descriptive statistics in the case of quantitative data (scales) part i
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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