descriptive statistics in the case of quantitative data (scales) part i
TRANSCRIPT
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Descriptive statistics in the case of quantitative data (scales)
Part I
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Descriptive statistics
Nominal level: Frequency, relative frequency,
distribution (Tables, charts), Mode
Ordinal Level Frequency, relative frequency,
distribution (Tables, charts), Mode, Median
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Symbols
Individual values of a variable x1,x2,…,xN
S: sum of the values
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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
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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)
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Measures of central tendency
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Mean Arithmetic Harmonic Geometric Quadratic
Measures of location Mode Median (Quantiles)
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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:
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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
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Properites of the Mean
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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
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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
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GROUPPED DATA BYA CATEGORICAL VARIABLE
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Calculate a value of a group
• Frequency (fj), relative Frequency (gj)
• Sum of values (Sj), relative sume of values (Zj)
• Group means
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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
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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
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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
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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
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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