can deduce that:
the bottom 25% of the values lie between 15 – 23 (as seen from the bottom whisker to Q1 )
the top 25% of the values lie between 50 – 58 (as seen from the top whisker to Q3)
the middle 50% of the values lie between 23 – 50 (as seen from Q1 to Q3)
The median (Q2 ) line is situated closer to the upper quartile (Q3), indicating that the top half of the values is grouped closer to Q2 than Q1 .
Quartiles & Box Plots (Explanation & Examples)
