1. Descriptive statistics in case of categorical data (scales)

2. Descriptive statistics
Nominal level:
Frequency, relative frequency, distribution (Tables, charts), Mode
Mode: Most frequent categeory

3. Descriptive statistics
Ordinal Level
Frequency, relative frequency, distribution (Tables, charts), Mode, Median
Median: The midpoint of the alternatives after they have been ordered from the smallest to the largest.

4. Descriptive statistics in case of quantitative data (scales)
Part I

5. Symbols Individual values of a scale variable S: sum of the values
x1,x2,…,xN
S: sum of the values

6. Example
In a group of friends the order of ages (year): 20, 20, 20, 25, 25, 32, 33, 33, 33, 33

7. Descriptive statistics
Scale level:
Frequency, relative frequency,
distribution (Tables, charts)
Measures of central tendencies: Mode, Median, Mean
Deviation and dispersion
Measures of the distribution shape (skewness, kurtosis)

8. Measures of central tendency

9. Arithmetic Harmonic Geometric Quadratic Mode Median (Quantiles) Mean
Measures of location
Arithmetic
Harmonic
Geometric
Quadratic
Mode
Median
(Quantiles)

10. Measures of location
The mode is the value of the observation that appears most frequently
Example
In a group of friends the order of ages (year):
20, 20, 20, 25, 25, 32, 33, 33, 33, 33
Mo=33 year
Problems

11. Measures of location
The median is the midpoint of the values after they have been ordered from the smallest to the largest.
If N (number of cases) is odd: the middle element in the ranked data
If N (number of cases) is even: the mean of the two middle elements in the ranked data
Example
In a group of friends the order of ages (year):
20, 20, 20, 25, 25, 32, 33, 33, 33, 33
Me=28,5 year

12. Mean
The mean is obtained by dividing the sum of all values by the number of values in the data set.
Calculation by individual cases:

13. Example
In a group of friends the order of ages (year): 20, 20, 20, 25, 25, 32, 33, 33, 33, 33

14. Properites of the Mean
What happens if we add the same number to all of our values or multiply all of them by the same number?
Measures of central tendency
Increasing all values by the same A number
Multiplying all values by the same A(≠0) number
Mean
Mean increases by A
Mean is multiplied by A
Mode
Mode increases by A
Mode is multiplied by A
Median
Median increases by A
Median is multiplied by A

15. Grouped data by a Categorical variable

16. Calculate a value of a group
Frequency (fj), relative Frequency (gj)
Sum of values (Sj), relative sume of values (Zj)
Group means

17. Sum of values
Relative sum of values
Areas
Sum of Water cons., m3 A
2349 B
5394 C
14109 D
7845
Total
29697
Areas
Water cons., % A
7,91 B
18,16 C
47,51 D
26,42
Total
100,00

18. Sum of Annual water cons., m3 Mean of annual water cons. , m3
Example
Groups Sj fj
Means
Areas
Sum of Annual water cons., m3
Number of households
Mean of annual water cons. , m3 A
2349 21
111,86 B
5394 46
117,26 C
14109
176
80,16 D
7845 75
104,60
Total
29697
318 …

19. The weighted mean The weighted mean is found by the formula
where is obtained by multiplying each data value by its weight and then adding the products.

20. Relationship betwwen the group menas and the grand mean
Calculation of group means:
Calculation of grand mean

21. Sum of Annual water cons., m3 Mean of annual water cons. , m3
Groups Sj fj
Means
Example
Areas
Sum of Annual water cons., m3
Number of households
Mean of annual water cons. , m3 A
2349 21
111,86 B
5394 46
117,26 C
14109
176
80,16 D
7845 75
104,60
Total
29697
318 …