Univariate descriptives
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1 Univariate descriptives Johan A. Elkink University College Dublin 18 September September /
2 Outline 1 Graphs for categorical variables 2 Graphs for scale variables 3 Frequency tables 4 Central tendency 5 Variation 18 September /
3 Introduction Outline Graphical presentations of single variables. Categorical Scale 18 September /
4 Introduction Outline Graphical presentations of single variables. Categorical pie-charts barplots Scale 18 September /
5 Introduction Outline Graphical presentations of single variables. Categorical pie-charts barplots Scale histogram density plot boxplot 18 September /
6 Outline Graphs for categorical variables 1 Graphs for categorical variables 2 Graphs for scale variables 3 Frequency tables 4 Central tendency 5 Variation 18 September /
7 Graphs for categorical variables Categorical variables For categorical variables, it is often useful to look at the number of cases or the proportion of cases in a particular category. Barplots and pie charts are useful for this. 18 September /
8 Barplot Graphs for categorical variables Fine Gael Labour Fianna Fail Sinn Fein Green Party Independent Other ( general election, 2011) 18 September /
9 Pie chart Graphs for categorical variables Sinn Fein Fianna Fail Green Party Independent Other Labour Fine Gael 18 September /
10 Exercise Graphs for categorical variables Using the data from the first lecture: Produce a pie chart for regime type Produce a bar chart for the cleavage variable 18 September /
11 Outline Graphs for scale variables 1 Graphs for categorical variables 2 Graphs for scale variables 3 Frequency tables 4 Central tendency 5 Variation 18 September /
12 Histogram Graphs for scale variables For continuous (or scale) variables, we often want to get an idea of the distribution of values. How many low, medium, high values? Histograms are useful to get an impression. 18 September /
13 Histogram Graphs for scale variables Histogram: bin the data using equal-distance cut-off points then produce a barplot of the number in each bin. 18 September /
14 Histogram Graphs for scale variables Frequency Probability ever vote for Labour (Irish Johan National A. Elkink Election (UCD) Study 2011) 18 September /
15 Density plot Graphs for scale variables A density plot is a smoothened version of a histogram, based on a non-parametric estimation of the shape of the distribution. Density Univariate Probability ever vote graphics for Labour 18 September /
16 Graphs for scale variables Distributions For graphs of distributions (histogram, density plot, boxplot, etc.) you want to get an impression of: the shape of the distribution; the center and spread of the distribution; the presence of outliers. (Moore 2003: 12) 18 September /
17 Boxplot Graphs for scale variables Another way of looking at the distribution of a continuous variable is to find out where the lowest 25% are located, where the lowest 50% are located, and where the top 25% are located. A plot that shows this is the boxplot. 18 September /
18 Boxplot Graphs for scale variables Probability to ever vote Fianna Fail 18 September /
19 Time plot Graphs for scale variables When data is measured over time, another useful plot is a time plot, to see trends over time. 18 September /
20 Time plot Graphs for scale variables When data is measured over time, another useful plot is a time plot, to see trends over time. Proportion democracies Year 18 September /
21 Exercise Exercise Using the data from the first lecture: Produce a histogram for the Polity IV democracy score Produce a boxplot for the number of battle deaths 18 September /
22 Exercise Exercise Open the demdev.dta data file in SPSS. Select only cases where the year is Produce the following plots of GDP per capita (laggdppc): 1 Histogram 2 Boxplot 3 Repeat for the logarithm of GDP 4 Repeat for energy usage (energy2) Describe the distribution in words. 18 September /
23 Outline Frequency tables 1 Graphs for categorical variables 2 Graphs for scale variables 3 Frequency tables 4 Central tendency 5 Variation 18 September /
24 Frequency tables Frequency tables Democracy is preferable to any other kind of government % In some circumstances, a non-democratic government can be preferable % For someone like me, it doesn t matter what kind of government we have % Missing / don t know % Total % 18 September /
25 Example Frequency tables September /
26 Frequency tables Example frequency table value count percentage total % 18 September /
27 Frequency tables Example summary Frequency x 18 September /
28 Outline Central tendency 1 Graphs for categorical variables 2 Graphs for scale variables 3 Frequency tables 4 Central tendency 5 Variation 18 September /
29 Central tendency Measures of central tendency Measures of central tendency provide information about the centre of a distribution, roughly put: what is a typical value for this variable? 18 September /
30 Central tendency Measures of central tendency Measures of central tendency provide information about the centre of a distribution, roughly put: what is a typical value for this variable? Different measures are available for different levels of measurement: mode median mean nominal x ordinal x x scale (x) x x 18 September /
31 Mode Central tendency The mode is the category with the highest frequency. 18 September /
32 Example Central tendency September /
33 Central tendency Example mode The mode is the value that most often occurs, i.e September /
34 Median Central tendency The median is the value where 50% of the cases has a lower value on this variable and 50% a higher value. 18 September /
35 Median Central tendency The median is the value where 50% of the cases has a lower value on this variable and 50% a higher value. If N is uneven: the middle value after sorting in ascending order. If N is even: the average of the two middle values after sorting in ascending order. 18 September /
36 Example Central tendency September /
37 Central tendency Example median First, sort the data: September /
38 Central tendency Example median First, sort the data: Then, find the centre: September /
39 Central tendency Example median First, sort the data: Then, find the centre: The median is = September /
40 Mean Central tendency The mean is the sum of all values, divided by the number of values. x = N i=1 x i N 18 September /
41 Example Central tendency September /
42 Central tendency Example mean x = = = September /
43 Exercise Central tendency Calculate appropriate measures of central tendency for each of the following variables related to party membership: years in years in gender class party gender class party M High 32 M Medium 17 M Low 32 M Low 50 M Low 25 M Medium 25 F High 12 F High 10 F Medium 21 F Medium 33 M Low 37 F Low 15 F Low 31 (Healey 1997: 81) 18 September /
44 Outliers Central tendency Outliers is a general term of values that are very far from the main values of the distribution. 18 September /
45 Central tendency Mean or median? For a symmetric distribution, median and mean are the same. 18 September /
46 Central tendency Mean or median? For a symmetric distribution, median and mean are the same. The more skewed the distribution, the more mean and median differ. 18 September /
47 Central tendency Mean or median? For a symmetric distribution, median and mean are the same. The more skewed the distribution, the more mean and median differ. Mean is sensitive to outliers, while the median is not. 18 September /
48 Central tendency Mean or median? For a symmetric distribution, median and mean are the same. The more skewed the distribution, the more mean and median differ. Mean is sensitive to outliers, while the median is not. Mean has better understood mathematical properties. 18 September /
49 Outline Variation 1 Graphs for categorical variables 2 Graphs for scale variables 3 Frequency tables 4 Central tendency 5 Variation 18 September /
50 Variation Measures of dispersion Measures of dispersion provide an indication of the amount of variation or heterogeneity in a variable. 18 September /
51 Range Variation The range is the highest value minus the lowest value. 18 September /
52 Range Variation The range is the highest value minus the lowest value. The interquartile range (IQR) is the range between the lowest 25% and the top 25%. 18 September /
53 Range Variation The range is the highest value minus the lowest value. The interquartile range (IQR) is the range between the lowest 25% and the top 25%. A boxplot typically provides the median and the IQR, with some indication of outliers. 18 September /
54 Example Variation September /
55 Example range Variation Range: 9 0 = September /
56 Variance Variation Var(x) = s 2 x = N i=1 (x i x) 2 N 18 September /
57 Variance Variation Var(x) = s 2 x = N i=1 (x i x) 2 Note that many software packages do not calculate this sample variance, but the unbiased estimator of the population variance: N ˆσ x 2 i=1 = (x i x) 2 N 1 N 18 September /
58 Example mean Variation x = = = September /
59 Variation Example variance x = = = 4.6 s 2 x = 1 N N (x x) 2 = 1 12 i=1 N (x 4.6) 2 i=1 18 September /
60 Variation Example variance sx 2 = 1 N (x x) 2 = 1 N 12 i=1 N (x 4.6) 2 i=1 = 1 12 ((1.4)2 + ( 2.6) 2 + ( 1.6) 2 + ( 4.6) 2 + (2.4) 2 + (4.4) 2 + (1.4) 2 + (2.4) 2 + (0.4) 2 + ( 1.6) 2 + (2.4) 2 + ( 4.6) 2 ) 18 September /
61 Variation Example variance s 2 x = 1 N N (x x) 2 = 1 12 i=1 N (x 4.6) 2 i=1 = 1 12 ((1.4)2 + ( 2.6) 2 + ( 1.6) 2 + ( 4.6) 2 + (2.4) 2 + (4.4) 2 + (1.4) 2 + (2.4) 2 + (0.4) 2 + ( 1.6) 2 + (2.4) 2 + ( 4.6) 2 ) = 1 ( ) = = September /
62 Variation Variance: sample data Country Asylum seekers (x x) (x x) 2 Denmark 2.3 Finland 3.6 Ireland 4.3 Norway 5.4 Netherlands 12.4 Belgium 16.0 Sweden 17.5 Germany 28.9 United Kingdom 30.5 France 50.1 Total s 2 = Ni=1 (x x) 2 N.. 18 September /
63 Variation Variance: sample data Country Asylum seekers (x x) (x x) 2 Denmark = 14.8 Finland = 13.5 Ireland = 12.8 Norway = 11.7 Netherlands = 4.7 Belgium = 1.1 Sweden = 0.4 Germany = 11.8 United Kingdom = 13.4 France = 33 Total s 2 = Ni=1 (x x) 2 N.. 18 September /
64 Variation Variance: sample data Country Asylum seekers (x x) (x x) 2 Denmark = 14.8 Finland = 13.5 Ireland = 12.8 Norway = 11.7 Netherlands = 4.7 Belgium = 1.1 Sweden = 0.4 Germany = 11.8 United Kingdom = 13.4 France = 33 Total s 2 = Ni=1 (x x) 2 N.. 18 September /
65 Variation Variance: sample data Country Asylum seekers (x x) (x x) 2 Denmark = 14.8 ( 14.8) 2 = Finland = 13.5 ( 13.5) 2 = Ireland = 12.8 ( 12.8) 2 = Norway = 11.7 ( 11.7) 2 = Netherlands = 4.7 ( 4.7) 2 = 22.1 Belgium = 1.1 ( 1.1) 2 = 1.2 Sweden = 0.4 (0.4) 2 = 0.2 Germany = 11.8 (11.8) 2 = United Kingdom = 13.4 (13.4) 2 = France = 33 (33) 2 = 1089 Total s 2 = Ni=1 (x x) 2 N.. 18 September /
66 Variation Variance: sample data Country Asylum seekers (x x) (x x) 2 Denmark = 14.8 ( 14.8) 2 = Finland = 13.5 ( 13.5) 2 = Ireland = 12.8 ( 12.8) 2 = Norway = 11.7 ( 11.7) 2 = Netherlands = 4.7 ( 4.7) 2 = 22.1 Belgium = 1.1 ( 1.1) 2 = 1.2 Sweden = 0.4 (0.4) 2 = 0.2 Germany = 11.8 (11.8) 2 = United Kingdom = 13.4 (13.4) 2 = France = 33 (33) 2 = 1089 Total s 2 = Ni=1 (x x) 2 N = = September /
67 Variation Standard deviation Standard deviation: s x = Var(x) = N i=1 (x i x) 2 N 18 September /
68 Variation Example variance s 2 x = 1 N N (x x) 2 = 1 12 i=1 N (x 4.6) 2 i=1 = 1 12 ((1.4)2 + ( 2.6) 2 + ( 1.6) 2 + ( 4.6) 2 + (2.4) 2 + (4.4) 2 + (1.4) 2 + (2.4) 2 + (0.4) 2 + ( 1.6) 2 + (2.4) 2 + ( 4.6) 2 ) = 1 ( ) = = September /
69 Variation Example standard deviation Standard deviation: s = s 2 = 7.9 = September /
70 Variation Variance: sample data Country Asylum seekers (x x) (x x) 2 Denmark = 14.8 ( 14.8) 2 = Finland = 13.5 ( 13.5) 2 = Ireland = 12.8 ( 12.8) 2 = Norway = 11.7 ( 11.7) 2 = Netherlands = 4.7 ( 4.7) 2 = 22.1 Belgium = 1.1 ( 1.1) 2 = 1.2 Sweden = 0.4 (0.4) 2 = 0.2 Germany = 11.8 (11.8) 2 = United Kingdom = 13.4 (13.4) 2 = France = 33 (33) 2 = 1089 Total s 2 Ni=1 (x x) = 2 N = = s = s 2 = = September /
71 Variation Variance and outliers The variance refers to the variation in the data around the mean. It is similarly sensitive to outliers - a few extreme values in the data can significantly increase the variance. 18 September /
72 Variation Example summary Frequency x 18 September /
73 Variation Example summary Frequency mean median mode x 18 September /
74 Exercise Exercise Open the demdev.dta data file in SPSS. Select only year Take a glance at the data. 2 Using SPSS, produce a histogram of GDP per capita (laggdppc). 3 Calculate the median, mean, range, variance, and standard deviation of GDP per capita. 4 Create a new variable that is the log of GDP per capita. 5 Repeat steps 2 and 3 on the logged variable. 6 Are there any extreme cases? What happens if you remove this? 18 September 2014 /
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