Univariate descriptive statistics

For univariate descriptive statistics, the following characteristics can be determined

• the mean

• the quadratic mean

• the geometric mean

• the harmonic mean

• the variance

• the empirical variance

• the moment of order alpha

• the standard deviation

• the empirical standard deviation

• the mode (and the modal class in the case of a continuous characteristic)

• the median (and the median class in the case of a continuous characteristic)

• the quartiles Q1 and Q3

• the absolute mean deviation

• the median absolute range

• the inter-quartile range

• the coefficient of variation

• the coefficient of skewness

• Fisher’s coefficient of skewness

• Yule’s coefficient of skewness

• Pearson’s coefficient of skewness

• Pearson’s coefficient of kurtosis

• Fisher’s kurtosis coefficient

Data entry is very simple as shown in the examples below.

Discrete Variables

Example 1: Statistical series

The following series represents the area (in \(m^2\)) of the nine apartments in a residence: 118 ; 70 ; 36 ; 84 ; 94 ; 144 ; 60 ; 48 ; 78

1. Determine the arithmetic mean and median of this distribution.

2. Calculate the following dispersion characteristics: mean absolute deviation from the mean and median, the standard deviation and the coefficient of variation.

Example 2: Statistical series in table form.

In a bookstore, 180 authors have been divided according to the number of textbooks they have written.

\(x_i\)	1	2	3	4	5	6	7
\(n_i\)	52	36	27	45	9	2	9

1. Determine the mode, median and quartiles \(Q_1\) and \(Q_3\).

2. Calculate the arithmetic mean, standard deviation and coefficient of variation of this series.

Continuous Variables

Example 3: Data grouped in classes of equal magnitude

The table below gives the distribution of the number of orders as a function of the amount of orders \(X\), for the last six months of GIE LIGGEEY.

\(X\)	\(1000 \leq X<1500\)	\(1500 \leq X<2000\)	\(2000 \leq X<2500\)	\(2500 \leq X<3000\)	\(3000 \leq X<3500\)	\(3500 \leq X<4000\)
Values	\(\ \ \ \ \ \ \ \ \ 4\ \ \ \ \ \ \ \ \)	\(\ \ \ \ \ \ \ \ \ 20\ \ \ \ \ \ \ \ \)	\(\ \ \ \ \ \ \ \ \ 24\ \ \ \ \ \ \ \ \)	\(\ \ \ \ \ \ \ \ \ 28\ \ \ \ \ \ \ \ \)	\(\ \ \ \ \ \ \ \ \ 22\ \ \ \ \ \ \ \ \)	\(\ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ \ \)

1. Determine the modal class, mode, median and quartiles \(Q_1\) and \(Q_3\).

2. Compute the centered moments of order \(2, 3\) and \(4\) of this distribution.

3. Calculate the Fisher skewness coefficient and the Pearson kurtosis coefficient.

Example 4: Data grouped into classes of unequal magnitude

The table below provides the percentage distribution of a municipality’s inhabitants according to the annual amount of their local taxes (in thousands of dollars)nof their local taxes (in thousands of francs).

Classes	[2 ; 4[	[4 ; 6[	[8 ; 9[	[9 ; 10[	[10 ; 12[	[12 ; 16[	[16 ; 20[	[20 ; 40[	[40 ; 60[	[60 ; 80[
Values	1	7	11	8	12	15	19	16	8	3

1. Determine the modal class, mode, median and quartiles \(Q_1\) and \(Q_3\).

2. Calculate the arithmetic mean of this series, the interquartile range, the variance and the coefficient of variation.