Numerical Summary Measures Boxplots Exploratory techniques for paired data Statistics in Geophysics: Descriptive Statistics II Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Background The numerical summaries presented in this section can be subdivided into measures of location, spread and shape Location refers to the central tendency of the data values Spread denotes the degree of variation or dispersion around the center Measures of shape tell you the amount and direction of departure from symmetry and how tall and sharp the central peak of the data is Let X be the variable of interest Suppose a sample of size n is given with observed values x1 , , xn Winter Term 2013/14 2/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Mode The mode, xmod , is the most frequently occurring value or category of X The mode is the most important measure of location for categorical variables The mode of the sample {1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17} is Given the list of data {1, 1, 2, 4, 4} the mode is not unique the data set may be said to be bimodal, while a set with more than two modes may be described as multimodal Winter Term 2013/14 3/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Arithmetic mean The arithmetic mean or average of a sample is x¯ = for which it holds that n n xi , i=1 n i=1 (xi − x¯) = For frequency data with different observed values a1 , , ak and relative frequencies f1 , , fk the mean is k x¯ = aj fj j=1 The mean is a meaningful measure for metric data It is not a robust statistic, meaning that it is strongly affected by outliers Winter Term 2013/14 4/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Median The sorted, or ranked, data values from a particular sample are called the order statistics of that sample Given x1 , x2 , , xn the order statistics x(1) , x(2) , , x(n) for this sample are the same numbers, sorted in ascending order Equal proportions of the data fall above and below the median, xmed Formally, xmed = x( n+1 ) if n is odd 2 (x(n/2) + x(n/2 +1) ) if n is even The median is a resistant measure of location and is meaningful for variables that possess at least an ordinal scale of measurement Winter Term 2013/14 5/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Quantiles A sample quantile, xp , is a number having the same units as the data, which exceeds that proportion of the data given by the subscript p, with < p < Computation: xp = x( np +1) if np is not an integer (x(np) + x(np+1) ) if np is an integer , where np is the largest integer not greater than np Commonly used quantiles: x0.5 = xmed ; x0.25 : first (or lower) quartile; x0.75 : third (or upper) quartile Winter Term 2013/14 6/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Variance The empirical variance of x1 , , xn is ˜s = n n (xi − x¯)2 i=1 Since E(˜s ) = σ (n − 1)/n, an unbiased estimator for the population variance, σ , is the sample variance s2 = n−1 n (xi − x¯)2 i=1 √ The standard deviation, s, is obtained as s = + s Both s and s are not resistant measures of dispersion Winter Term 2013/14 7/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Variance decomposition k groups (x11 , x21 , , xn1 ,1 ), · · · , (x1k , x2k , , xnk ,k ) with nj x¯j = nj and ˜sn2j = nj with n = k j=1 nj (j = 1, , k) nj (xij − x¯j )2 , (j = 1, , k) i=1 Then ˜sn2 = xij , i=1 n k nj (¯ xj − x¯)2 + j=1 and x¯ = Winter Term 2013/14 n n k ¯j j=1 nj x 8/29 k nj ˜sn2j j=1 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Coefficient of variation The coefficient of variation is a normalized measure of dispersion of a frequency distribution It is defined as v= s , x¯ x¯ > The CV is independent of scale and can be used to compare different dispersions Winter Term 2013/14 9/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Range The range of a set of data is the difference between the largest and smallest values, x(n) − x(1) It is the size of the smallest interval which contains all the data and provides an indication of statistical dispersion The range can sometimes be misleading when there are extremely high or low values Example: The range of the sample {8, 11, 5, 9, 7, 6, 3616} is 3616 − = 3611 Winter Term 2013/14 10/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Skewness and kurtosis III To remove the bias in g1 and g2 corrections need to be applied The sample skewness, G1 , and kurtosis, G2 , are defined as G1 = n(n − 1) g1 n−2 and G2 = n−1 [(n+1)g2 +6] (n − 2)(n − 3) G1 = for symmetric distributions; G1 > (G1 < 0) for distributions that are right-skewed (left-skewed) G2 = for mesokurtic distributions; G2 > (G2 < 0) for distributions that are leptokurtic (platykurtic) Winter Term 2013/14 15/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Graphical summary of location measures The boxplot, or box-and-whisker plot, is a very widely used graphical tool It is a simple plot of five numbers: the minimum, x(1) , the lower quartile, x0.25 , the median, x0.5 , the upper quartile, x0.75 , and the maximum, x(n) Using these five numbers, the boxplot presents a sketch of the distribution of the underlying data Winter Term 2013/14 16/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data 40 30 20 10 Temperature in degrees Fahrenheit 50 Boxplot: Example II 10 20 30 40 Temperature in degrees Fahrenheit Figure: Boxplot for the January 1987 Ithaca (left) and Canandaigua (right) maximum temperature data (n = 31) Winter Term 2013/14 17/29 50 Numerical Summary Measures Boxplots Exploratory techniques for paired data Boxplot: modified version The following quantities (called fences) can be used for identifying extreme values in the tails of the distribution: lower inner fence: x0.25 − 1.5 × IQR; upper inner fence: x0.75 + 1.5 × IQR; lower outer fence: x0.25 − × IQR; upper outer fence: x0.75 + × IQR Outlier detection criteria: A point beyond an inner fence on either side is considered a mild outlier A point beyond an outer fence is considered an extreme outlier Winter Term 2013/14 18/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Design of a boxplot * Extreme outlier o (Mild) Outlier Whisker Third Quartile Median First Quartile Minimal value which is no outlier Winter Term 2013/14 19/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Boxplot: Example II ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Figure: Boxplot for the earthquake magnitudes in South Carolina, 1987-1996 (n = 4843) Winter Term 2013/14 20/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data 25 20 15 Miles Per Gallon 30 Boxplots for variables by group 10 ● Number of Cylinders Figure: Boxplot of miles per gallon by car cylinder for car mileage data (n = 32) Winter Term 2013/14 21/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Scatterplots ● ● ● ● ●● ● ● ●● ● ● ● ●● ●●● ●● ● ●● ● ● ● ●● ●● ●● ●●● ●● ● ●●●● ●● ●● ● ● ●●● ●●● ● ● ●●●● ●●●●●●● ● ●● ●●● ● ● ●● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●●●●● ●● ●●●● ● ● ●●●●●●● ●●● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ●●●●●●●●●● ● ● ●● ● 5.5 Petal.Length ● ● ●● ● ● ●● ●●●● ● ●●●● ● ●● ● ●● ● ● ●●●● ● ●● ● ● ● ● ● ●●● ●● ● ●● ● ●● ●● ● ● ● ●●● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●●●●●● ● ● ● ● ●● ● ● ●●●●● ●●●●●●● ● ● ●●●● ●●● ● ●● ● ●●● ●● ● ●● ● ● ●●●● ●●● ● ●● ● ●●●●● ● ●●●● ●● ●●● ● ●●● ●●●● ● ● ● ●● ● ● ● ●●●●●● ● ●●●● ● ● ●●●●●●● ●●●● ● ● ●● ●●● ●● ● ● 6.5 ● ●●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ●●● ●● ●●● ●●●● ● ●● ● ●● ● ● ● ●●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ●●● ● ● ●● ● ●● ● ●● ● ●●●●●●● ● ●● ● ● ● ● ● ● ● ●● ●●●● ● ● ● ● ●●● ● ●●● ●●●● ● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ●●●● ●●●●●●●● ● ● ●● ● ● 7.5 6.5 5.5 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ●● ●● ●●● ●● ●● ● ● ● ● ●● ●●● ● ● ●● ● ●●●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ●● ●●●●●●● ●●●● ●●● ● ●●●● ●●●●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●●●● ●● ● ● ● ● ● ● ●●● ● ● ●● ●● ● ●●● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ●● ● ●● ●●● ●● ●●● ● ●●● ●● ● ●●●●●●●●●● ● ●●● ● ● ●●●●● ● ● ●● ●●●●●● ● ● ● ●●● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● Sepal.Width ● ● ● ● ● ● ●●●● ● ● ● ●●●● ●●●●● ●● ●●●● ●● ●●●● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●●● ● ● ● ●●● ● ● ●● ●● ● ● ● ●● ● ● ●● ●●● ● ● ●● ● ●● ●● ●●● ● ●● ●●● ●●●● ●● ●● ● ●● ●●●●● ● ● ●●● ●●●●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● 4.5 ● ●● ● ● ● ● ●● ●●●● ●●● ● ●● ●●● ● ●● ● ● ● 2.0 2.5 3.0 3.5 4.0 ● 0.5 1.0 1.5 2.0 2.5 ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ●● ● ● ● ● ● ● ●● ●●● ● ●● ●● ●● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ●●● ● ●● ●●●●● ●●●●● ●● ●● ● ●● ● ●● ● ●● ● ●● ● ● 7.5 0.5 1.0 1.5 2.0 2.5 ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ●● ●●● ●● ● ●● ● ● ●● ●●● ● ●●●● ● ● ●● ● ● ● ●●● ● ● ● ●●● ● ●● ●●● ● ● ●● ● ● ● ●●● ● ● ●●● ● ● ● ● ●● ● ● Sepal.Length ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ●●● ●● ● ● ●●● ●● ● ●● ● ● ●●● ● ● ●● ● ● ● ●●● ● ●● ●●● ● ● ● ●●●● ●●●● ● ● ● ● ● ● ● ● ●● ● ● ●●● ●● ● ● ● ●●●●● ●● ●● ● ●● ● ● ●● ● ● ● ●● ● ● 4.5 2.0 2.5 3.0 3.5 4.0 Petal.Width ● ● ●●●●● ●●●● ●●●● ●● ●● ●●●● ● ●● Figure: Scatterplot matrix of iris data (n = 150) Winter Term 2013/14 22/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Pearson correlation Often an abbreviated, single valued measure of association between two variables is needed The term correlation coefficient is used to mean the Pearson product-moment coefficient of linear correlation between two variables X and Y Formally, rXY = where n−1 and Y n−1 n−1 n i=1 (xi n i=1 (xi n i=1 (xi − x¯)(yi − y¯ ) − x¯)2 n−1 n i=1 (yi , − y¯ )2 − x¯)(yi − y¯ ) is the sample covariance of X The heart of the Pearson correlation is the covariance between X and Y in the numerator The denominator is in effect just a scaling constant Winter Term 2013/14 23/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Pearson correlation II −1 ≤ rXY ≤ Interpretation: rXY > 0: positive linear correlation rXY < 0: negative linear correlation rXY = 0: no linear correlation It is computationally easier to calculate rXY = n i=1 xi yi n i=1 xi Winter Term 2013/14 − n¯ x2 24/29 − n¯ x y¯ n i=1 yi − n¯ y2 Numerical Summary Measures Boxplots Exploratory techniques for paired data Spearman rank correlation A robust measure of association is the Spearman rank correlation coefficient The Spearman correlation is simply the Pearson correlation coefficient computed using the ranks of the data Formally, rSP = (rank(xi ) − rankX )(rank(yi ) − rankY ) (rank(xi ) − rankX )2 (rank(yi ) − rankY , )2 where rankX and rankY are the averages of the ranks of X and Y , respectively The Spearman correlation can be used for variables that are measured on an ordinal scale Winter Term 2013/14 25/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Spearman rank correlation II In cases of ties (a particular data value appears more than once) all of these equal values are assigned their average rank −1 ≤ rSP ≤ Interpretation: rSP > 0: Y tends to increase when X increases rSP < 0: Y tends to decrease when X increases rSP = 0: No tendency for Y to either increase or decrease when X increases If there are no ties, rSP can be computed as di2 , (n2 − 1)n where di is the difference in ranks between the ith pair of data values rSP = − Winter Term 2013/14 26/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Association between categorical variables Suppose two variables X and Y with observed tuples (x1 , y1 ), , (xn , yn ) are given The k (k ≤ n) different characteristics of X are denoted by a1 , , ak The m (m ≤ n) different characteristics of Y are denoted by b1 , , bm a1 a2 ak b1 n11 n21 bm n1m n2m n1 n2 nk1 n.1 nkm n.m nk n Table: (k × m)-contingency table of absolute frequencies for two categorical variables X and Y Winter Term 2013/14 27/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Association between categorical variables II The conditional frequency distribution of Y given X = , Y |X = , is nim ni1 , , fY (bm |ai ) = fY (b1 |ai ) = ni ni The conditional frequency distribution of X given Y = bj , X |Y = bj , is n1j nkj fX (a1 |bj ) = , , fX (ak |bj ) = n.j n.j Postulate of empirical independence: n ˜ij n.j ni n.j = ⇒n ˜ij = , ni n n where n ˜ij is the absolute frequency one would expect under the assumption of no association between X and Y Winter Term 2013/14 28/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Association between categorical variables III Association measure: k m χ2 = i=1 j=1 (nij − n ˜ij )2 , n ˜ij χ2 ∈ [0, ∞) Contingency coefficient: K= χ2 , n + χ2 which can take values between and Kmax = with M = min{k, m} The adjusted contingency coefficient is K∗ = K , Kmax Winter Term 2013/14 K ∗ ∈ [0, 1] 29/29 (M − 1)/M [...]... in the numerator The denominator is in effect just a scaling constant Winter Term 2013/14 23/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Pearson correlation II −1 ≤ rXY ≤ 1 Interpretation: rXY > 0: positive linear correlation rXY < 0: negative linear correlation rXY = 0: no linear correlation It is computationally easier to calculate rXY = n i=1 xi yi n 2 i=1 xi Winter... modified version The following quantities (called fences) can be used for identifying extreme values in the tails of the distribution: lower inner fence: x0.25 − 1.5 × IQR; upper inner fence: x0.75 + 1.5 × IQR; lower outer fence: x0.25 − 3 × IQR; upper outer fence: x0.75 + 3 × IQR Outlier detection criteria: A point beyond an inner fence on either side is considered a mild outlier A point beyond an outer... techniques for paired data Spearman rank correlation II In cases of ties (a particular data value appears more than once) all of these equal values are assigned their average rank −1 ≤ rSP ≤ 1 Interpretation: rSP > 0: Y tends to increase when X increases rSP < 0: Y tends to decrease when X increases rSP = 0: No tendency for Y to either increase or decrease when X increases If there are no ties, rSP can be computed... Exploratory techniques for paired data Location Spread Shape Interquartile range The most common resistant measure of dispersion is the interquartile range (IQR) The IQR is defined as IQR = x0.75 − x0.25 The IQR is a good index of the spread in the central part of a data set, since it simply specifies the range of the central 50% of the data Winter Term 2013/14 11/29 Numerical Summary Measures Boxplots... data are measured by the kurtosis Higher values indicate a higher, sharper peak; lower values indicate a lower, less distinct peak Winter Term 2013/14 13/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Skewness and kurtosis II The moment coefficients of skewness, g1 , and kurtosis, g2 , are typically defined as g1 = m3 3/2 m2 and g2 = m4 −3 , m22 where... Figure: Boxplot for the earthquake magnitudes in South Carolina, 1987-1996 (n = 4843) Winter Term 2013/14 20/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data 25 20 15 Miles Per Gallon 30 Boxplots for variables by group 10 ● 4 6 8 Number of Cylinders Figure: Boxplot of miles per gallon by car cylinder for car mileage data (n = 32) Winter Term 2013/14 21/29 Numerical Summary... n is defined as n 1 (xi − x¯)r mr = n i=1 The sample central moments are not unbiased estimates of the population central moments Winter Term 2013/14 14/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Location Spread Shape Skewness and kurtosis III To remove the bias in g1 and g2 corrections need to be applied The sample skewness, G1 , and kurtosis, G2 , are defined as... of the distribution of the underlying data Winter Term 2013/14 16/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data 40 30 20 10 Temperature in degrees Fahrenheit 50 Boxplot: Example II 10 20 30 40 Temperature in degrees Fahrenheit Figure: Boxplot for the January 1987 Ithaca (left) and Canandaigua (right) maximum temperature data (n = 31) Winter Term 2013/14 17/29 50 Numerical... The median absolute deviation (MAD) is easiest to understand by imagining the transformation yi = |xi − x0.5 | The MAD is then just the median of the transformed (yi ) values: MAD = median(yi ) = median|xi − x0.5 | The MAD is analogous to computation of the standard deviation, but using operations that do not emphasize outlying data Winter Term 2013/14 12/29 Numerical Summary Measures Boxplots Exploratory... empirical independence: n ˜ij n.j ni n.j = ⇒n ˜ij = , ni n n where n ˜ij is the absolute frequency one would expect under the assumption of no association between X and Y Winter Term 2013/14 28/29 Numerical Summary Measures Boxplots Exploratory techniques for paired data Association between categorical variables III Association measure: k m χ2 = i=1 j=1 (nij − n ˜ij )2 , n ˜ij χ2 ∈ [0, ∞) Contingency ... −1 ≤ rSP ≤ Interpretation: rSP > 0: Y tends to increase when X increases rSP < 0: Y tends to decrease when X increases rSP = 0: No tendency for Y to either increase or decrease when X increases... modified version The following quantities (called fences) can be used for identifying extreme values in the tails of the distribution: lower inner fence: x0.25 − 1.5 × IQR; upper inner fence: x0.75 +... of the Pearson correlation is the covariance between X and Y in the numerator The denominator is in effect just a scaling constant Winter Term 2013/14 23/29 Numerical Summary Measures Boxplots