1. Frequency distributions and graphic representations
Chapter 2
Frequency distributions and graphic representations

2. Introduction Data description
How can we represent and provide information about them in the most convenient form?
This depends on their number. To work with a large number of data (we can not obtain all the information at a glance) use two basic descriptive strategies:
Frequency tables
Graphic representations

3. Selecting the type of variable
CUALITATIVE OR NOMINAL
ORDINAL (CUASICUANTITATIVE)
DISCRETE CUANTITATIVE
CONTINUE CUANTITATIVE

4. The best election VARIABLE: Alcoholism
Cualitative (2 categories): Abstemious/Drinker
Ordinal: Abstemious/Moderate drinker/Heavy drinker
C. Discrete: Number of drinks and cocktails drunk for a certain time interval, for example, one week.
C. Continua: Liters of alcohol ingested per week.

5. 1. Cualitative variables (nominal or categorical)
What is your marital status?
Single 1
Marital
Status
Married 2
Widow/er 3

6. Cualitative variables (nominal or categorical)
A) Binary, binomial or dichotomous
B) Polytomous or Multinomial

7. Xi : Are the different values of the variable “Marital status": Single, Married and Widowed
Sex: Male and Female
Gender: Male / Fem.
Smoking: Yes / No
Sex relation: Yes / No
Media: Radio / TV / Press
Decriminalization of abortion: Against / Indifferent / A favor
Sick: psychotic / neurotic / Organic
Political opinion: Center / Left / Right
Skills: Ambidextrous / Left / Right
Political parties: PP / IU / PSOE / PA
Type of Drug: Cocaine, Marijuana, Heroin

8. Marital Status S M W

9. Structure of a frequency distribution
Frequency distribution: data tables in which represent at least:
The values of the variable (Xi)
The frequencies of these values(fi)

10. Structure of qualitative or categorical data distribution table
Different values of the variable “marital status”
Number of times that each value appears Xi fi S 18 M 9 W 3

11. a) Absolute frequency: fi
Number of subjects that belong
in each category (f1 , f2 , f3) fi
f1 + f2 + f3 = n n
Total number of subjects

12. Absolute frequency: fiXi fi
The categories should
X1 = S
f1 = 18
1. Be clearly defined
X2 = M
f2 = 9
2. Be mutually exclusive
X3 = W
f3 = 3
3. Be exhaustives
Total
n = 30

13. b) Relative frequency or proportion: fri
The relative frequency is the ratio of the absolute frequency and n
Reports on the importance that has a value within the set to which it belongs:
Example:
n= 100
n= 50
X1-X10; CI= 120
fi= 10
fi = 10
fr= 0’1 (10%)
fr= 0’2 (20%)

14. Relative frequency: fri
To find the value:

15. Properties
∑ fri = fr1 + fr2 + fr3 = 1 1ª
0 ≤ fri ≤ 1 2ª

16. Example 0, 6 0, 3 0, 1 1 Xi fi fri S f1 = 18 M f2 = 9 W f3 = 3 T
n = 30
0, 6
0, 3
0, 1 1

17. c) Percentages: %i
The percentage is equal to the relative frequency multiplied by 100
%i = fri x x 100

18. Properties ∑ %i = %1 + %2 + %3 = 100% 1ª 2ª 0 ≤ %i ≤ 100 60 30 10 100Xi fi
fri %i S
f1 = 18
0,6 M
f2 = 9
0.3 W
f3 = 3
0.1 T
n = 30 1 60 30 10
100

19. Graphic representation: Bar Chart
More frequency, more bar height
ORDINATE
ABSCISSA

20. Ciclograma or Pie chart
MARITAL STATUS

21. Complete the graphic representation
Reproduces the frequency distribution table of a qualitative variable (4 categories) based on its incomplete graphical representation, knowing that 25% of the data have the value 2.
Complete the graphic representation

22. 2. Ordinal variables (cuasicuantitatives)
Degree of
Responsibility
Too high 1
High 2
Medium 3
Low 4 5
Too Low

23. 2. Ordinal variables Socioeconomic status: High / Medium / Low
Level of agreement: Poor / Fair / Moderate / Substantial / Excellent
Responsibility level: Very High / High / Medium / Low / Very Low
Social class: High / Medium / Low
Skill level: High / Medium / Low
Concentration ability: Poor / Fair / Moderate / Substantial / Excellent
Degree of emotional distress: Very High / High / Medium / Low / Very Low
Income Level: High / Medium / Low

24. Degree of responsibility1 2 3 4 5

25. What we already know ... Xi fi fri %i Very high 6 0,075 7,5 High 10
0,125
12,5
Medium 48
0.600 60
Low 12
0,150 15
Very low 4
0,050 5
Total 80 1
100

26. d) Cumulative frequency: Fi
It allows consider each value not isolated from the others, but in conjunction with them Xi fi
fri %i Fi
Very high 6
0,075
7,5 F1
High 10
0,125
12,5 F2
Medium 48
0.600 60 F3
Low 12
0,150 15 F4
Very low 4
0,050 5 F5
Total 80 1
100
F1 = f1
F2 = F1 + f2
F3 = F2 + f3
F4 = F3 + f4
F5 = F4 + f5

27. An alternative F1 = f1 F2 = f1 + f2 F3 = f1 + f2 + f3Xi fi
fri %i Fi
Very high 6
0,075
7,5
High 10
0,125
12,5 16
Medium 48
0.600 60 64
Low 12
0,150 15 76
Very low 4
0,050 5 80
Total 1
100
An alternative
F1 = f1
F2 = f1 + f2
F3 = f1 + f2 + f3
F4 = f1 + f2 + f3 + f4
F5 = f1 + f2 + f3 + f4 + f5

28. e) Cumulative relative frequency (or cumulative proportion: Fri
Ratio of cumulative frequency of a value Xi and the total frequency Xi fi
fri %i Fi
Fri
Very high 6
0,075
7,5
High 10
0,125
12,5 16
0,200
Medium 48
0.600 60 64
0,800
Low 12
0,150 15 76
0,950
Very low 4
0,050 5 80 1
Total
100
Fr1 = fr1 = 0,075
Fr2 = Fr1 + fr2 = 0, ,125 = 0,20
Fr3 = Fr2 + fr3 = 0,20 + 0,60 = 0,80
Fr4 = Fr3 + fr4 = 0,80 + 0,15 = 0,95
Fr5 = Fr4 + fr5 = 0,95 + 0,05 = 1

29. f) Cumulative Percentages: %aiXi fi
fri %i Fi
Fri
%ai
Very high 6
0,075
7,5
High 10
0,125
12,5 16
0,200 20
Medium 48
0.600 60 64
0,800 80
Low 12
0,150 15 76
0,950 95
Very low 4
0,050 5 1
100
Total
%a1 = %1 = 7,5%
%a2 = %a1 + %2 = 7,5 + 12,5 = 20%
%a3 = %a2 + %3 = = 80%
%a4 = %a3 + %4 = = 95%
%a5 = %a4 + %5 = = 100%

30. Bar chart

31. Cumulative bar chart

32. 3. Discrete quantitative variable
Number of children of Seville’s families
20 married couples are selected from the city of Seville and asked about the number of children they have. Their answers are:
2, 1, 0, 3, 2, 2, 3, 1, 1, 0, 1, 2, 1, 2, 0, 2, 4, 2, 3, 1

33. Frequency table Raw Data Cumulative data Xi fi fri %i Fi Fri %ai 33
0.15 15 1 6
0.30 30 9
0.45 45 2 7
0.35 35 16
0.80 80 19
0.95 95 4
0.05 5 20
100
Total

34. Bar Chart

35. Cumulative Bar chart

36. 4. Continuous quantitative variable
Attention test’s scores 22 17 6 7 18 10 23 11 19 24 13 14 25 26 28 15 20 30 16 32 21

38. Histogram