box plot or whisker plot

Acharya Narendra Deva University of Agriculture and Technology,
Kumarganj, Ayodhya (U.P.) 224229
Submitted to :
Dr. Vishal Mehta
Assistant Professor
Department of Agricultural Statistics
Submitted by:
 Shubham Patel (ID NO. -: H-10325/18/22)
Course name – Stat 502 (3+1)
Course Title - Statistical Methods in Applied Science
Session – 2022-23
M.Sc. (Horti.) Vegetable Science 1st year
College of Agriculture

Content
Sr. no. Content name
1. Definition
2. How to read box plot
3. Exeption
3. Parts of box plot
4. Formula of quartiles
5. Box plot distribution
6. Uses of Box Plot
7. How to compare box plot
8. Examples for raw frequency data no -1
9. Examples for raw frequency data no -2
10. Example for discreate frequency data
11. Example for continuous frequency data

Box plot
• The method to summarize a set of data that is measured using
an interval scale is called a box plot.
or
• A box plot is a graphical way that summarizes the important
aspects of the distribution of numeric data.
• It is also referred to as a Box-and-Whisker Plot as it displays
the data in a box-and-whiskers format.

• Box plots can be drawn either vertically or
horizontally
Diagram of box plot

How to Read a Box Plot
• A boxplot is a way to show a five number summary in a chart.
• The main part of the chart (the “box”) shows where the middle
portion of the data is: the interquartile range.
• At the ends of the box, you” find the first quartile (the 25% mark) and
the third quartile (the 75% mark).
• The far left of the chart (at the end of the left “whisker”) is the
minimum (the smallest number in the set) and the far right is the
maximum (the largest number in the set).
• Finally, the median is represented by a vertical bar in the center of
the box.
• Box plots aren’t used that much in real life. However, they can be a
useful tool for getting a quick summary of data

Exception
• If your data set has outliers (values that are very high
or very low and fall far outside the other values of the
data set), the box and whiskers chart may not show
the minimum or maximum value. Instead, the ends of
the whiskers represent one and a half times
the interquartile range

Parts of Box Plots
1. Minimum: The minimum value in the given dataset
2. First Quartile (Q1): The first quartile is the median of the lower half of
the data set.
3. Median: The median is the middle value of the dataset, which divides
the given dataset into two equal parts. The median is considered as
the second quartile.
4. Third Quartile (Q3): The third quartile is the median of the upper half
of the data.
5. Maximum: The maximum value in the given dataset.

Cont…
Apart from these five terms, the other terms used in the box
plot are:
• Interquartile Range (IQR): The difference between the third quartile
and first quartile is known as the interquartile range. (i.e.) IQR = Q3-Q1
• Outlier: The data that falls on the far left or right side of the ordered
data is tested to be the outliers. Generally, the outliers fall more than
the specified distance from the first and third quartile.

Quartiles formula
For Raw & descreate data set :-
• Q1 =
𝑛+1
4
𝑡ℎ
item value
• Median or Q2 =
𝑛
2
𝑡ℎ
item value
• Q3 = 3 ×
𝑛+1
4
𝑡ℎ
item value
Where is ,
N = number of observations
L= lower limit of class interval
CF = cumulative frequency from upper one
F = frequency of selected class interval
H= class interval

Cont..
• For Continuous data set :-
Q1 = 𝐿 +
𝑛
4
−𝐶𝐹
𝐹
× ℎ
Q2= 𝐿 +
𝑛
2
−𝐶𝐹
𝐹
× ℎ
Q3 = 𝐿 +
3𝑛
4
−𝐶𝐹
𝐹
× ℎ
Where is ,
N = number of observations
L= lower limit of class interval
CF = cumulative frequency from upper one
F = frequency of selected class interval
H= class interval

Boxplot Distribution
• The box plot distribution will explain how tightly the data is grouped,
how the data is skewed, and also about the symmetry of data.
• Positively Skewed: If the distance from the median to the maximum
is greater than the distance from the median to the minimum, then the
box plot is positively skewed.

Cont..
• Negatively Skewed: If the distance from the median to
minimum is greater than the distance from the median to the
maximum, then the box plot is negatively skewed.
• Symmetric: The box plot is said to be symmetric if the
median is equidistant from the maximum and minimum
values

Box plot on a normal distribution

Uses of box Plot
• Box plots are widely used in statistics, process improvement,
scientific research, economics, and in social and human sciences.
• Mainly used to explore data as well as to present the data in an easy
and understandable manner.
• Box plots provide a visual summary of the data with
which we can quickly identify the average value of the
data, how dispersed the data is, whether the data is
skewed or not (skewness).
• The Median gives you the average value of the data.

• Box Plots shows Skewness of the data
• The dispersion or spread of data can be visualized by the
minimum and maximum values which are found at the end of
the whiskers.
• The Box plot gives us the idea of about the Outliers which are
the points which are numerically distant from the rest of the data
Cont…

How to compare box plots
• As we have discussed at the beginning of the article that box
plots make comparing characteristics of data between
categories very easy. Let us have a look at how we can
compare different box plots and derive statistical conclusions
from them.
• Let us take the below two plots as an example:-

• Compare the Medians — If the median line of a box plot lies
outside the box of the other box plot with which it is being
compared, then we can say that there is likely to be a
difference between the two groups. Here the Median line of
the plot B lies outside the box of Plot A.
• Compare the Dispersion or Spread of data — The Inter
Quartile range (length of the box) gives us an idea about how
dispersed the data is. Here Plot A has a longer length than
Plot B which means that the dispersion of data is more in plot
A as compared to plot B. The length of whiskers also gives an
idea of the overall spread of data. The extreme values
(minimum &maximum) gives the range of data distribution.
Larger the range more scattered the data. Here Plot A has a
larger range than Plot B.

• Comparing Outliers — The outliers gives the idea of
unusual data values which are distant from the rest of the
data. More number of Outliers means the prediction will be
more uncertain. We can be more confident while predicting
the values for a plot which has less or no outliers.
• Compare Skewness — Skewness gives us the direction
and the magnitude of the lack of symmetry. We have
discussed above how to identify skewness. Here Plot A is
Positive or Right Skewed and Plot B is Negative or Left
Skewed.
Cont…

Examples for raw data
Q.1- Draw the box plot from given data .
Given- 20, 28, 40, 12, 30, 15, 50
Solution- Firstly arrange the data in accending order.
12, 15, 20, 28, 30, 40, 50
• Number of observations (n) = 7
• Q1 =
𝑛+1
4
𝑡ℎ
item value
=
7+1
4
𝑡ℎ
=
8
4
𝑡ℎ
= 2nd item
Q1 = 15

• Q2 or mean =
𝑛
2
𝑡ℎ
item value
=
7+1
2
𝑡ℎ
=
8
2
th
= 4th item value
= Q2 = 28

• Q3 = 3 ×
𝑛+1
4
𝑡ℎ
item value
= 3 ×
7+1
4
𝑡ℎ
= 3 ×
8
4
= (3 × 2 )th
= 6th value
= Q3 = 40

 Q1= 15
 Q2= 28
 Q3 = 40
 Minimum value = 12
 Maximum value = 50

The box plot are normally distributed

Examples for raw data
Q.1- Draw the box plot from given data .
Given- 64, 25, 52, 32, 48, 29, 57, 21
Solution- Firstly arrange the data in accending order.
21, 25, 29, 32, 48, 52, 57, 64
• Number of observations (n) = 8
• Q1 =
𝑛+1
4
𝑡ℎ
item value
=
8+1
4
𝑡ℎ
=
9
4
𝑡ℎ
= 2.25th item

= 2nd item +
1
4
× (3rd item – 2nd item )
= 25+
1
4
× (29-25)
= 25+
1
4
× 4
25+
4
4
= 25+1
Q1 = 26

• Q2 or mean =
𝑛
2
𝑡ℎ
item value
=
8+1
2
𝑡ℎ
=
9
2
th
= 4.5th item value
= 4th item +
1
2
× (5th item – 4th item )
= 32+
1
2
× (48-32)
= 32+
1
2
× 16
= 32+
16
2
= 32+8
Q2 = 40

• Q3 = 3 ×
𝑛+1
4
𝑡ℎ
item value
= 3 ×
8+1
4
𝑡ℎ
= 3 ×
9
4
= (3 × 2.25 )th
= 6.75th value
= 6th item +
3
4
× (7th item – 6th item )
= 52 +
3
4
× (57-52)

= 52 +
3
4
× 5
= 52 +
15
4
= 52+ 3.75
Q3 = 55.75
 Q1= 26
 Q2= 40
 Q3 = 55.75

Q. 2- Draw the box plot for given discreate frequency data .
Given-
Solution-
• Number of observations (N) = 43
X 10 20 30 40 50 60
F 4 7 15 8 7 2
X F CF
10 4 4
20 7 11
30 15 26
40 8 34
50 7 41
60 2 43
N = 43
Example for discreate frequency data

• Q1=
𝑛+1
4
𝑡ℎ
item
=
43+1
4
𝑡ℎ
=
44
4
𝑡ℎ
= 11th item
now see the 11th item value in the table
Here we found the the Q1 = 20

• Q2 =
𝑛+1
2
𝑡ℎ
item
=
43+1
2
𝑡ℎ
=
44
2
𝑡ℎ
= 22th item

• Q3 = 3 ×
𝑛+1
4
𝑡ℎ
item
= 3 ×
43+1
4
𝑡ℎ
= 3 ×
44
4
𝑡ℎ
= 3 × 11
= 33th
 Q1= 20
 Q2= 30
 Q3 = 40

Example for continuous frequency data
Q.3 – Draw the box plot for given data .
Given
Solution =
Number of observations (N) = 67
C.I. 10-20 20-30 30-40 40-50 50-60 60-70 70-80
F 12 19 5 10 9 6 6
C.I. F CF
10-20 12 12
20-30 19 31
30-40 5 36
40-50 10 46
50-60 9 55
60-70 6 61
70-80 6 67
N= 67

• Q1 =
𝑛
4
𝑡ℎ
item
=
67
4
𝑡ℎ
= 16.75th item
Thus the Q1 lies in the class 20-30 ; & the corresponding value are –
L= 20,
𝒏
𝟒
= 16.75 , CF = 12 , F = 19 , h= 10
Q 1 = 𝐿 +
𝑛
4
−𝐶𝐹
𝐹
× ℎ
= 20+
16.75−12
12
× 10

=20 +
4.75
19
× 10
= 20 +
47.5
19
= 20+ 2.5
Q1= 22.5
• Q2 =
𝒏
𝟐
𝒕𝒉
item
=
67
2
𝑡ℎ
= 33.5th item

Thus Q2 lies in the class 30-40; & the corresponding value are
L= 30,
𝑛
2
= 33.5 , CF = 31 , F = 5 , h= 10
Q2= 𝑳 +
𝑛
2
−𝐶𝐹
𝐹
× 𝒉
= 30+
33.5−31
5
× 10
= 30+
2.5
5
× 10
= 30+ 2.5 × 2
= 30 + 5
Q2 = 35

• Q3 =
𝟑𝒏
𝟒
𝒕𝒉
item
=
3×67
4
𝑡ℎ
=
201
4
th
= 50.25th
Thus Q3 lies in the class 50-60; & the corresponding value are
L= 50,
3𝑛
4
= 50.25 , CF = 46 , F = 9 , h= 10
Q3 = 𝑳 +
𝟑𝒏
𝟒
−𝑪𝑭
𝑭
× 𝒉
= 50+
50.25−46
9
× 10

= 50+
4.25
9
× 10
= 50+
42.5
9
= 50+4.722
Q3= 54.722
 Q1= 22.5
 Q2= 35
 Q3 = 54.722

box plot or whisker plot

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