PEOPLE’S DEMOCRATIC REPUBLIC of ALGERIA
MINISTRY of HIGHER EDUCATION and SCIENTIFIC RESEARCH
NATIONAL POLYTECHNIC INSTITUTE MALEK BENNABI of CONSTANTINE
COURSE HANDOUT
PROBABILITY and STATISTICS
Mohamed BOUKELOUA
Academic year 2023/2024Contents
Introduction 3
Part I: Descriptive Statistics 4
1 Statistical series with one character 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Types of characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Statistical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Discrete quantitative case . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Continuous quantitative case . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Qualitative case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Representation of a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Discrete quantitative case . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Continuous quantitative case . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Qualitative case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Parameters of a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Location parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 Dispersion parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Variance and standard deviation of a character . . . . . . . . . . . . . 19
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Statistical series with two characters 26
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Distributions and characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Marginal distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Marginal characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 28
Marginal mean and variance . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 Conditional distribution . . . . . . . . . . . . . . . . . . . . . . . . . 30
12.2.4 Conditional Characteristics . . . . . . . . . . . . . . . . . . . . . . . 31
Conditional mean of X given Y = yj . . . . . . . . . . . . . . . . . . 31
Conditional variance of X given Y = yj . . . . . . . . . . . . . . . . . 32
2.3 Covariance of two characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 Properties of the covariance . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 Correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 Fitting of type Y = aX + b . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.2 Fitting of type Y = BaX . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Part II: Probability 50
3 Introduction to probability calculus 51
3.1 Reminders on combinatorial analysis . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 k−permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.3 Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Probabilities of events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Study of the equiprobability . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.3 General definition of a probability . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Conditional probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.5 Law of total probabilities and chain rule . . . . . . . . . . . . . . . . 57
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Solutions to the exercises 64
4.1 Solutions to the exercises of chapter 1 . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Solutions to the exercises of chapter 2 . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Solutions to the exercises of chapter 3 . . . . . . . . . . . . . . . . . . . . . . 96
2Introduction
In the present course, we study fundamental notions in the descriptive statistics and the
probability theory, in accordance with the programs of mathematics of the preparatory
classes leading to the engineering institutes, the common cores of sciences and technology,
mathematics and computer science and the sciences of matter. The course is divided into two
parts. In the first part, we focus on the descriptive statistics. We start, in the first chapter,
by the statistical series with one character. After giving the definitions of the three types
of characters, namely, the qualitative character, the discrete quantitative character and the
continuous quantitative one, we explain how to differentiate between them. Then, we study
each type in detail. More precisely, we present, in each case, the numeric and the graphic
methods of representation of the statistical series. For the quantitative characters, we also
discuss the central tendency and the dispersion parameters such as the arithmetic mean, the
quantiles, the mode, the variance and the standard deviation. Each concept is illustrated
by some practical examples. In the second chapter, we move on into statistical series with
two characters. First, we study the marginal distributions and their characteristics. Then,
we focus on conditional distributions, conditional means and conditional variances. We also
define the covariance and the correlation coefficient of two characters and we present their
properties. Finally, we consider the different types of fittings such as the linear fitting, the
exponential fitting and the power fitting. As in the first chapter, we give many examples of
application to illustrate the treated notions. The second part of the course is devoted to an
introduction to the probability calculus. We start by a remainder on combinatorial analysis,
where we recall the notions of k−permutations, permutations and combinations. Then, we
define the notion of probability and we present its properties. We also study the conditional
probabilities and we give some fundamental formulas such as the Bayes formula, the law of
total probability and the chain rule. Each chapter in this course is followed by some exercises
of application, that we advise to treat before checking their solutions which are given at the
end of the course.
3Descriptive Statistics
Part I
Descriptive Statistics
4Chapter 1
Statistical series with one character
1.1 Introduction
1.1.1 Generalities
Descriptive statistics is a collection of methods used to describe, summarize, interpret and
analyse datasets which can be found in a given study. It helps analysts to better understand
the data and to draw conclusions from them. The datasets may be treated using tables,
graphs and numerical characteristics such as the mean, the variance, the quantiles, etc. The
statistical analysis may be univariate or multivariate. Univariate analysis focuses on one
character of the data. The main aspects of interest in this framework are the distribution,
the central tendency and the dispersion. Furthermore, multivariate analysis focuses on the
relationship between two or more characters. The main aspects in this framework are the
covariance, the coefficient of correlation and the conditional distributions. An other important topic in descriptive statistics is the regression. This notion deals with the possibility to
establish an equation that links two (or more) variables. Such an equation may be linear,
exponential, polynomial or may have other forms.
1.1.2 Definitions
We will start with some basic definitions of descriptive statistics.
Population
The population is a set of similar items on which the statistical study is based. The number
of elements within a population is called the size of the population.
Sample
A sample is a subset of the population having the same characteristics as it. Samples are
used when the population sizes are too large so as it becomes impossible to include all possible observations. A sample should represent the population as a whole and not reflect any
bias toward a specific attribute.
5Statistical unit
Each element in the population is called a statistical unit or an individual.
Statistical character
The character is a particular feature of the observations, in which the statistical study is
interested.
Modalities of a character
The modalities of a character are the different situations taken by this character.
We will illustrate theses definitions by some examples.
Example 1:
The study of the blood group of 150 students in a university.
In this situation:
- The population is comprised of the 150 students of the university. Each student is a
statistical unit.
- The character is the blood group.
- The modalities of this character are A, B, AB and O.
Example 2:
The study of the number of children in 60 families of a city.
In this situation:
- The population is comprised of the 60 families of the city. Each family is a statistical unit.
- The character is the number of children.
- The modalities of this character are for example: 0, 1, 2, 3, 4 and 5.
Example 3:
The study of the size of 200 students in a university.
For this example:
- The population consists of the 200 students of the university. Each student is a statistical
unit.
- The character is the size.
- The modalities of this character may be any values between 1.50 and 1.90 m. Instead of
using all the 200 sizes of the students, it is preferable to group them into classes such as
[1.50, 1.60[, [1.60, 1.65[, [1.65, 1.75[, [1.75, 1.85[ and [1.85, 1.90[.
1.1.3 Types of characters
There are two types of characters: qualitative characters and quantitative characters.
Qualitative character
They are measures of "types" and may be represented by names or symbols. They are related to categorical variables. The modalities of a qualitative character are words or symbols.
Qualitative characters may also be represented by number codes.
6Quantitative character
They are measures of values or counts and are expressed as numbers. They are related to
numeric variables. Quantitative characters may be discrete or continuous.
- Discrete quantitative character (or discrete statistical variable): It is a variable that takes
on distinct and countable values. The set of values of such a variable is finite or countable
(at most countable). The modalities are distinct numbers.
- Continuous quantitative character (or continuous statistical variable): It is a variable that
takes on an infinite number of possible values within a given range. The set of values of such
a variable is infinite and uncountable. The modalities are intervals called "Class intervals".
For the previous examples, we have
- In example 1: The character (blood type) is qualitative.
- In example 2: The character (number of children) is discrete quantitative.
- In example 3: The character (size) is continuous quantitative.
1.2 Statistical series
Consider a statistical population including n individuals. Assume that we are interested in
a statistical character related to this population, with k modalities M1, M2, . . . , Mk.
Definition 1. (Absolute frequency)
The absolute frequency of the modality Mi (1 ≤ i ≤ k) is the number of individuals corresponding to this modality. It is noted ni.
Definition 2. (Relative frequency)
The relative frequency of the modality Mi (1 ≤ i ≤ k) is the proportion of individuals
corresponding to this modality and it is noted fi. So, we have
fi = ni
n
, ∀ i ∈ {1, . . . , k}.
Remark 1.
We have
• Pk i=1 ni = n.
• ∀ i ∈ {1, . . . , k}, 0 ≤ fi ≤ 1.
• Pk i=1 fi = 1.
Definition 3. (Statistical series)
The set {(M1, n1), (M2, n2), . . . , (Mk, nk)} is called a statistical series. It is generally represented by a statistical table. When the character is quantitative, its values (modalities) are
sorted in ascending order.
In the sequel, we will study in detail the different types of characters, using some examples.
71.2.1 Discrete quantitative case
Example 2 (continued):
The study of the number of children in 60 families of a city gave the following results.
Number of children Number of families
0 5
1 10
2 11
3 18
4 11
5 5
This statistical series can be represented by the following statistical table.
xi ni fi
0 5 0.083
1 10 0.167
2 11 0.183
3 18 0.3
4 11 0.183
5 5 0.083
Total 60 1
The (xi)1≤i≤6 are the values of the studied statistical variable X (the number of children).
fi = ni
n
=
ni
60, ∀i ∈ {1, 2, . . . , 6}.
1.2.2 Continuous quantitative case
Example 3 (continued):
The study of the size (in m) of 200 students in a university gave the following results.
Class intervals Number of students
[1.50, 1.60[ 20
[1.60, 1.65[ 45
[1.65, 1.75[ 85
[1.75, 1.85[ 40
[1.85, 1.90[ 10
In this continuous quantitative case, the statistical table has the following form.
8ei
1.50
ni fi
160
1.60
20 0.1
165
1.65
45 0.225
175
1.75
85 0.425
185
1.85
40 0.2
190
1.90
10 0.05
Total 200 1
The (ei)0≤i≤5 are the limits of the class intervals.
For all i ∈ {1, 2, . . . , 5}, fi = ni
n
=
ni
200
.
1.2.3 Qualitative case
Example 1 (continued):
The study of the blood group of 150 students in a university gave the following results.
Blood group Number of students
A 45
B 25
AB 9
O 71
This statistical series can be represented by the following statistical table.
Modalities ni fi
A 45 0.3
B 25 0.167
AB 9 0.06
O 71 0.473
Total 150 1
fi = ni
n
=
ni
150, ∀i ∈ {1, 2, 3, 4}.
1.3 Representation of a series
In this section, we will study some graphical representations of a statistical series for the
different types of characters.
91.3.1 Discrete quantitative case
Let X be a discrete statistical variable taking the values {x1, x2, . . . , xk}, with x1 < x2 <
· · · < xk. For all i ∈ {1, 2, . . . , k}, we denote by ni (resp. fi) the absolute (resp. the relative)
frequency of xi. In this case, the statistical series can be represented by two types of graphics:
The differential diagram and the integral diagram.
Differential diagram (Line graph)
The line graph consists of vertical lines representing the values of X. The height of the line
corresponding to xi is determined by either ni or fi.
Integral diagram (Cumulative frequency curve)
To define this graphic, we need first to define the cumulative absolute frequencies and the
cumulative relative frequencies.
Definition 4. (Cumulative absolute frequency)
For all i ∈ {1, 2, . . . , k}, the cumulative absolute frequency of the ith value xi of X is defined
by
Ni =
iXj
=1
nj.
Definition 5. (Cumulative relative frequency)
For all i ∈ {1, 2, . . . , k}, the cumulative relative frequency of the ith value xi of X is defined
by
Fi =
iXj
=1
fj = Ni
n
.
Definition 6. (The empirical cumulative distribution function)
The empirical cumulative distribution function (ECDF) of X is the function F : R −→ [0, 1]
defined for all x ∈ R by
F (x) =
0 if x < x1
Fi if x ∈ [xi, xi+1[, for 1 ≤ i ≤ k − 1
1 if x ≥ xk.
The cumulative frequency curve is the graph of the ECDF.
Remark 2.
The ECDF F satisfies the following properties:
• ∀ x ∈ R, 0 ≤ F (x) ≤ 1.
• F is a non-decreasing right continuous function.
• lim
x→−∞ F (x) = 0 and limx→+∞ F (x) = 1.
10Now, we will represent our statistical series of Example 2 (Number of children) using the
above graphics.
Example 2 (continued):
The line graph (using the relative frequencies) of this statistical series is as follows.
- To draw the cumulative frequency curve of this series, we need to calculate the cumulative relative frequencies (Fi)1≤i≤6.
xi ni fi Ni
0
Fi 0
0 5 0.083 5
5
0.083
0.083
1 10 0.167 15
15
0.25
0.25
2 11 0.183 26
26
0.433
0.433
3 18 0.3 44
44
0.733
0.733
4 11 0.183 55
55
0.916
0.916
5 5 0.083 60
60
1 1
Total 60 1
So, the cumulative frequency curve is as follows.
111.3.2 Continuous quantitative case
Let X be a continuous statistical variable. We assume that the class intervals of X are
[e0, e1[, [e1, e2[, . . . , [ek−1, ek[. For all i ∈ {1, 2, . . . , k}, we denote by ni (resp. fi) the
absolute (resp. the relative) frequency of the class [ei−1, ei[. In this case, the statistical
series can be represented by two types of graphics: The differential diagram and the integral
diagram.
Differential diagram (Histogram)
A histogram consists of bars that correspond to the class intervals. For all i ∈ {1, 2, . . . , k},
the height of the ith bar is hi = fi
di , where di = ei − ei−1 denotes the magnitude of the ith
class [ei−1, ei[. An important consideration for this concept is that the area of each bar is
proportional to the corresponding relative frequency.
Remark 3.
If all the classes have the same magnitude, we can take hi = fi for all i ∈ {1, 2, . . . , k}.
Integral diagram (Cumulative frequency curve)
The cumulative absolute and relative frequencies can be defined in the same way as in the
discrete case. The cumulative frequency curve is the graph of the ECDF, which is defined
in the continuous case as follows.
Definition 7.
The empirical cumulative distribution function (ECDF) of the continuous variable X is the
12function F : R −→ [0, 1] defined for all x ∈ R by
F(x) =
0 if x < e0
Fi−1 + fi
ei − ei−1
(x − ei−1) if x ∈ [ei−1, ei[, for 1 ≤ i ≤ k − 1
1 if x ≥ ek,
with F0 = 0.
F is a piecewise linear function and it satisfies the same properties of the ECDF in the
discrete case except the fact that it is continuous on R (and not only right continuous).
Now, we will represent our statistical series of Example 3 (Size of students) using the above
graphics.
Example 3 (continued):
To draw the graphical representations of this statistical series, we need to calculate the
magnitudes (di)1≤i≤5 and the cumulative relative frequencies (Fi)1≤i≤5.
ei
1.50
ni fi Ni
0
Fi 0
di fi
di
160
1.60
20 0.1 20
20
0.1
0.1
0.1 1
165
1.65
45 0.225 65
65
0.325
0.325
0.05 4.5
175
1.75
85 0.425 150
150
0.75
0.75
0.1 4.25
185
1.85
40 0.2 190
190
0.95
0.95
0.1 2
190
1.90
10 0.05 200
200
1 1
0.05 1
Total 200 1
The histogram of our statistical series is as follows.
13We can also add the frequency polygon by joining the midpoints of the tops of the rectangles.
We plot also the previous and next points on the x−axis to start and end the polygon. These
two points correspond respectively to e0 − d1
2
and ek + dk
2
.
Moreover, the cumulative frequency curve is as follows.
141.3.3 Qualitative case
In this case, a statistical series can be represented by two types of graphics: The bar chart
and the pie chart.
Bar chart
This graphic consists of bars representing the modalities of the character. The height of each
bar is determined by either the absolute frequency or the relative frequency of the respective
modality.
Pie chart
A pie chart is a circle partitioned into segments, where each of the segments represents a
modality. The size of each segment depends upon the relative frequency and is determined
by the angle θi = fi × 360◦.
We will represent our statistical series of Example 1 (Blood group) using these graphics.
Example 1 (continued):
The bar chart (using the relative frequencies) of this statistical series is as follows.
15- To draw the pie chart of this series, we need to calculate the angle θi = fi × 360◦ for all
i ∈ {1, 2, 3, 4}.
Modalities ni fi θi
A 45 0.3 108◦
B 25 0.167 60.12◦
AB 9 0.06 21.6◦
O 71 0.473 170.28◦
Total 150 1 360◦
So, the pie chart is as follows.
161.4 Parameters of a series
In this section, we will study some parameters that measure the central tendency and the
dispersion of a statistical series with a quantitative character. We will deal with the discrete
and the continuous cases separately.
1.4.1 Location parameters
Let {(x1, n1), (x2, n2), . . . , (xk, nk)} (with x1 < x2 < · · · < xk) be a statistical series corresponding to a discrete statistical variable X.
Mode
The mode of X, denoted by M, is the value(s) having the largest absolute frequency. The
mode may not be unique.
Arithmetic mean
The arithmetic mean of X is defined by
X = 1
n
kXi
=1
nixi =
kXi
=1
fixi.
Remark 4.
If we use a transformation Y = aX + b with a, b ∈ R, then Y = aX + b.
Indeed, we have for all i ∈ {1, . . . , k} yi = axi + b, then
Y = 1
n
kXi
=1
niyi
=
1 n
kXi
=1
ni(axi + b)
=
1 n
kXi
=1
(anixi + bni)
=
1 n
kXi
=1
anixi +
1 n
kXi
=1
bni
= a
1 n
kXi
=1
nixi! + b n1 Xi=1 k ni!
= aX + b × n
n
= aX + b.
17Median
Let p ∈ [0, 1], the quantile of order p (or pth quantile) of X is the value xp of X which divides
the dataset in two parts such that p−proportion of the data are less than or equal to xp and
(1 − p)−proportion of the data are greater than xp. In other words
x
p = inf {x ∈ R/F (x) ≥ p},
where F is the ECDF of X.
-Particular cases:
For p = 0.5, x0.5 is called the median of X, denoted by Med.
For p = 0.25, x0.25 is called the first quartile of X, denoted by Q1.
For p = 0.75, x0.75 is called the third quartile of X, denoted by Q3.
These parameters can be defined in the same way in the case of a continuous statistical
variable.
Let {([e0, e1[, n1), ([e1, e2[, n2), . . . , ([ek−1, ek[, nk)} be a statistical series corresponding to a
continuous statistical variable X.
Arithmetic mean
The arithmetic mean of X is defined by
X = 1
n
kXi
=1
nici =
kXi
=1
fici,
where ci = ei−1 + ei
2
is the centre of the ith class [ei−1, ei[.
Modal class
The modal class of X, denoted by M, is the class(es) that correspond(s) to the largest ni/di
(or fi/di). It may not be unique.
Median
The quantiles are defined in the same way as in the discrete case. To calculate them, we
use the method of linear interpolation. For example, to calculate the median, we determine
i such that Fi−1 ≤ 0.5 < Fi which means that Med ∈ [ei−1, ei[, then we apply the formula
Med − ei−1
ei − ei−1
=
0.5 − F (ei−1)
F (ei) − F (ei−1) =⇒ Med = ei−1 + (ei − ei−1) × F0(.e5i)−−FF(e(ie−i−1)1).
For any p ∈ [0, 1], we apply the same method to calculate the quantile xp, using the appropriate proportion p.
1.4.2 Dispersion parameters
Dispersion parameters are statistical parameters that describe the dispersion of the observations around any particular value.
18Variance and standard deviation of a character
The variance of X is defined by
V ar(X) = 1
n
kXi
=1
nixi − X2 =
kXi
=1
fixi − X2 .
It measures of how far the dataset is spread out from their average value.
The variance is always non-negative and the standard deviation of X is defined by
σX = pV ar(X).
The standard deviation has the same unit of measurement as the data whereas the unit of
the variance is the square of the units of the observations.
Remark 5.
i) We have
V ar(X) = 1
n
kXi
=1
nix2 i − X2 =
kXi
=1
fix2 i − X2 .
Indeed
V ar(X) = 1
n
kXi
=1
nixi − X2
=
1 n
kXi
=1
nix2 i − 2xiX + X2
=
1 n
kXi
=1
nix2 i −
kXi
=1
2nixiX +
kXi
=1
niX2!
=
1 n
kXi
=1
nix2 i − 2X
n
kXi
=1
nixi +
X2
n
kXi
=1
ni
=
1 n
kXi
=1
nix2 i − 2X2 + X2
n
× n
=
1 n
kXi
=1
nix2 i − X2 .
ii) If Y = aX + b with a, b ∈ R, then V ar(Y ) = a2V ar(X) and σY = |a|σX.
19Indeed
V ar(Y ) = 1
n
kXi
=1
niyi − Y 2
=
1 n
kXi
=1
niaxi + b − aX − b2
=
1 n
kXi
=1
nia2xi − X2
=
a2
n
kXi
=1
nixi − X2
= a2V ar(X)
and
σY = pV ar(Y ) = pa2V ar(X) = |a|σX.
Range
The range of X is defined as the difference between the maximum and minimum value of X.
R = max
1≤i≤k
xi − min
1≤i≤k
xi.
Interquartile range
The interquartile range of X is defined as the difference between the first and the third
quartiles of X.
IQ = Q3 − Q1.
Now, we will calculate the central tendency and the dispersion parameters of the statistical
series of Example 2 (Number of children).
These parameters can be defined in the same way in the continuous quantitative case.
Let {([e0, e1[, n1), ([e1, e2[, n2), . . . , ([ek−1, ek[, nk)} be a statistical series corresponding to a
continuous statistical variable X.
Variance and standard deviation
The variance of X is defined by
V ar(X) = 1
n
kXi
=1
nici − X2 = 1
n
kXi
=1
nic2 i − X2
and the standard deviation of X is defined by
σX = pV ar(X).
20Range
The range of X is defined by
R = ek − e0.
Interquartile range
The interquartile range of X is defined by
IQ = Q3 − Q1.
Now, we will calculate the central tendency and the dispersion parameters of the statistical
series of Example 2 (Number of children).
Example 2 (continued):
To calculate the parameters of this statistical series, we need to add the following columns
in the statistical table.
xi ni fi Ni
0
Fi0
nixi nix2 i
0 5 0.083 5
5
0.083
0.083
0 0
1 10 0.167 15
15
0.25
0.25
10 10
2 11 0.183 26
26
0.433
0.433
22 44
3 18 0.3 44
44
0.733
0.733
54 162
4 11 0.183 55
55
0.916
0.916
44 176
5 5 0.083 60
60
1 1
25 125
Total 60 1 155 517
Total/n 2.583 8.617
- The central tendency parameters are:
• The arithmetic mean
X = 1
n
6Xi
=1
nixi =
155
60
= 2.583
• The mode M = 3 because the largest ni is 18.
• The median:
We have 0.433 < 0.5 < 0.733, then Med = 3.
• The quartiles:
We remark that 0.25 is on the line between the values 1 and 2, so we take the smallest value
Q1 = 1.
We have 0.733 < 0.75 < 0.916, then Q3 = 4.
- Furthermore, the dispersion parameters are:
21