Análisis datos: momentos, tendencia central y dispersión - Prof. Salafranca, Apuntes de Psicología

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UNIT 5

PART 5.3:

QUANTITATIVE VARIABLES

(INTERVAL AND RATIO SCALES)

Recommended readings

• (^) Solanas, A., Salafranca, Ll., Fauquet, J. y Núñez, M. I. (2005).

Estadística descriptiva en ciencias del comportamiento. Madrid:

Thomson.

 Where: p1 4/2 EST

 What: Capítulo 8

• (^) Guàrdia, J., Freixa, M., Peró, M. y Turbany, J. (2008). Análisis de

datos en psicología. 2ª edición. Madrid: DELTA Publicaciones.

 Where: p1 4/2 ANA

 What: Capítulo 3

• (^) Peró, M., Leiva, D., Guàrdia, J. y Solanas, A. (Eds.) (2012).

Estadística aplicada a las ciencias sociales mediante R y R-

Commander. Madrid: Garceta.

• (^) Useful for describing variables measured in an

interval or ratio scale in terms of location, central

tendency, scatter, shape and presence of outliers.

• (^) Such data can also be analyzed using indices and

plots for lower scales.

• (^) These indices use all data and are, thus sensitive

to outliers (i.e., not resistant).

• (^) Recommended for symmetric distributions and

lack of outliers.

Indices based on moments

Arithmetic mean The center of gravity or the balance between all values. Parameter Statistic

Central tendency measures



   

i N i n

i i

i i

x x x N n

Generalized means

a) Quadratic mean

b) Harmonic mean

Useful for distributions which

have been transformed using the

inverse for making them

symmetric.

c) Geometric mean

Central tendency measures

2 1 1 i n i i Q x n     1 1 1 i n i (^) i x H (^) n    

log log

i n

i

i

x G n



Resistant central tendency measures

Mode (Mo) – Most frequent value. Mind continuous variables and multimodal

distributions!

Median (Md) – Value that divides the distribution in to two halves of equal size

Trimean:

Quartiles mean:

Interquartile mean:

Trimmed mean and winsorized mean.

Q 1 2 · Q 2 Q 3

Trimean

2 1 3 Q Q Q   i i i

n

x x

Mid

( p 251 ) ( p 7 5)



Central tendency measures: Example

Comparison between the two villages:

indices based on moments and position are

both applicable.

• (^) Scatter: dispersion, heterogeneity of the values, taking as a reference an indicator of central tendency (e.g., arithmetic mean).
• (^) Assessment is done considering the distance with respect to the central value. Distances are defined as differences.
• (^) Measures of scatter are absolute (with measurement units; variance, standard deviation) and relative (coefficient of variation). Measures of scatter

Standard deviation: same measurement unit as

the variable:

Measures of scatter: Absolute             2 2 1 1 ( ) ( ) ; i N i n i i i i n x x x s N n

Coefficient of variation:  (^) Quotient: SD / absolute value of the mean.  (^) Enables comparisons between variables in different measurement units or the same one, but with markedly different values. Measures of scatter: Relative    ^ ^      ; 100 s s CV CV X X

Measures of scatter

Comparison between the two villages:

indices based on moments and position are

both applicable.

 (^) The deviation with respect to a theoretical symmetric model is assessed.  (^) The symmetry axis is defined by the mean.

Pearson’s third coefficient :

Measures of shape: skewness    

/ /

i n

i

i

i n

i

i

x X n x X n 

                                



Other ways to assess skewness *

Positively skewed

Symmetric

Negatively skewed

*Not always!!!!

Measures of shape: skewness Mode  Median  Mean Mode  Median  Mean Mode  Median  Mean

Measures of shape: skewness

Negative asymmetry Symmetry Positive asymmetry

Si T 1 > 0 Positive asymmetry

Si T 1 = 0 Symmetry

Si T 1 < 0 Negative asymmetry

 

/ S x x n S T

i n

i

 i

   