1. Chapter 4
Fundamental statistical characteristics II: Dispersion and form measurements

2. Fundamental statistical characteristics
Group indexes
Central tendency
Variability (Dispersion)
Bias (Asymmetry)
Skewness (Kurtosis)
Individual indexes
Position
Centiles (Ci)
Percentiles (Pi)
Quartiles (Qi)
Raw scores (Xi)
Differentials scores (xi)
Standard scores (Zi)

3. How is the data arranged with respect to the distribution center?
How far or together are the data from each other?
Variability or dispersion indexes At S2
C.V Q S

4. How are the data arranged with respect to the rest?
Are data piled at one end?
Bias or Asymmetry indexes g1

5. Which form is the distribution?
Is it flattened or sharp?
Skewness or Kurtosis indexes g2

6. Variability or dispersion indexes

7. Example A 4 10 12 14 20 B 11 13 C
104
110
112
114
120

8. 4 10 12 14 20 A 10 11 12 14 14 B
104
110
112
114
120 C

9. Variability quantifiers5
V = 0 5

10. Variability quantifiers1 2 5 8 9 10 1 2 5 8 9 10

11. SEMIINTERQUARTILE AMPLITUDEAt
TOTAL AMPLITUDE Q
SEMIINTERQUARTILE AMPLITUDE
INDEXES S2
VARIANCE S
STANDARD DEVIATION

12. a) The total amplitude (or Range)
It is the distance between the maximum and minimum value of a data set.
Advantage: easy to calculate.
Disadvantages:
It only uses two sample data, so it is very sensitive to extreme values and not to the average values. It is not stable.
It is not independent of the sample sizes (AT obtained in samples of different sizes are not directly comparable).
AT = XMáx – XMin

13. Total amplitude calculationB)
AT = 13 – 7 = 6 C)
B = C
A > B y C

14. A) B) C)

15. b) The semiinterquartile amplitude
It is the semidistance between quartile 3 and quartile 1
It is usually calculated:
When we only want to consider the central scores of the distribution.
When we can’t use the mean.

16. Calculation example Data analysis:
A) Unbalance to the student of Psychology.
B) It is an essential tool for Psychology.

17. Unbalance to the student
Essential tool A B XA fA FA XB fB FB 1 35 5 2 40 75 20 25 3
150 65 4 30
180 80
145
200 55
CALCULATE Q

18. A: Data analysis unbalance to the student of Psychology
Centile
Position
Cum. Frec.
Value 25
50,25
Between 50 and 51
2+0,25(2-2) = 2
Centile
Position
Cum. Frec.
Value 75
150,75
Between 150 and 151
3+0,75(4-3) = 3,75

19. B: Data analysis is an essential tool for Psychology
Centile
Position
Cum.Frec.
Value 25
50,25
Between 50 and 51
2+0,25(2-2) = 3
Centile
Position
Cum.Frec.
Value 75
150,75
Between 150 and 151
5+0,75(5-5) = 5

20. Calculate the semiinterquartile deviation (Q) in the following distributionXi
fri Fi
%ai 21 23 40 24
0,12 32 25
110 26
0,20 29 30
100

21. c) The variance and d) the standard deviation
To what point are people in relation to the representative person of the population?
We are interested in what are the approximate average distance between every subject and the representative person 21

22. Same mass above and below the average: to solve this problem we square.
Because of the discrepancies sum, as more data will mean more differences: to avoid this influence of n, divided by n.

23. Other ways of calculating the variance derived from this formula

24. Type I distribution Small data set
3 – 6 – 9 – 12 – 15 Xi 3
3 – 9 = -6 36 6
6 – 9 = -3 9
9 – 9 = 0 12
12 – 9 = 3 15
15 – 9 = 6
Total 90

25. Variance is expressed in units squared, and this does not usually use.
2 onions are ok, but 2 onions squared have no sense!
In order to the dispersion is also expressed in the same units as the variable in its origin we do the square root (obtaining the standard deviation).

26. Variance
Standard desviation

27. With another derived formulaXi X2 3 9 6 36 81 12
144 15
225
Total
495

28. Type II distribution Big data setXi fi
fiXi 3 1
3-5 = -2 4 7 28
4-5 = -1 5 6 30
5-5 = 0 18
6-5 = 1 21
7-5 = 2 12
Total 20
100 10 26

29. Type II distribution Big data setXi fi
fiXi 3 1
3-5 = -2 4 7 28
4-5 = -1 5 6 30
5-5 = 0 18
6-5 = 1 21
7-5 = 2 12
Total 20
100 10 26

30. Xi fi
fi Xi
fiX2 3 1 9 4 7 28
112 5 6 30
150 18
108 21
147
Total 20
100
491

31. The variance and the standard deviation
It is the mean squared differences (squared high) respect to the mean:
If the data are grouped by frequency (big data set):
Provide an absolute value of variability 31

32. In a data set we have calculated distances to the mean and squaring these. The result is:
4 – 9 – 4 – 1 – 0 – 25 – 9
A) Calculate the standard deviation.
B) Reproduce the original distancies.
C) Supossing that mean is 9, elaborate the frequency table.

33. Other dispersion indexes

34. e) The Quasivariance and f) The Standard Quasideviation
It can be used when you want to make a more accurate estimate of the variance and the standard deviation of the population.
When the sample is small, the difference is significant.

35. The quasivariance Formula: If we have frequency table:
If we have the variance:

36. The standard quasideviation
Formula:

37. From the previous example
The variance: from 18 to 22.5
The s.d.: from 4.2 to 4.74

38. Variance and standard deviation properties
1. Both S2 and S are essentially non-negative values :
2. It is not calculable or not recommended when it is not calculable or the mean is not considered a good measure of central tendency.
S2 ≥ 0
S ≥ 0

39. Variance and standard deviation properties
3. The s.d. is expressed in the same units for which data are expressed.
4. Both variance (S2) and standard deviation (S) from the sample are lower than variance ( 2) and standard deviation ( ) from the population:
S2 < 2
S <

40. g) Pearson’s variation coefficient
Ex: age
A difference of 2 years of age may be a lot or little:
78-80 age range
1-3 age range
Does this difference of two years have the same connotations?
There is a psychological difference that the numbers can not detect. We, as psychologists (in the near future) have to interpret it. 40

41. Pearson’s variation coefficient
Called ‘Relative variability coefficient’ too.
Symbolically :

42. It is preferable to use the CV before the S when we want to compare the dispersions of two or more distributions of data.
Large units produce larger differences. This is reflected in the mean.
When the means are similar, it is more simple and equally valid comparison in terms of S (calculating the CV does not bring anything new).

43. a) S1= 2 b) S2= 2 c) S3=10
What do you think about the degree of variability of these groups?

44. Mean= 5
Mean = 50
Mean = 50
Larger or smaller units are reflected in the means
CV1= 40 b) CV2= 4 c) CV3=20

45. Mean = 5
Mean = 50
Mean = 50
S1= 2 b) S2= 2 c) S3=10
CV1= 40 b) CV2= 4 c) CV3=20
When the means are equal, CV does not add anything (same conclusions)

46. In A there is more variation
Example 1: We performed an experiment on reaction times to two stimuli A and B in a sample of subjects. The results were as follows:
Mean Stand. Desv. A B
Which of A or B show more variation?
In A there is more variation

47. In group A there is more variation
Example 2: We used the same test to two groups of students A and B. The results have been:
Mean Stand. Desv.
Grupo A
Grupo B
Which group have higher dispersion?
In group A there is more variation

48. Form measurements

49. Bias or asimmetry measurements
Two distributions with the same mean and the same dispersion can be, in terms of shape, totally different.
These measures tell us in which distribution side there is a greater dispersion.

50. g1 g1 < 0 Negative asymmetric g1 = 0 Symmetric g1 > 0 Positive50

51. Skewness or Kurtosis measurements
Refers to the distributions degree of pointing or slenderness.
The skewness indicates a marked contrast between the high central frequency and the rest.

52. Skewness measurements(Kurtosis)
g2 < 0 Platykurtic
g2 = 0 Mesokurtic
g2 > 0 Leptokurtic g2 52

53. Example 1
The next set of data corresponds to a symmetric distribution with mean equal to 5. Substitute the Xs for their values: 1 3 4 X

54. Solution Xi fi
fiXi 1 3 9 4 2 8 6 12 7 21
Total 60

55. Example 2: Two distributions with equal means and standard deviations do not necessarily have the same form Xi fi 12 40 11 10 13 30 20 14 15

56. Simmetry
(it is not necessary)

57. Example Xi fi 1 2 3 4 6 7