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 x1,x2,…,xN

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

year

xSN

ii

274333333333225252020201

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.

N

S

N

xx

N

ii

1

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

year

N

xx

N

ii

a

4,2710

33333333322525202020

1

Properites of the Mean

Measures of locationThe mode is the value of the observation that appears most frequently

ExampleIn 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 locationThe 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

ExampleIn 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 BYA CATEGORICAL VARIABLE

Calculate a value of a group

• Frequency (fj), relative Frequency (gj)

• Sum of values (Sj), relative sume of values (Zj)

• Group means

AreasSum of Water

cons., m3

A 2349B 5394C 14109D 7845

Total 29697

Areas Water cons., %A 7,91B 18,16C 47,51D 26,42

Total 100,00

Sum of valuesRelative sum of values

S

SZ jj

4

1jjSS

Example

339,93318

29697m

N

Sx

AreasSum of Annual water cons., m3

Number of households

Mean of annual water cons. , m3

A 2349 21 111,86B 5394 46 117,26C 14109 176 80,16D 7845 75 104,60Total 29697 318 …

fj MeansGroups Sj

Nfk

jj

1

4

1jjSS

k

jjxx

1

j

j

j

f

ii

j f

S

f

xx

j

1

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.

N

xf

f

xf

fff

xfxfxfx

k

iii

k

ii

k

iii

N

NN

1

1

1

21

2211

k

iiixf

1

Relationship betwwen the group menas and the grand mean

k

j j

j

k

jj

k

jj

k

jj

k

jjjk

jj

k

jjj

x

S

S

f

S

N

Sx

xgf

xf

N

Sx

1

1

1

1

1

1

1

Calculation of group means:

j

j

j

f

ii

j f

S

f

xx

j

1

Calculation of grand mean

j

jj

jjj

x

Sf

xfS

Korábbi példa

3

3

39,93318

60,1047516,8017626,1174686,11121

39,93318

29697

mx

mx

a

a

339,9360,104318

7516,80

318

17626,117

318

4686,111

318

21mxa

AreasSum of Annual water cons., m3

Number of households

Mean of annual water cons. , m3

A 2349 21 111,86B 5394 46 117,26C 14109 176 80,16D 7845 75 104,60

Total 29697 318 …

fj MeansGroups Sj