Unformatted text preview: Lecture 4 Summarization of Data: Algebraic Approach There are three aspects we look at 1. Where does the centre of the data lie? Measures of Central Tendency 2. How is the data spread across the centre? Measures of Dispersion. 3. Is the data symmetric? Measures of skewness. Central tendency • The mean: Given a numerical data set {x1, x2, x3, … xn}, the Arithmetic Mean is given by _ x = x1 +x2 +x3 +….xn _
, i.e x = Σ xi/n n Other (rarely used) means • Geometric mean G.M = (x1*x2*x3*…xn)1/n • Harmonic Mean H.M = n/(1/x1 +1/x2 +1/x3+…1/xn) Applications: (i) Average interest rate (ii) Average Speed • The median: This can be used when the data is ordinal. The median is defined to be the value of Q2 such that half the observations are below Q2 and half are above Q2 If we ranked the data as x(1), x(2),…. x(n), where x(1) ≤ x(2) ≤ x(3)≤…….≤x(n), then if n is odd, then the median is x({n+1}/2). When n is even, then the median is ½[ x(n/2) +x((n+1)/2)] • The mode: This is used when we have grouped or frequency data. The mode is the one with the highest frequency Measures of Dispersion • Range = Maximum –Minimum • Quartiles: The lower quartile Q1 is the value which has 25% of the observations smaller than it. The upper quartile Q3 splits the data 75%:25% • The inter
quartile range: Q3
Q1 • Percentiles: The pth percentile is one that divides the data in the ratio p:100
p 100 Graphical Representation of Quartiles Quartiles
Cumulative Distribution of Weights 80
75% Cumulative Proportion (%) 60
50% 40
25% 20 lower quartile median 0 20 0 upper quartile 40 Weights (lbs) 60 80 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit Variance and Standard Deviation • The variance is defined to be the mean of the squared difference from the average The population variance is given by σ2n = Σ1n(xi
x)2/n • The variance = Average of squares
Square of average • The sample variance is given by σ2n
1 = Σ(xi
x)2/(n
1) why? • The standard deviation = sqrt(variance) Mean and Variance of affine transformations. Suppose y = Ax +B • Mean(y) = A*Mean(x) +B • Variance(y) = A2 Variance(x) • Standard deviation(y) = AS.D(x) Applications: The disappearance of the 400 hitter in baseball Conventional explanations: • Genesis myth • External Arguments • Internal Arguments None of these seem to be correct. League Averages in the 20th century A.L N.L 1901
1910 0.251 0.253 1911
1920 0.259 0.257 1921
1930 0.286 0.288 1931
1940 0.279 0.272 1941
1950 0.260 0.260 1951
1960 0.257 0.260 1961
1970 0.245 0.253 1971
1980 0.258 0.256 1981
1990 0.262 0.254 Standard deviation of batting averages Other Graphical data summaries: 1. Box
Plots 2. The Q
Q plot 3. Scatter Plot (for Bivariate Data) 4. Scatter Plot Matrix 5. Time Series Data Algebraic Measures of Association 1. Dependence and Relative Risk 2. Correlation ...
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 Winter '08
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