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

Descriptive statistics Nominal level: Frequency, relative frequency, distribution (Tables, charts), Mode Ordinal Level Frequency, relative frequency, distribution (Tables, charts), Mode, Median

Symbols Individual values of a variable x 1,x 2,…,x N S: sum of the values

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

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

Measures of central tendency

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

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:

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

Properites of the Mean

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

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

GROUPPED DATA BY A CATEGORICAL VARIABLE

Calculate a value of a group Frequency (f j ), relative Frequency (g j ) Sum of values (S j ), relative sume of values (Z j ) Group means

Areas Sum of Water cons., m 3 A2349 B5394 C14109 D7845 Total29697 AreasWater cons., % A7,91 B18,16 C47,51 D26,42 Total100,00 Sum of values Relative sum of values

Example Areas Sum of Annual water cons., m3 Number of households Mean of annual water cons., m3 A ,86 B ,26 C ,16 D ,60 Total … fjfj MeansGroupsSj

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.

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

Korábbi példa Areas Sum of Annual water cons., m3 Number of households Mean of annual water cons., m3 A ,86 B ,26 C ,16 D ,60 Total … fjfj MeansGroupsSj