Probability and Statistics (MA-2011)
Chapter-1
1.1 Population, sample and processes
Population: Scientists and Engineers depend on a collection of facts or data. An
investigation is mainly focused on a well-defined collection of objects consisting a
population of interest. In fact, a large number of well-defined objects or data from which
the observation or investigation is to be done is called the population.
Sample: A small amount of objects or data is selected from the population that caries the
behavior of the population is called the sample. Sample is a subset of population.
For example, let a size of 10 arbitrary screws be selected from a lot of 1000 screws to
study the defectiveness, then population size is 1000 and the sample size is 10.
Variable: A variable is any characteristic whose value may change from one object to
another in the population. For example: for a company, the variables �, �, � are
� = the brand of the object
� = quality of the object;
� = number of defective objects etc.
Data can be observed into 3 types in the sense of variable such as
a. Univariate: When the observations are made on a single variable, the data set is called
An univariate data.
For example The sample of pulse rates (heart beats per minute) for the patients recently
admitted to an adult ICU is a numerical univariate data set: 85, 81,80, 102, 115, 95, 90,
105.
b. Bivariate: When the observations are made on each of two variables, the data set is
called bivariate.
For example The sample of saturation (the amount of oxygen bound to hemoglobin in the
blood measured in percentage of the maximum binding capacity) and pulse rates (heart
beats per minute) for the patients recently admitted to an adult ICU is a numerical
bivariate data set: (85, 92), (81,87), (80,85), (95,102), (94,115), (95,120), (90.95),
(98,102) .
c. Multivariate: When the observations are made on more than one variables, the data set
is called multivariate. Univariate, bivariate are particular cases of multivariate.
, For example: The sample of maximum blood pressure, saturation (the amount of oxygen
bound to hemoglobin in the blood measured in percentage of the maximum binding
capacity) and pulse rates (heart beats per minute) for the patients recently admitted to an
adult ICU is a numerical trivariate data set: (130, 85, 92), (115, 81,87), (140, 80,85), (102,
95,102), (148, 94,115), (205, 95,120), (145, 90.95), (125, 98,102) .
In general, if �1, �2 , �3 , ⋯ are different type of categories of the object, then multivariate
data set is of form
� = �1 × �2 × �3 × ⋯ = (�1 , �2 , ⋯: �� ∈ �� , � = 1,2, ⋯) .
1.2 Pictorial and tabular methods in Descriptive statistics
Given a data set consisting of � observations on some variable � , the individual
observations will be denoted by �1 , �2 , ⋯, �� . The subscript bears not relation to the
magnitude of a particular observation. Thus �1 will not in general be the smallest
observation in the set, not will �� typically be the largest. In many applications, �1 will
be the first observation gathered by the experimenter, �2 the second, and so on. The �th
observation in the data set will be denoted by �� .
a. Stem-and-leaf display
Consider a numerical data set �1 , �2 , ⋯, �� for which each �� consists at least two
digits. A quick way to obtain the informative visual representation of the data set is to
construct stem-and-leaf display as follows.
i. Select one or more leading digits for the stem values. The trailing digits becomes
the leaves.
ii. List possible stem values in a vertical column.
iii.Record the leaf for each observation beside the corresponding stem value.
iv.Indicate the units for the stem and leaves someplace in the display.
Example: For a random sample fo lengths of golf courses (yard) that have been
designated by Golf Megazine as among the most challenging in the United States.
Express the sample of 40 courses
6435, 6526, 7131, 6790, 6464, 6527, 6605, 6433, 6694, 6506, 6890. 6583, 6870, 6770,
6614, 6873, 6900, 6700, 6927, 7051, 6850, 7011, 6798, 7040, 7050, 7022, 6770, 6936,
6904, 7169, 7168, 6745, 7105, 7005, 7280, 6470, 6713, 7209, 7113, 7165
by stem-and-leaf display.
Solution: Among the sample of 40 courses, the shortest is 6433 yards long, and the
longest is 7280. Taking Thousand-Tundred digits for stem, and Tens-Ones digits for leaf,
we obtain the stem-and-leaf display with two-digit leaves shown in Figure-(a).
Chapter-1
1.1 Population, sample and processes
Population: Scientists and Engineers depend on a collection of facts or data. An
investigation is mainly focused on a well-defined collection of objects consisting a
population of interest. In fact, a large number of well-defined objects or data from which
the observation or investigation is to be done is called the population.
Sample: A small amount of objects or data is selected from the population that caries the
behavior of the population is called the sample. Sample is a subset of population.
For example, let a size of 10 arbitrary screws be selected from a lot of 1000 screws to
study the defectiveness, then population size is 1000 and the sample size is 10.
Variable: A variable is any characteristic whose value may change from one object to
another in the population. For example: for a company, the variables �, �, � are
� = the brand of the object
� = quality of the object;
� = number of defective objects etc.
Data can be observed into 3 types in the sense of variable such as
a. Univariate: When the observations are made on a single variable, the data set is called
An univariate data.
For example The sample of pulse rates (heart beats per minute) for the patients recently
admitted to an adult ICU is a numerical univariate data set: 85, 81,80, 102, 115, 95, 90,
105.
b. Bivariate: When the observations are made on each of two variables, the data set is
called bivariate.
For example The sample of saturation (the amount of oxygen bound to hemoglobin in the
blood measured in percentage of the maximum binding capacity) and pulse rates (heart
beats per minute) for the patients recently admitted to an adult ICU is a numerical
bivariate data set: (85, 92), (81,87), (80,85), (95,102), (94,115), (95,120), (90.95),
(98,102) .
c. Multivariate: When the observations are made on more than one variables, the data set
is called multivariate. Univariate, bivariate are particular cases of multivariate.
, For example: The sample of maximum blood pressure, saturation (the amount of oxygen
bound to hemoglobin in the blood measured in percentage of the maximum binding
capacity) and pulse rates (heart beats per minute) for the patients recently admitted to an
adult ICU is a numerical trivariate data set: (130, 85, 92), (115, 81,87), (140, 80,85), (102,
95,102), (148, 94,115), (205, 95,120), (145, 90.95), (125, 98,102) .
In general, if �1, �2 , �3 , ⋯ are different type of categories of the object, then multivariate
data set is of form
� = �1 × �2 × �3 × ⋯ = (�1 , �2 , ⋯: �� ∈ �� , � = 1,2, ⋯) .
1.2 Pictorial and tabular methods in Descriptive statistics
Given a data set consisting of � observations on some variable � , the individual
observations will be denoted by �1 , �2 , ⋯, �� . The subscript bears not relation to the
magnitude of a particular observation. Thus �1 will not in general be the smallest
observation in the set, not will �� typically be the largest. In many applications, �1 will
be the first observation gathered by the experimenter, �2 the second, and so on. The �th
observation in the data set will be denoted by �� .
a. Stem-and-leaf display
Consider a numerical data set �1 , �2 , ⋯, �� for which each �� consists at least two
digits. A quick way to obtain the informative visual representation of the data set is to
construct stem-and-leaf display as follows.
i. Select one or more leading digits for the stem values. The trailing digits becomes
the leaves.
ii. List possible stem values in a vertical column.
iii.Record the leaf for each observation beside the corresponding stem value.
iv.Indicate the units for the stem and leaves someplace in the display.
Example: For a random sample fo lengths of golf courses (yard) that have been
designated by Golf Megazine as among the most challenging in the United States.
Express the sample of 40 courses
6435, 6526, 7131, 6790, 6464, 6527, 6605, 6433, 6694, 6506, 6890. 6583, 6870, 6770,
6614, 6873, 6900, 6700, 6927, 7051, 6850, 7011, 6798, 7040, 7050, 7022, 6770, 6936,
6904, 7169, 7168, 6745, 7105, 7005, 7280, 6470, 6713, 7209, 7113, 7165
by stem-and-leaf display.
Solution: Among the sample of 40 courses, the shortest is 6433 yards long, and the
longest is 7280. Taking Thousand-Tundred digits for stem, and Tens-Ones digits for leaf,
we obtain the stem-and-leaf display with two-digit leaves shown in Figure-(a).
