Chapter 1 Handout: Descriptive Statistics

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1 Preview Chapter 1 Handout: Descriptive Statistics Describing a Single Data Variable o Introduction to Distributions o Measure of the Distribution Center: Mean (Average) o Measures of the Distribution Spread: Variance and Standard Deviation o Histogram: Visual Illustration of a Data Variable s Distribution Describing the Relationship between wo Data Variables o Scatter Diagram: Visual Illustration of How wo Data Variables Are Related o Correlation and Independence of wo Variables o Measures of Correlation: Covariance and the Correlation Coefficient o Correlation and Causation Arithmetic of Means, Variances, and Covariances Descriptive Statistics: Describing a Single Variable Question: June is the wettest month of the summer, April is the wettest month of the year, he summer of 212 was the hottest on record, etc. How can we assess statements like this? Amherst Monthly Precipitation Data: 191 to 2 Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2 Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

3 3 Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Measure of the Distribution Center: Mean (Average) Question: In the 2 th century, what has been the wettest summer month? Mean for June = 1 Equations for the Mean Mean[ x ] where = t 1 2 t 1 x x = x x x = otal number of Observations x 1 = Value of the first observation (June 191) =.75 x 2 = Value of the second observation (June 192) = 4.54 x 3 = Value of the second observation (June 193) = x = Value of the last (N th ) observation (June 2) = 7.99 otal Number of Observations

4 4 Computing the Monthly Mean Precipitation for Summer Months Getting Started in EViews Access the EViews workfile. Amherst Weather Data hen: In the File Download window: Click Open. (Note that different browsers may present you with a slightly different screen to open the workfile.) Next, we instruct EViews to calculate the means: In the Workfile window: Highlight year by clicking on it; then, while depressing <Ctrl>, click on month and precip to highlight them also. In the Workfile window: Double click on any of the highlighted variables. A new list now pops up: Click Open Group. A spreadsheet including the variables Year, Month, and Precip for all the months appears. In the Group window: Click View; then click Descriptive Stats, and then Individual Samples. Descriptive statistics for all the months of the twentieth century now appear. We only want to consider one month at a time. We want to compute the mean for June and then for July and then for August. Let us see how to do this. In the Group window: Click Sample. In the Sample window: Enter month=6 in the If condition (optional) text area to restrict the sample to the sixth month, June, only. Click OK. Descriptive statistics for the 1 Junes appear in the Group window. Record the mean. In the Group window: Click Sample. o In the Sample window: Enter month=7 in the If condition (optional) text area to restrict the sample to July only. o Click OK. Descriptive statistics for the 1 Julys appear in the Group window. Record the mean. In the Group window: Click Sample. o In the Sample window: Enter month=8 in the If condition (optional) text area to restrict the sample to August only. o Click OK. Descriptive statistics for the 1 Augusts appear in the Group window. Record the mean. NB: his last step is critical. In the Group window: Click Sample. o In the Sample window: Clear the If condition (optional) text area by deleting month=8; otherwise the restriction, month=8, will remain in effect if you ask EViews to perform any more computations. Last, do not forget to close the file: In the EViews window: Click File, then Exit. In the Workfile window: Click No in response to the save changes made to workfile. Jun Jul Aug Mean Based on the mean, proved to be the wettest month of the summer in the 2 th century.

5 5 Measures of the Distribution Spread: Variance and Standard Deviation Growing Season Precipitation Year Apr May Jun Jul Aug Mean Question: Which growing season was better? Calculating the Variance: he Steps For each month, calculate the amount by which that month s precipitation deviates from the mean. For each month, square the deviation. Calculate the average of the squared deviations; that is, sum the squared deviations and divide by the total number of months, 5 in this case. Standard Deviation: Square Root of the Variance Calculating the Variance for 1998 Month Precipitation Mean Deviation From Mean Squared Deviation Apr = May = Jun = Jul = Aug = Sum of Squared Deviations Variance = Sum of Squared Deviations = = = Standard deviation = Variance = Calculating the Variance for 1951 Month Precipitation Mean Deviation From Mean Squared Deviation Apr = May = Jun = Jul = Aug = Sum of Squared Deviations Variance = Sum of Squared Deviations = = = Standard deviation = Variance = Variance Summary: Small spread Large spread All deviations are Some deviations are All squared deviations are Some squared deviations are Variance Variance

6 6 Equations for the Variance and Standard Deviation ( x1mean[ x]) ( x2 Mean[ x]) ( x Mean[ x]) Var[ x] ( x1x) ( x2 x) ( x x) ( xt x) t 1 where otal Number of Observations x Mean[ x] Mean of x SD[ x] Var[ x ] Why do we take the trouble of calculating the standard deviation? Histogram: Visual Illustration of a Variable s Distribution of Values Each bar of the histogram reports on the number of months in which precipitation fell within the specified range. In years, there was less than 1 inch of rain during September. In years, there was between 1 and 2 inches of rain during September. In years, there was between 2 and 3 inches of rain during September. Descriptive Statistics: he Relationship between wo Variables Scatter Diagram: Visual Illustration of How wo Variables Are Related Monthly Percentage Growth Rate of Dow Jones Industrial Average Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Monthly Percentage Growth Rate of NASDAQ Composite Average Year Jan Feb Mar Apr Feb 2 2 Nasdaq Jan 1987 Each point on the scatter diagram represents the growth rate of the Dow and the growth rate of the Nasdaq for one specific month. Dow Jones Oct 2 2

7 7 Independence and Correlation Not Independent or Correlated Independent or Uncorrelated One variable us predict the other One variable us predict the other Measure of Correlation: Covariance of wo Variables For each month, calculate the amount by which variable x deviates from its mean and the amount by which variable y deviates from its mean. For each month, multiply x s deviation by y s deviation. Calculate the average of these products; that is, sum the products of the deviations and divide by the number of months. x xy yx xy y x xy y t1 Scatter Diagram and Covariance Quadrant I: x t > x and y t > y (x t x) and (y t y ) (x t x)(y t y ) Quadrant II: x t < x and y t > y (x t x) and (y t y ) (x t x)(y t y ) Quadrant III: x t < x and y t < y (x t x) and (y t y ) (x t x)(y t y ) Quadrant IV: x t > x and y t < y (x t x) and (y t y ) (x t x)(y t y ) Scatter Diagrams Deviations from Means Quadrant II (x i x )_ (y i y )_ (x i x )(y i y ) _ Quadrant III (x i -x )(y i -y ) _ (y i y ) Quadrant I (x i x )_ (y i y )_ (x i x )(y i y ) _ x xy y t (x i - x ) Quadrant IV (x i x )_ (y i y )_ (x i x )_ (y i y )_ (x i x )(y i y ) _ t Deviations From Means 2 Nasdaq Deviations From Means 2 Nasdaq 1 1 Dow Jones Precipitation Not Independent: (Positively) Correlated Cov = 2 Independent: Uncorrelated Cov =

8 8 We can use statistical to calculate the variances and covariances: Getting Started in EViews Access the EViews workfile. hen: Stock Market Data In the File Download window: Click Open. (Note that different browsers may present you with a slightly different screen to open the workfile.) Next, we instruct EViews to calculate the covariance of Amherst precipitation and the Nasdaq growth rate: In the Workfile window: Highlight djgrowth by clicking on it; then while depressing <Ctrl> click on nasdaqgrowth to highlight it. In the Workfile window: Double click on any of the highlighted variables. A new list now pops up: Click Open Group. A spreadsheet including the variables Precip and NasdaqGrowth appears. In the Group window: Click View, and then click Covariance Analysis In the Covariance Analysis window: Be certain that the Covariance checkbox is selected; then, click OK. Last, close the file: In the EViews window: Click File, then Exit. In the Workfile window: Click No in response to the save changes made to the workfile. he Downside of Covariance: he Covariance has no natural range. he covariance has no natural range: its magnitude depends on the units used. x1x y1 y x2 x y2 y x x y y t1 x xy y t t where otal Number of Observations Cov[x, y] = x Mean[ x] Mean of x y Mean[ y] Mean of y Question: What would happen to the value of the covariance if we measured precipitation in centimeters rather than inches? Letting x t represent precipitation: x t s up by a x up by a (x t - x ) s up by a Cov[x, y] up by a

9 9 Correlation Coefficient: CorrCoef[ x, y] Var[ x] Var[ y] Similarity between the Covariance and Correlation Coefficient he sign of covariance and the sign of the correlation coefficient are. Differences between the Covariance and the Correlation Coefficient Correlation Coefficient Is Unaffected by the Choice of Units Again, suppose that we measure rainfall in centimeters rather than inches: (x t - x ) s up by a Cov[x, y] = Error! Var[ x ] t1 (x t - x) 2 s up by a 2 ( xt x) Var[x] up by Var[x] up by Cov[x, y] up by CorrCoef[ x, y] Var[ x] Var[ y] CorrCoef[x, y] is.

10 1 Correlation Coefficient Has a Natural Range: 1 to +1 Unlike covariance, the correlation coefficient has a limited range; the correlation coefficient must lie between 1 and +1. Perfect Negative Correlation Independent Perfect Positive Correlation CorrCoef 1 +1 Example of Perfect Positive Correlation: y t = x t for each i = 1, 2,, N y = x y t y = x t x for each i = 1, 2,, N (y t y ) 2 = (x t x )(y t y ) = = ( yt y) 2 Var[y] = ( xt x)( yt y) Cov[x, y] = CorrCoef[ x, y] Var[ x] Var[ y ] = = = Example of Perfect Negative Correlation: y t = x t for each i = 1, 2,, N y = x y t y = ( x t x) for each i = 1, 2,, N (y t y ) 2 = (x t x )(y t y ) = = ( yt y) 2 Var[y] = ( xt x)( yt y) Cov[x, y] = CorrCoef[ x, y] Var[ x] Var[ y ] = = =

11 11 Statistical software allows us to calculate the correlation coefficient easily: Stock Market CorrCoef[Dow Jones Growth Rate, NASDAQ Growth Rate] CorrCoef[NASDAQ Growth Rate, Amherst Precipitation] = = Scatter Diagrams Deviations from Means Deviations From Means 2 Nasdaq Deviations From Means 2 Nasdaq 1 1 Dow Jones Precipitation Not Independent: (Positively) Correlated Independent: Uncorrelated Knowing the value of one variable Knowing the value of one variable Help us predict the value of the other does not help us predict the value of the other Cov = Cov =.91 CorrCoef = CorrCoef = Correlation and Causation: Correlation does not necessarily imply causation. Arithmetic of Means, Variances, and Covariances Mean of the sum of a constant and a variable: Mean[c + x] = c + Mean[x] Mean of the product of a constant and a variable: Mean[cx] = c Mean[x] Mean of the sum of two variables: Mean[x + y] = Mean[x] + Mean[y] Variance of the sum of a constant and a variable: Var[c + x] = Var[x] Variance of the product of a constant and a variable: Var[cx] = c 2 Var[x] Variance of the sum of two variables: Var[x + y] = Var[x] + 2Cov[x, y] + Var[y] Variance of the sum of two independent variables: Var[x + y] = Var[x] + Var[y] Covariance of the sum of a constant and a variable: Cov[c + x, y] = Cov[x, y] Covariance of the product of a constant and a variable: Cov[cx, y] = ccov[x, y]

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