Anscombe_Quartet
Anscombe’s quartet
Vinod Basalalli
2017-04-28
Anscombes quartet is a set of 4 \(x,y\) data sets that were published by Francis Anscombe in a 1973 paper [Graphs in statistical analysis]
(https://www.jstor.org/stable/2682899?seq=1#page_scan_tab_contents).
## x1 x2 x3 x4 y1 y2 y3 y4
## 1 10 10 10 8 8.04 9.14 7.46 6.58
## 2 8 8 8 8 6.95 8.14 6.77 5.76
## 3 13 13 13 8 7.58 8.74 12.74 7.71
## 4 9 9 9 8 8.81 8.77 7.11 8.84
## 5 11 11 11 8 8.33 9.26 7.81 8.47
## 6 14 14 14 8 9.96 8.10 8.84 7.04
## 7 6 6 6 8 7.24 6.13 6.08 5.25
## 8 4 4 4 19 4.26 3.10 5.39 12.50
## 9 12 12 12 8 10.84 9.13 8.15 5.56
## 10 7 7 7 8 4.82 7.26 6.42 7.91
## 11 5 5 5 8 5.68 4.74 5.73 6.89Calculating the mean, variance, for each column and the correlation between each pair (eg. x1 and y1, x2 and y2, etc)
## x1 x2 x3 x4 y1 y2
## nobs 11.000000 11.000000 11.000000 11.000000 11.000000 11.000000
## NAs 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
## Minimum 4.000000 4.000000 4.000000 8.000000 4.260000 3.100000
## Maximum 14.000000 14.000000 14.000000 19.000000 10.840000 9.260000
## 1. Quartile 6.500000 6.500000 6.500000 8.000000 6.315000 6.695000
## 3. Quartile 11.500000 11.500000 11.500000 8.000000 8.570000 8.950000
## Mean 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909
## Median 9.000000 9.000000 9.000000 8.000000 7.580000 8.140000
## Sum 99.000000 99.000000 99.000000 99.000000 82.510000 82.510000
## SE Mean 1.000000 1.000000 1.000000 1.000000 0.612541 0.612568
## LCL Mean 6.771861 6.771861 6.771861 6.771861 6.136083 6.136024
## UCL Mean 11.228139 11.228139 11.228139 11.228139 8.865735 8.865795
## Variance 11.000000 11.000000 11.000000 11.000000 4.127269 4.127629
## Stdev 3.316625 3.316625 3.316625 3.316625 2.031568 2.031657
## Skewness 0.000000 0.000000 0.000000 2.466911 -0.048374 -0.978693
## Kurtosis -1.528926 -1.528926 -1.528926 4.520661 -1.199123 -0.514319
## y3 y4
## nobs 11.000000 11.000000
## NAs 0.000000 0.000000
## Minimum 5.390000 5.250000
## Maximum 12.740000 12.500000
## 1. Quartile 6.250000 6.170000
## 3. Quartile 7.980000 8.190000
## Mean 7.500000 7.500909
## Median 7.110000 7.040000
## Sum 82.500000 82.510000
## SE Mean 0.612196 0.612242
## LCL Mean 6.135943 6.136748
## UCL Mean 8.864057 8.865070
## Variance 4.122620 4.123249
## Stdev 2.030424 2.030579
## Skewness 1.380120 1.120774
## Kurtosis 1.240044 0.628751## x1 x2 x3 x4 y1 y2 y3 y4
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909## x1 x2 x3 x4 y1 y2 y3
## 11.000000 11.000000 11.000000 11.000000 4.127269 4.127629 4.122620
## y4
## 4.123249##
## Title:
## Pearson's Correlation Test
##
## Test Results:
## PARAMETER:
## Degrees of Freedom: 9
## SAMPLE ESTIMATES:
## Correlation: 0.8164
## STATISTIC:
## t: 4.2415
## P VALUE:
## Alternative Two-Sided: 0.00217
## Alternative Less: 0.9989
## Alternative Greater: 0.001085
## CONFIDENCE INTERVAL:
## Two-Sided: 0.4244, 0.9507
## Less: -1, 0.9388
## Greater: 0.5113, 1
##
## Description:
## Fri Apr 28 16:04:01 2017##
## Title:
## Pearson's Correlation Test
##
## Test Results:
## PARAMETER:
## Degrees of Freedom: 9
## SAMPLE ESTIMATES:
## Correlation: 0.8162
## STATISTIC:
## t: 4.2386
## P VALUE:
## Alternative Two-Sided: 0.002179
## Alternative Less: 0.9989
## Alternative Greater: 0.001089
## CONFIDENCE INTERVAL:
## Two-Sided: 0.4239, 0.9506
## Less: -1, 0.9387
## Greater: 0.5109, 1
##
## Description:
## Fri Apr 28 16:04:01 2017##
## Title:
## Pearson's Correlation Test
##
## Test Results:
## PARAMETER:
## Degrees of Freedom: 9
## SAMPLE ESTIMATES:
## Correlation: 0.8163
## STATISTIC:
## t: 4.2394
## P VALUE:
## Alternative Two-Sided: 0.002176
## Alternative Less: 0.9989
## Alternative Greater: 0.001088
## CONFIDENCE INTERVAL:
## Two-Sided: 0.4241, 0.9507
## Less: -1, 0.9387
## Greater: 0.511, 1
##
## Description:
## Fri Apr 28 16:04:01 2017##
## Title:
## Pearson's Correlation Test
##
## Test Results:
## PARAMETER:
## Degrees of Freedom: 9
## SAMPLE ESTIMATES:
## Correlation: 0.8165
## STATISTIC:
## t: 4.243
## P VALUE:
## Alternative Two-Sided: 0.002165
## Alternative Less: 0.9989
## Alternative Greater: 0.001082
## CONFIDENCE INTERVAL:
## Two-Sided: 0.4246, 0.9507
## Less: -1, 0.9388
## Greater: 0.5115, 1
##
## Description:
## Fri Apr 28 16:04:01 2017## x1 x2 x3 x4
## Min. : 4.0 Min. : 4.0 Min. : 4.0 Min. : 8
## 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 8
## Median : 9.0 Median : 9.0 Median : 9.0 Median : 8
## Mean : 9.0 Mean : 9.0 Mean : 9.0 Mean : 9
## 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.: 8
## Max. :14.0 Max. :14.0 Max. :14.0 Max. :19
## y1 y2 y3 y4
## Min. : 4.260 Min. :3.100 Min. : 5.39 Min. : 5.250
## 1st Qu.: 6.315 1st Qu.:6.695 1st Qu.: 6.25 1st Qu.: 6.170
## Median : 7.580 Median :8.140 Median : 7.11 Median : 7.040
## Mean : 7.501 Mean :7.501 Mean : 7.50 Mean : 7.501
## 3rd Qu.: 8.570 3rd Qu.:8.950 3rd Qu.: 7.98 3rd Qu.: 8.190
## Max. :10.840 Max. :9.260 Max. :12.74 Max. :12.500Scatter plots for each \(x, y\) pair of data.
Changing the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
Fitting a linear model to each data set using the lm() function.
Combining the last two tasks.
Comparing the model fits for each model object.
Analysis of Variance Table
Response: data\(x1 Df Sum Sq Mean Sq F value Pr(>F) data\)y1 1 73.32 73.320 17.99 0.00217 ** Residuals 9 36.68 4.076
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1 
Analysis of Variance Table
Response: data\(x2 Df Sum Sq Mean Sq F value Pr(>F) data\)y2 1 73.287 73.287 17.966 0.002179 ** Residuals 9 36.713 4.079
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1 
Analysis of Variance Table
Response: data\(x3 Df Sum Sq Mean Sq F value Pr(>F) data\)y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1 
Analysis of Variance Table
Response: data\(x4 Df Sum Sq Mean Sq F value Pr(>F) data\)y4 1 73.338 73.338 18.003 0.002165 ** Residuals 9 36.662 4.074
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1 
