Box Whisker Plot
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Dimitris Charalampidis
charalampidisdimitris.com
Box Whisker Plot
Box Whisker Diagram Concept of a box whisker plot A box whisker plot (also known as boxplot) is a way of visualizing a dataset’s certain characteristics. More precisely, through it one can directly see the set’s: •
Minimum value
min
•
Maximum value
max
•
Lower quartile
Q1
•
Upper Quartile
Q3
•
Median
Med
Having these values, it is evident that through the boxplot one can easily also determine the values of: •
Range
R = max − min
•
Interquartile range
IQR = Q3 − Q1
So how will a boxplot look like and how can we draw it? Drawing a box whisker plot is a quite easy process as long as you know the previously mentioned characteristics of the dataset you are working with. Then, all you need is simply a line of numbers and a pen. Example
min
Q1
Med
Q3
max
Dimitris Charalampidis
charalampidisdimitris.com
Box Whisker Plot
Outliers Sometimes, a set contains some values that are considered significantly far enough from the rest of the set’s values. The presence of these values can often lead to misinterpretations regarding the way the set behaves. Therefore, it is strongly needed not only to identify these values but to be able to demonstrate them in a diagram. Regarding their calculation we can use the following rules: A set’s value is considered to be an outlier if it is either Q3 + 1.5 IQR or
Q1 − 1.5 IQR A box plot can be used to present the outliers as dots
Q1 Med
min
Q3
max
The dots indicate the presence of outliers
Distribution of the set’s values One extra information we can get from a boxplot is regarding the way the set’s values are distributed. If you get a closer look in the way a boxplot is constructed, you will notice that the range is divided into 4 smaller (not necessarily equal) segments. In each one of these segments we can find the 25% of the dataset’s size.
25%
25% 25%
25%
min − Q1 : 25% of
size
Q1 − Med : 25% of
size
− Q3 : 25% of
size
− max : 25% of
size
Med Q3
Dimitris Charalampidis
charalampidisdimitris.com
Box Whisker Plot
Calculating the mean of a set using information taken from the boxplot As we learned in this presentation’s first slide, through a box and whisker diagram we can only see directly the values of min, lower quartile, median, upper quartile and max and through them we can also calculate the values of range and interquartile range. There will be some exercises thought, in which we will be given a boxplot and we will be asked to calculate the set’s arithmetic mean. To do that we will need to think of the previous property we mentioned about the way the values are distributed and using that, create a table of grouped data. Then by calculating the mid-interval values we would proceed normally in finding the set’s arithmetic mean
Example In the given box whisker diagram data taken from a sample of 60 students regarding their grades are demonstrated. Calculate the arithmetic mean of the sample’s grades Solution Intervals 27 – 62 62 – 70 70 – 87 87 - 94
Mid-interval values 44.5 66 78.5 90.5
Frequency 15 15 15 15
25 60 = 15 100 mean =
( 44.5 15) + ( 66 15) + ( 78.5 15) + ( 90.5 15 ) = 69.875 60
You can also remember that the mean will be equal to the average of the mid-interval values
44.5 + 66 + 78.5 + 90.5 = 69.875 4
27
62
70
87
94
charalampidisdimitris.com
Box Whisker Plot
Box Whisker Diagram Concept of a box whisker plot A box whisker plot (also known as boxplot) is a way of visualizing a dataset’s certain characteristics. More precisely, through it one can directly see the set’s: •
Minimum value
min
•
Maximum value
max
•
Lower quartile
Q1
•
Upper Quartile
Q3
•
Median
Med
Having these values, it is evident that through the boxplot one can easily also determine the values of: •
Range
R = max − min
•
Interquartile range
IQR = Q3 − Q1
So how will a boxplot look like and how can we draw it? Drawing a box whisker plot is a quite easy process as long as you know the previously mentioned characteristics of the dataset you are working with. Then, all you need is simply a line of numbers and a pen. Example
min
Q1
Med
Q3
max
Dimitris Charalampidis
charalampidisdimitris.com
Box Whisker Plot
Outliers Sometimes, a set contains some values that are considered significantly far enough from the rest of the set’s values. The presence of these values can often lead to misinterpretations regarding the way the set behaves. Therefore, it is strongly needed not only to identify these values but to be able to demonstrate them in a diagram. Regarding their calculation we can use the following rules: A set’s value is considered to be an outlier if it is either Q3 + 1.5 IQR or
Q1 − 1.5 IQR A box plot can be used to present the outliers as dots
Q1 Med
min
Q3
max
The dots indicate the presence of outliers
Distribution of the set’s values One extra information we can get from a boxplot is regarding the way the set’s values are distributed. If you get a closer look in the way a boxplot is constructed, you will notice that the range is divided into 4 smaller (not necessarily equal) segments. In each one of these segments we can find the 25% of the dataset’s size.
25%
25% 25%
25%
min − Q1 : 25% of
size
Q1 − Med : 25% of
size
− Q3 : 25% of
size
− max : 25% of
size
Med Q3
Dimitris Charalampidis
charalampidisdimitris.com
Box Whisker Plot
Calculating the mean of a set using information taken from the boxplot As we learned in this presentation’s first slide, through a box and whisker diagram we can only see directly the values of min, lower quartile, median, upper quartile and max and through them we can also calculate the values of range and interquartile range. There will be some exercises thought, in which we will be given a boxplot and we will be asked to calculate the set’s arithmetic mean. To do that we will need to think of the previous property we mentioned about the way the values are distributed and using that, create a table of grouped data. Then by calculating the mid-interval values we would proceed normally in finding the set’s arithmetic mean
Example In the given box whisker diagram data taken from a sample of 60 students regarding their grades are demonstrated. Calculate the arithmetic mean of the sample’s grades Solution Intervals 27 – 62 62 – 70 70 – 87 87 - 94
Mid-interval values 44.5 66 78.5 90.5
Frequency 15 15 15 15
25 60 = 15 100 mean =
( 44.5 15) + ( 66 15) + ( 78.5 15) + ( 90.5 15 ) = 69.875 60
You can also remember that the mean will be equal to the average of the mid-interval values
44.5 + 66 + 78.5 + 90.5 = 69.875 4
27
62
70
87
94
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