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Attached pdf of this Article as SPOILER at the top! Happy learning!
Hi All,
I have recently uploaded a video on YouTube to discuss Box and Whisker Plot in Detail:
Following is covered in the video
¤ What is Box and Whisker Plot? ¤ Five Number Summary: Basics of Box and Whisker Plot. ¤ Insights from Box and Whisker Plot. (Top and Bottom Values) ¤ Inter Quartile Range, Outliers and Modified Box and Whisker Plot ¤ Comparing two Box and Whisker Plots ¤ Skewness in Box and Whisker Plot
What is Box and Whisker Plot?
A Box and Whisker plot is a graphical method of representing the data in Quartiles. This also helps in easily identifying the locality, spread and skewness in the dataset.
Example: Let's say the marks of the students in a class (from 0 to 100) are given to us in the form of below Box and Whisker Plot.
Attachment:
Box and Whisker Image-1.jpg [ 20.21 KiB | Viewed 1445 times ]
The left whisker in above Box and Whisker Plot is called as Lower Whisker The right whisker in above Box and Whisker Plot is called as Upper Whisker The box in above Box and Whisker Plot is called as Box The line in between the Box is as the Median of the dataset.
Five Number Summary: Basics of Box and Whisker Plot
Five number summary of a Box and Whisker Plot is given by following values
¤ Minimum ¤ Q1 or First Quartile or Lower Quartile ¤ Median or Q2 or Second Quartile or Middle Quartile ¤ Q3 or Third Quartile or Upper Quartile ¤ Maximum
Attachment:
Box and Whisker Plot five point summary.jpg [ 29.37 KiB | Viewed 1457 times ]
In Above Box and Whisker Plot:
¤ Minimum : Is the starting point from left of lower whisker = 20 ¤ Q1 : Is the start of the box from left = 30 ¤ Median or Q2 : Is the line in between the box = 40 ¤ Q3 : Is the end of the box = 80 ¤ Maximum : Is the ending point of upper whisker = 100
Q 1: Marks of 11 students in a class test are given below: (Min Marks =0, Max Marks = 100)
55, 10, 64, 88, 40, 65, 50, 52, 45, 58, 100
1.1 Make a box and Whisker Plot representing this data set. 1.2 Find the Highest score. 1.3 Find the Lowest score. 1.4 Find the Range. 1.5 Find the value of Q1 1.6 Find the value of Q2 1.7 Find the value of Q3
Sol 1: Let's start by arranging the terms in ascending order, we get 10, 40, 45, 50, 52, 55, 58, 64, 65, 88, 100
As there are 11 terms so median will be the \(\frac{11+1}{2}\) = \(6^{th}\) term => Median = 55
So, the set will be divided into three parts First Half = 10, 40, 45, 50, 52 and median of first half = Q1 = \(3^{rd}\) term = 45 Q2 = Median = 55 Second Half = 58, 64, 65, 88, 100 and median of second half = Q3 = \(3^{rd}\) term = 65
So, the Box and Whisker Plot has Min = 10 (starting of lower whisker) Q1 = 45 (start of the box) Q2 = 55 (place of line inside the box) Q3 = 65 (end of the box) Max = 100 (end of upper whisker)
1.1 So, the Box and Whisker Plot will be as below:
Attachment:
Box and Whisker Example 1.jpg [ 11.3 KiB | Viewed 1441 times ]
1.2 Find the Highest score - 100 1.3 Find the Lowest score - 10 1.4 Find the Range = 100 - 10 = 90 1.5 Find the value of Q1 = 45 1.6 Find the value of Q2 = 55 1.7 Find the value of Q3 = 65
Insights from Box and Whisker Plot. (Top and Bottom Values)
Bottom 25% of Values are between Min to Q1 Bottom 50% of Values are between Min to Q2 Bottom 75% of Values are between Min to Q3 Middle 50% values are between Q1 to Q3 Top 25% of Values are between Q3 to Max Top 50% of Values are between Q2 to Max Top 75% of Values are between Q1 to Max
Attachment:
Top and bottom Values.jpg [ 34.71 KiB | Viewed 1451 times ]
Q2. Answer the following questions based on the information given in the Box and Whisker Plot below:
2.1 What was the lowest temperature across the days? 2.2 What was the Highest temperature across the days? 2.3 What was the Median temperature across the days? 2.4 More than 50% of the days had a temperature of more than 42F? True/False? 2.5 Colder 25% of the days had a temperature between? 2.6 Hotter 25% of the days had a temperature between? 2.7 Middle 50% of the days had a temperature between? 2.8 Majority of the days had temperature of more than 45F? True/False? 2.9 At least 75% of the days had temperature less than 75F? True/False?
Attachment:
Box and Whisker Example 2.jpg [ 13.08 KiB | Viewed 1449 times ]
2.1 What was the lowest temperature across the days? : 20F as Min value is 20 2.2 What was the Highest temperature across the days? : 80F as Max value is 80 2.3 What was the Median temperature across the days? : 80F as Median value is 50 2.4 More than 50% of the days had a temperature of more than 42F? True/False? : True as top 50% values lies between Median and Max and Median is 50F 2.5 Colder 25% of the days had a temperature between? : 20(Min) and 40(Q1) 2.6 Hotter 25% of the days had a temperature between? : 70(Q3) and 80(Max) 2.7 Middle 50% of the days had a temperature between? : 40(Q1) and 70(Q3) 2.8 Majority of the days had temperature of more than 45F? True/False? : True as top 50% values lies between Median and Max and Median is 50F and 45F is to the left of Median 2.9 At least 75% of the days had temperature less than 75F? True/False? : True as bottom 75% values lies between Min and Q3 and Q3 is 70F and 75F is to the right of Q3
Inter Quartile Range, Outliers and Modified Box and Whisker Plot
Inter Quartile Range = Q3 - Q1
Q3. Marks of 11 students in a class test are given below. (Min marks = 0, Max Marks = 100). Find the IQR of the set.
55, 10, 64, 88, 40, 65, 50, 52, 45, 58, 100
Attachment:
Box and Whisker Example 3.jpg [ 12.8 KiB | Viewed 1440 times ]
So, Any value < 15 and > 95 is an outlier => 10 and 100 are outliers
Modified Box and Whisker Plot: If we remove the outlier values from the set and redraw the Box and Whisker plot then the new plot is called as a Modified Box and Whisker Plot.
After removing 10 and 100 following is the Modified Box and Whisker Plot
Attachment:
Modified Box and Whisker Example 3.jpg [ 12.35 KiB | Viewed 1442 times ]
Comparing two Box and Whisker Plots
Let's say Marks of students from Class A and Class B are given in terms of Box and Whisker Plots as below:
Attachment:
Comparing two box and whisker plots.jpg [ 18.76 KiB | Viewed 1456 times ]
Insights from these Box and Whisker Plots
¤ Highest Marks scored (90) by Class A students is more than the Highest Marks scored (80) by Class B students ¤ Lowest Marks scored (20) by Class B students is lower than the Lowest Marks scored (30) by Class A students ¤ Median Marks scored (70) by Class A students is more than the Median Marks scored (50) by Class B students ¤ 75% of the students in class A have score ≥ 60 marks ¤ 75% of the students in class B have score ≥ 35 marks
based on Above information we can conclude that Students from Class A on an average have scored more than Students from Class B.
Skewness in Box and Whisker Plot
Consider Following Box and Whisker Plots
Attachment:
Skewness in Box and Whisker Plot.jpg [ 10.94 KiB | Viewed 1424 times ]
Box and Whisker Plot A
Positive Skewed or Right Skewed: Length of Right Whisker is Larger than the length of Left Whisker, indicating that the graph is skewed towards right.
Box and Whisker Plot B
Zero Skewed or Normal Distribution: Length of Right Whisker is equal to the length of Left Whisker, indicating that the graph is not skewed.
Box and Whisker Plot C
Negative Skewed or Left Skewed: Length of Left Whisker is Larger than the length of Right Whisker, indicating that the graph is skewed towards left.
Attachment:
Skewness in Box and Whisker Plot Explained.jpg [ 34.58 KiB | Viewed 1590 times ]