MEANING OF THE DISPERSION AND ITS MEASURES IN SECONDARY EDUCATION

ICOTS-7, 2006: Ortega Moya and Estepa Castro
MEANING OF THE DISPERSION AND ITS MEASURES IN SECONDARY
EDUCATION

Juan Ortega Moya
UNED, Spain
Antonio Estepa Castro
University of Jaén, Spain
JOMOYA@telefonica.net

In this paper we present an onto-semiotic macroscopic analysis of the measures of dispersion:
range, interquartile range, average deviation, variance, standard deviation and coefficient of
variation by following the theoretical framework of the Theory of Semiotic Functions. This
research has been carried out with a sample of textbooks from the most representative publishers
used by Spanish second-cycle Secondary students of 15 and 16 years of age. The paper finishes
by presenting several useful conclusions for the planning of the teaching process and for the
research on the issue.

INTRODUCTION
During the first years of instruction, the teaching of statistics focuses on the topics of
location and variability. Although one admits that in the last decades of the XXth century
researchers on statistical education have paid great attention to measures of centralization (Moore,
1990; Shaughnessy, 1997), tendencies are changing and at the beginning of the XXIst century
variation is seen as the be all and end all of statistics (Shaughnessy and Ciancetta, 2001).
According to Moore (1990), “the ability to deal intelligently with variation and uncertainty is the
goal of instruction about data and chance.”
Convinced of the importance of variation, we endeavours to inquire into the nature of
measures of dispersion, to describe the obstacles that students come across when trying to
understand these measures and to look for adequate sequences of instruction in order to improve
both the capacity to teach and learn them.
In this study an epistemological analysis has been carried out of measures of dispersion.
We have based this analysis on a selection of textbooks of the second cycle of secondary
education (15 and 16 year-old students). By looking at different textbooks one is provided with a
variety of the components of the meaning of measures of dispersion from which one can conclude
that some of these present certain tendencies and deficiencies (Cobo and Batanero, 2004). They
also allow us to see how to guide and design the study of training processes.

THEORETICAL FRAMEWORK
This study comes within the Ontological-Semiotical Approach of mathematical
instruction and cognition (OSA), also known as Theory of Semiotical Functions (TSF), developed
by J. D. Godino and his collaborators, who have been developing it for more than ten years
(Godino, 2003). The OSA adopts anthropological, ecological and systemic theories about
mathematics; it considers mathematics as the outcome of human activity employed to resolve
problematic situations, which could be either external or internal to mathematics itself. As a result
of this activity mathematical objects emerge as entities that are used to resolve problematic
situations. In the OSA, the analysis of mathematical activity is carried out by introducing six
types of primary entities:

Situations: Problems which are more or less open and that induce one to mathematical
activity. An example would be to determine the maximum error of a set of measures.

Actions: What subjects carry out when they try to resolve problems and what become
routine actions with the practice (operations, algorithms, procedures). It is common
practice in statistics to calculate a measure of dispersion for a set of data.

Language: Ostensive elements of mathematical activity, words and expressions, symbols,
formulations, equations, graphics, tables, ...

Concepts: Definitions, notions and mathematical ideas.

Properties: Characteristics of mathematical objects.

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ICOTS-7, 2006: Ortega Moya and Estepa Castro

Arguments: Reasoning that relates the previous elements of meaning and justifies their
properties.

When a process of instruction is planned on a mathematical subject, one must begin by
specifying “what this topic means as far as mathematical and didactic institutions are concerned”
(Font, 2004). This means that the epistemical dimension of the object in question is drawn up. In
order to do this, one has to look at mathematical textbooks, curricular orientations, historical
sources and, in general, what “experts” consider to be the operative and discursive practices
inherent in the object. All this amounts to the institutional meaning of reference of the object. In
the OSA does not exists a only institutional meaning of reference. On the contrary, a variety of
meanings are possible depending on the institution which imposes these problems and where the
activities are being carried out to solve them.
At Secondary Education, curricular approaches, textbooks and the “knowledge of
teachers” describe functionality, an operating strategy and a representation of measures of
dispersion which characterise their meaning at this educational level and which may prove
different to those held by a university institution (Estepa and Ortega, 2006).

METHOD
The aim of present study is to characterise the institutional meaning of measures of
dispersion, Range (R), Interquartile Range (IR), Average Deviation (AD), Variance (V), Standard
Deviation (S) and Coefficient of Variation (CV), which are introduced in Spanish textbooks in the
3rd and the 4th years of Compulsory Secondary Education.
In order to carry out this analysis, a selection of textbooks from the most widely used
publishers in Spain have been chosen (see appendix). Specific issues on statistics have been
selected from each textbook and within these issues the sections which refer to measures of
dispersion have been analysed, as well as the activities proposed in examples and exercises. Every
section has been divided in units and sub-units, which were in turn assigned to the
aforementioned categories of primary entities (situations, actions, language, concepts, properties
and arguments).

RESULTS
Situations
S1. Variation by ranges: Finding the maximum difference in a set or subset of data.
S2. Variation by deviations: Measuring the variation around a centre.
S3. Global comparisons. Two kinds:
S3a. Comparing the variation of two or more distributions measured in the same magnitude.
S3b. Comparing the variation the two more distributions measured in different magnitude.
S4. External local comparisons: Comparing the relative position of data in different distributions
S6. Inverse problem: Generating data, identifing graphs, tables,...from information about its
variation.
(See Estepa and Ortega (2006) for the codification and other kinds of situations.)

Table 1: Frequency of the situations found and the publishers in which they appear


SITUATIONS
S1 S2 S3a
S3b S4 S6
Anaya
5 56 16 1 0 0
Santillana 24
47
22
0
0
0
Bruño
15
19 9 0 0 1
McGrawHill 11
19
11
0
0
0
USE
Edelvives
13
56 2 6 0 0
HO
Oxford 11
29
10
0
0
0
PUBLISHING
SM
9 17 5 0 0 2
Casals 10
42
10
0
1
2
Total situations
98
285
85
7
1
5
Total
Publishing
Houses 8 8 8 2 1 3

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ICOTS-7, 2006: Ortega Moya and Estepa Castro
We can deduce, observing Table 1, that the situations: S1, S2 and S3a are present at all
publishers and the rest almost are not used.

Actions
We have analyzed the following actions (techniques, algorithms, procedures, ...):
A1. Calculating measures of spread of a series of data (without tabulation).
A2. Calculating measures of spread of tabulated data (not grouped).
A3. Calculating measures of spread of tabulated data, grouped in intervals.
A4. Calculating measures of spread with calculator and/or computer.
A5. Calculating measures of spread of transformed data (brief calculation).
A6. Calculating measures of spread of data displayed graphically.
A8. Calculating measures of dispersion without the data (from information based on properties,
definitions, formulas, etc.).
A9. Making inverse calculations with dispersion measures.
A10. Calculating and interpreting the interval (x − σ
k , x + σ
k ) and the percentage of data that
it contains.
A11. Representing graphs that contain information on dispersion.
A12. Standardizing data.

Table 2: Frequency of actions analyzed and the publishers in which they appear

PUBLISHING
ACTIONS
HOUSE
A1 A2 A3 A4 A5 A6 A8 A9 A10
A11
A12
Anaya
30
16
17 6 0 4 9 0 3 0 0
Santillana
61
17 2 10 0 9 0 0 4 1 0
Bruño
29 0 7 2 0 0 0 0 0 2 0
McGrawH
11 9 14 7 0 0 0 0 0 0 0
Edelvives
81
23 9 1 0 2 0 0 0 0 0
Oxford
25
10 9 5 0 14 0 0 2 0 0
SM
17 3 7 1 0 0 0 2 4 0 0
Casals
18
16 4 18 1 6 0 2 9 0 1
TOTALS 272
94 69 50 1 35 9 4 22 3 1

Other kinds of actions are been found in Estepa and Ortega (2006).

Language
We have found a wide diversity of elements of expression, representation and
communication which conform the language. It can make the learning of the mathematical objects
related to variation difficult.

Table 3: Frequency of language items

PUBLISHING
LANGUAGE
HOUSE
Words/Expresions Symbols Formulas Graphs
Tables
Anaya
12
6 10 15 61
Santillana
11 3
3
11
61
Bruño
13
4 3 5 35
McGrawH
14
5 4 8 25
Edelvives
18
6 5 2 69
Oxford
15
10 18 5 35
SM
11
3 4 7 19
Casals 12 5
6
17
47

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ICOTS-7, 2006: Ortega Moya and Estepa Castro
Concepts. Definitions
Not all the definitions appear in all the books. We can see the frequency of the definitions
found in Table 4.

Table 4: Frequency of the definitions analyzed

CONCEPT DEFINITION
F
Dispersion
CD.R. Variation of the data around an average (Referential variation).
8
(CD)
CD.I. Variation between data (Intrinsic variation).
3
Range
CR.N. Range = max – min
8
(CR)
CR.T Length of interval where data are.
1
Interquartile
CIR.D Difference between the third quartile and the first quartile.
1
Range
(CIR)
CIR.R Range of the middle 50% of the data.
1
CAD.N. Average of the absolute deviations of the data with respect to the mean
Average
5
of that data set.
Deviation
CAD.T Average of the distances of the data with respect to the mean of that data
(CAD)
2
set.
CV.M. Average of the squared deviations of the observations from their mean.
8
CV.T Average of the squared distances of the observations from their mean.
1
Variance
CV.S2 The variance is the square of the standard deviation.
1
(CV)
2
2
CV.MM σ
= a − a
2
2
1
1/ 2
⎛⎛
⎞
⎞
2
CS.M Formula,
Standard
∑(x − x

3
i
) ⋅n /
⎜
⎜⎜⎜
i ⎟
⎟ N ⎟⎟
⎝⎝ i
⎠
⎠
Deviation
(CS)
CS.V2 The standard deviation is the square root of the variance.
8
2
CS.MM Formula,
a − a
1
2
1
CCV.C The coefficient of variation of Pearson is the quotient between the
Coefficient of
2
standard deviation and the mean.
Variation
(CCV)
CCV.100 CV = (σ / x 100
)·
or CV = (σ / x ) ⋅100
1

Properties

Three kinds of properties are considered: numerical, algebraic and statistical.
NP1. The ranges, the average deviation and the standard deviation are measured in the same units
as the data. The variance is measured in square units of the data. The coefficient of variation
is a-dimensional
NP2. All measure of variation is essentially nonnegative.
NP5. The standard deviation of n data is bounded both at the top and at the bottom.
AP4. If there is a linear transformation, the dispersion measures behave thus: R(axi + b) =
|a|·R(xi); IR(axi + b) = |a|·IR(xi) ; AM(axi + b) = |a|·AM(xi) ; V(axi + b) = a2·V(xi); S(axi + b)
= |a|·S(xi)
AP6. The formulas CV.MM and CS.MM are more used than CV.M and CS.M respectively.
SP1. The sum of the deviations with respect to the mean is zero.
2
2
2
SP2. König’s theorem (property of deviations): ∑(x − x = ∑
−
−
−
,∀ ∈
i
)
(x a
i
) n(x a) a R
SP3. All the observations are equal if and only if the variation is null.
SP4. A greater value of the dispersion measures means greater dispersion.
SP5. In the calculation of the average deviation, variance, standard deviation and coefficient of
variation take part all the values that the variable takes. On the contrary, in the calculation of
the ranges not all the data take part.
SP8. The ranges are measures of the dispersion based on the order of the data.
SP9. The ranges measure the dispersion without reference to averages. The standard deviation
and the variance measure the dispersion of the observations around the mean.

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ICOTS-7, 2006: Ortega Moya and Estepa Castro
SP10. The coefficient of variation is useful to compare the variation of variables measured in
different units or that take measured values in different magnitudes
R
SP15. For any distribution is true the inequality: AM ≤ S ≤
.
2
SP16. The ranges, deviations and CV only are calculable for cuantitive variables.

Table 5: Frequency of actions analyzed and the publishers in which they appear

PROPERTIES

NUMERICAL ALGEBRAICAL
STATISTICAL
1 2 5
4
6
1 2 3 4 5 8 9 10
15
16
Anaya
3 0 0
2
2
1 3 0 6 0 0 0 2 0 1
Santillana
2 3 0
1
0
7 0 0 12 0 0 0 0 0 3
Bruño
2 0 0
0
0
1 0 1 11 1 0 0 0 0 0
McGraw
2 0 0
0
0
0 0 0 14 1 0 1 0 0 0
Edelvives
1 0 0
0
0
1 0 0 8 0 2 1 0 0 0
Oxford
0 2 1
4
1
1 0 3 12 0 0 0 0 2 0
SM
1 1 0
2
0
0 1 0 3 0 0 0 0 0 0
Casals 2 0 0
2
0
3 0 0 12 0 0 0 0 0 7
TOTAL 13 6 1 11
3 14 4 4 78 2 2 2 2 2 11

Arguments
The previous elements of meaning are leagued and related by means of arguments and
reasonings in the textbooks analysed. The kinds of arguments that we have found, can be
classified in the following way:

Table 6: Frequency of arguments found

Anaya
Santillana
Bruño
McGraw
Edelvives
Oxford
SM
Casals
Total
Empirical 15
19
5
15
9
13 7 17 100
Deductive 2
2
5
1
1
2 2 7 22

DISCUSSION
In general, the following conclusions can be drawn:
• Only a few situations can be considered as true problems, looking for almost exclusively
the dominion of skills of calculation.
• The techniques needed for the resolution of problems are mostly developed in numerical
contexts. Graphs are rarely used to represent or interpret dispersion, and, in several
textbooks, the use of formulas is avoided.
• All textbooks use words and expressions whose meaning can be confusing for students.
For example, dispersion, difference, error, distance, homogeneity, heterogeneity,
representativeness, concentration, etc.
• Confusion is introduced, sometimes deliberately, between referential and intrinsic
dispersion, as well as between absolute and relative dispersion.
• There is no unanimity in the sequencing of contents dealing with dispersion. Some
publishers include dispersion in 2nd E.S.O. (13 year old), whereas others postpone it to 4th
E.S.O. (16 year old).
• Range, variance, and standard deviation are studied in all the analyzed textbooks. It’s not
the same with interquartile range or the coefficient of variation.
• Finally, we have to emphasize the overuse of empiric reasoning, generally based upon
examples chosen at random, impairing deductive ways of arguing.

Our results will be useful for the study, from the didactic point of view, that we will make
in Compulsory Secondary Education. It can also be useful to plan the teaching of this topic, since

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ICOTS-7, 2006: Ortega Moya and Estepa Castro
we have detected many important elements of meaning. Finally, it can be useful for designing and
planning of educative research in this topic.

ACKNOWLEDGEMENT
This research has been funded by “Programa de Promoción General del Conocimiento.”
Dirección General de Investigación. Ministerio de Ciencia y Tecnología (Spain), project
BSO2003-06331/PSCE.

REFERENCES
Cobo, B. and Batanero, C. (2004). Significado de la media en los libros de texto de secundaria.
Enseñanza de las Ciencias, 22(1), 5-18.
Estepa, A. and Ortega, J. (2006). The meaning of the statistics variation in university text books.
In A. Rossman and B. Chance (Eds.), Proceedings of the Seventh International Conference on
Teaching Statistics, Salvador, Brazil. Voorburg: The Netherlands: International Statistical
Institute.
Font, V. (2004). La dimensión dual “personal/institucional” y el problema del encaje de los
objetos personales del profesorado en la Teoría de las Funciones Semióticas. First
International Congress on Applications and Developments of the TFS. Jaén. Spain.
Godino, J. D. (2003). Teoría de las Funciones Semióticas. Un enfoque ontológico-semiótico de la
cognición e instrucción matemática. University of Granada. Spain.
http://www.ugr.es/~jgodino/indice_eos.htm
Moore, D. S. (1990). Uncertainty. In L. Steen (Ed.), On the Shoulders of Giants: New Approaches
to Numeracy. Washington, D.C.: National Academy Press.
Shaughnessy, J. M. (1997). Missed opportunities in research on the teaching and learning of data
and chance. In F. Biddulph and K. Carr (Eds.), People in mathematics education.
Proceedings of the Twentieth Annual Conference of the Mathematics Education Research
Group of Australasia Incorporated, 1, 6-22. Waikato, New Zealand: Mathematics Education
Research Group of Australasia.
Shaughnessy, J. M. and Ciancetta, M. (2001). Conflict between students’ personal theories and
actual data: The spectre of variation. Paper presented at the Second Roundtable Conference
on Research on Statistics Teaching and Learning. Armidale, New South Wales, Australia.

APPENDIX: TEXTBOOKS ANALYZED
[1] Colera, J., Gaztelu, I., Oliveira, M. J. and García, R. (2002). Matemáticas 3ª ESO. Anaya.
[2] Colera, J., Gaztelu, I., Oliveira, M. J. and Martínez, M. M. (2003)/ Matemáticas 4ª ESO-A.
Anaya.
[3] Colera, J., Gaztelu, I., Oliveira, M. J., García, R. and Martínez, M. M. (2003). Matemáticas 4ª
ESO-B. Anaya.
[4] Vizmanos, J. R. and Anzola, M. (1999). Algoritmo 2000 (Matemáticas 3ºESO, 4ª ESO-A & 4º
ESO-B). Sm.
[5] Sánchez, J. L. and Vera, J. (1998). Matemáticas 3ª ESO, 4º ESO-A & 4º ESO-B. Oxford Ed.
[6] Lazcano, I. and Sanz, J. F. (1998). Matemáticas 3º ESO & 4º ESO (Proyecto Adara).
Edelvives.
[7] Arias, J. M. and Pérez, S. A. (1995). Matemáticas 3º ESO. Casals.
[8] Arias, J. M., Carpintero, E. and Sanz, F. J. (1996). Matemáticas 4º ESO-A & Matemáticas 4º
ESO-B. Casals.
[9] Alvarez, M., Ovejero, M. J., Bartomeu, C., Guiteras, J. M. Jané, A. and Moreno, J. (1998).
Matemáticas 3ª ESO. McGraw-Hill.
[10] Amigo, C., Peña, P., Pérez, A., Rodríguez, A., Sivit, F., Asencio, M. J., and Vicente, E.
(1997). Matemáticas 4ª ESO-A & Matemáticas 4º ESO-B. McGraw-Hill.
[11] Almodóvar, J. A., García, P., Gil, J. Vázquez, C., Santos, D. and Nortes, A. (1999). Órbita
2000 (Matemáticas 3º ESO, 4º ESO-A & 4º ESO-B). Santillana.
[12] Miñano, A. and Ródenas, J. A. (1998). Matemáticas 3º ESO. Bruño.

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