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Descriptive Univariate Statistics
Data. (Example 1991 U.S. General Social Survey with sample size N = 1517)
Variable X
EDUC
Variable Y
PAEDUC
Variable Z
MAEDUC
Individual 1 x 1 = 12 99 12 ...
Individual 2 x 2 = 20 20 18 ...
Individual 3 x 3 = 20 16 14 ...
Individual 4 x 4 = 20 20 20 ...
We pick out the Variable X = EDUC (Highest Years of Education Completed, 7 Missings).
Mean.
The arithmetic mean of a variable is the sum of the values divided by the number of
(nonmissing) values.
x N
x N i 1
i = 12.
Variance.
The mean of the squared deviations of values from the mean.
N i 1 2 i (^2) x x N 1
s ( ) = 8.
Standard Deviation.
Standard deviation, a measure of spread, is the square root of the sum of the squared deviations
of the values from the mean divided by (N-1).
s = s 2 = 2.98.
Computer Output: EDUC N of cases 1510 Minimum 0. Maximum 20. Range 20. Median 12. Mean 12. Standard Dev 2. Variance 8. Skewness(G1) -0. Kurtosis(G2) 0. Median. The median estimates the center of a distribution. If the data are sorted in increasing order, the median is the value above which half of the values fall. Skewness. A measure of the symmetry of a distribution about its mean. If skewness is nonzero, the distribution is asymmetric. A positive value indicates a long right tail; a negative value, a long left tail. Kurtosis. A value of kurtosis greater than 0 indicates that the variable has longer tails than those for a normal distribution (leptokurtic shape); less than 0 indicates that the distribution is flatter than a normal distribution (platykurtic shape).
Multivariate Data Matrix
Variable 1 Variable 2 Variable 3 ... Individual 1 2 2.56 56. ... Individual 2 3 -0.26 100. ... Individual 3 1 6.21 999 ... Individual 4 2 999 106. ...
Typical Scatter Plots for Different Correlations
r = 0.00 (no correlation):
VAR(1)
VAR(2)
r = 0.20 (weak correlation):
VAR(1)
VAR(2)
r = 0.90 (very high correlation): -3 -2 -1 0 1 2 3 4 VAR(1)
VAR(2)
Example: Scatterplot of Education (EDUC) and Father’s Education (PAEDUC). r = 0. 0 5 10 15 20 25 PAEDUC 0 5 10 15 20 25 E D U C
Correlation Coefficient For the multivariate analysis of data, it is of primary relevance to find concise ways to describe how strong variables are associated. What do we mean by “association”? Two variables, e.g. the education of an individual (X) and his parents’ education (Y), are strongly positively associated when in the population a high level of an individual’s education typically coincides with a high level of parent education, whereas a low level of the individual’s education typically coincides with a low level of parent education. Alternatively, we would say that the variables are weakly associated when, for the individuals with high parent education level, their education level could be low or high with equal probability. Weak or no association would mean that we cannot draw any conclusions about an individuals typical education level, although we know his parents’ education level. An example for strongly negatively associated variables could be tobacco consumption (X) and lung volume (Y): typically, although not necessarily for all individuals, people who smoke a lot can be expected to have a smaller lung volume than people who are nonsmokers or only smoke occasionally. The bivariate (that is, combined!) distribution of two variables X and Y can be displayed by a scatter plot (see examples). The correlation coefficient r (or sometimes rxy) of two variables X and Y is a measure for the level of linear association of these variables. It reflects the abovementioned notion of the term “association”. The correlation r has a value between –1 and 1. If there is no association between X and Y, we have r = 0.00 (“X and Y are uncorrelated”). Typically, a correlation of 0.20 is regarded to be low, a correlation of 0.50 is regarded to be medium, and a correlation of 0.80 is regarded to be high. A correlation of r = 1.00 means a perfect positive association, a correlation of r = -1.00 means a perfect negative association. Another measure of association is the covariance Cov(X,Y) between X and Y. Similar to correlation, a zero covariance reflects no association, a high covariance reflects a higher degree of association, and so on. The correlation r is a normed covariance Cov(X, Y). Sometimes we also write Corr(X, Y) instead of rxy. Formula: Covariance between X and Y. Cov(X,Y) =
N i 1 N 1 xi^ x y^ i y
Formula: Correlation rxy between X and Y. rxy =
sx sy
Cov ( X , Y )
-1.00 rxy 1.
