Ahead of the 2012 election, most polls were favoring Obama very slightly, but they were also all well within the margin of error indicating that Obama was not statistically likely to win, and the poll results may be due to random chance. However, political statistician Nate Silver. confidently stated that he believed Obama would win the election and that his claims were statistically backed by the same polls that were inconclusive. We will examine Silver’s methods and how he came to his conclusions.
url <- 'https://en.wikipedia.org/wiki/Nationwide_opinion_polling_for_the_2012_United_States_presidential_election#Two-way_race'
xpath <- '//*[@id="mw-content-text"]/div[1]/table[2]'
table <- url %>% read_html() %>% html_nodes(xpath = xpath) %>% html_table(fill = T)
table <- table[[1]]
table <- table[2:nrow(table),]
knitr::kable(table)| Poll source | Date | Barack Obama.mw-parser-output .nobold{font-weight:normal}Democratic | Mitt RomneyRepublican | Leading by % | Sample Size* | Margin of Error (MoE) | |
|---|---|---|---|---|---|---|---|
| 2 | Rasmussen Tracking | November 2–4, 2012 | 48% | 49% | 1 | 1,500 LV | ±3.0 |
| 3 | CNN/Opinion Research | November 2–4, 2012 | 50% | 48% | 2 | 918 RV | ±3% |
| 4 | CNN/Opinion Research | November 2–4, 2012 | 49% | 49% | Tie | 693 LV | ±3.5% |
| 5 | Gallup Tracking | November 1–4, 2012 | 49% | 46% | 3 | 2,854 RV | ±2% |
| 6 | Gallup Tracking | November 1–4, 2012 | 49% | 50% | 1 | 2,551 LV | ±2% |
| 7 | Democracy Corps | November 1–4, 2012 | 49% | 45% | 4 | 1,080 LV | ±3.0% |
| 8 | Ipsos/Reuters | October 31–November 4, 2012 | 48% | 42% | 6 | 5,158 RV | ±3.0% |
| 9 | Ipsos/Reuters | October 31–November 4, 2012 | 48% | 47% | 1 | 3,805 LV | ±3.4% |
| 10 | Angus Reid Public Opinion[permanent dead link] | November 1–3, 2012 | 51% | 48% | 3 | 1,050 LV | ±3.1% |
| 11 | PPP/Americans United for Change | November 1–3, 2012 | 50% | 47% | 3 | 1,200 LV | ±2.8% |
| 12 | NBC News/Wall Street Journal | November 1–3, 2012 | 48% | 47% | 1 | 1,475 LV | ±2.6% |
| 13 | Rasmussen Tracking | November 1–3, 2012 | 49% | 49% | Tie | 3,000(?) LV | ±2.5% |
| 14 | Pew Research | October 31–November 3, 2012 | 49% | 42% | 7 | 3,151 RV | ±2.0% |
| 15 | Pew Research | October 31–November 3, 2012 | 48% | 45% | 3 | 2,709 LV | ±2.2% |
| 16 | YouGov | October 31–November 3, 2012 | 49% | 47% | 2 | 36,472 LV | ? |
| 17 | ABC News/Washington Post | October 31–November 3, 2012 | 49% | 48% | 1 | 2,069 LV | ±2.5% |
| 18 | PPP/Americans United for Change | October 30– November 1, 2012 | 49% | 48% | 1 | 1,200 LV | ±2.8% |
| 19 | Politico/George Washington University/Battleground | October 29–November 1, 2012 | 48% | 49% | 1 | 1,000 LV | ±3.1% |
| 20 | Washington Times/JZ Analytics | October 29–31, 2012 | 49% | 49% | Tie | 800 LV | ±3.5% |
| 21 | ABC News/Wash Post | October 28–31, 2012 | 49% | 48% | 1 | 1,293 LV | ±3% |
| 22 | UPI/CVOTER | October 15–31, 2012 | 49% | 48% | 1 | 3,633 LV | ±3.5% (?) |
| 23 | Fox News | October 28–30, 2012 | 46% | 45% | 1 | 1,230 RV | ±3% |
| 24 | Fox News | October 28–30, 2012 | 46% | 46% | Tie | 1,128 LV | ±3% |
| 25 | JZAnalytics | October 28–30, 2012 | 45% | 48% | 3 | 1,015 LV | ±3.1% |
| 26 | High Point University | October 22–30, 2012 | 46% | 43% | 3 | 805 RV | ±3.5% |
| 27 | Rasmussen Tracking | October 27–29, 2012 | 47% | 49% | 2 | 1,500 LV | ±3.0% |
| 28 | YouGov/The Economist | October 26–28, 2012 | 48% | 46% | 2 | 757 RV | ±4.6% |
| 29 | YouGov/The Economist | October 26–28, 2012 | 48% | 47% | 1 | 688 LV | ±4.6% |
| 30 | United Technologies/National Journal | October 25–28, 2012 | 50% | 45% | 5 | 713 LV | ±4.4% |
| 31 | CBS News/New York Times | October 25–28, 2012 | 48% | 47% | 1 | 563 LV | ±4.0% |
| 32 | ABC News/Washington Post | October 25–28, 2012 | 49% | 49% | Tie | 1,259 LV | ±3.5% |
| 33 | DailyKos/PPP/SEIU | October 25–28, 2012 | 49% | 49% | Tie | 1,400 LV | ±2.6% |
| 34 | Pew Research | October 24–28, 2012 | 47% | 45% | 2 | 1,678 RV | ±2.8% |
| 35 | Pew Research | October 24–28, 2012 | 47% | 47% | Tie | 1,495 LV | ±2.9% |
| 36 | Ipsos/Reuters | October 24–28, 2012 | 51% | 39% | 12 | 1,133 RV | ±3.3% |
| 37 | Ipsos/Reuters | October 24–28, 2012 | 49% | 46% | 3 | 795 LV | ±4.0% |
| 38 | Gallup Tracking | October 22–28, 2012 | 48% | 48% | Tie | 3,050 RV | ±2% |
| 39 | Gallup Tracking | October 22–28, 2012 | 46% | 51% | 5 | 2,700 LV | ±2% |
| 40 | Investor’s Business Daily/TIPP | October 22–27, 2012 | 45% | 44% | 1 | 942 LV | ±3.5% |
| 41 | National Public Radio | October 23–25, 2012 | 47% | 48% | 1 | 1,000 LV | ±3.1% |
| 42 | Politico/George Washington University/Battleground | October 22–25, 2012 | 49% | 48% | 1 | 1,000 LV | ±3.1% |
| 43 | PPP/Americans United for Change | October 22–24, 2012 | 49% | 48% | 1 | 1,200 LV | ±2.8% |
| 44 | ABC News/Washington Post | October 21–24, 2012 | 47% | 50% | 3 | 1,386 LV | ±3.0% |
| 45 | Associated Press/GFK | October 19–23, 2012 | 48% | 40% | 8 | 1,186 adults | ±3.5% |
| 46 | Associated Press/GFK | October 19–23, 2012 | 45% | 44% | 1 | 1,041 RV | ±3.5% |
| 47 | Associated Press/GFK | October 19–23, 2012 | 45% | 47% | 2 | 839 LV | ±4.2% |
| 48 | IBD/TIPP | October 17–22, 2012 | 47% | 45% | 2 | 938 LV | ±3.5% |
| 49 | Rasmussen Tracking | October 20–22, 2012 | 46% | 50% | 4 | 1,500 LV | ±3% |
| 50 | ABC News/Wash Post | October 18–21, 2012 | 49% | 48% | 1 | 1,376 LV | ±3% |
| 51 | Monmouth/SurveyUSA/Braun | October 18–21, 2012 | 45% | 48% | 3 | 1,402 LV | ±2.6% |
| 52 | Gallup Tracking | October 14–20, 2012 | 45% | 52% | 7 | 2,700 LV | ±2% |
| 53 | Gallup Tracking | October 14–20, 2012 | 46% | 49% | 3 | 3,050 RV | ±2% |
| 54 | Washington Times/JZ Analytics | October 18–20, 2012 | 50% | 47% | 3 | 800 LV | ±3.5% |
| 55 | CBS News | October 17–20, 2012 | 48% | 46% | 2 | 790 LV | ±4% |
| 56 | NBC News/Wall Street Journal | October 17–20, 2012 | 47% | 47% | Tie | 816 LV | ±3.4% |
| 57 | NBC News/Wall Street Journal | October 17–20, 2012 | 49% | 44% | 5 | 1,000 RV | ±3.1% |
| 58 | PPP/Americans United for Change | October 17–19, 2012 | 49% | 47% | 2 | 1,200 LV | ±2.8% |
| 59 | Politico/George Washington University/Battleground | October 15–18, 2012 | 47% | 49% | 2 | 1,000 LV | ±3.1% |
| 60 | UConn/Hartford Courant | October 11–16, 2012 | 48% | 45% | 3 | 1,023 LV | ±3% |
| 61 | Rasmussen Reports | October 13–15, 2012 | 47% | 49% | 2 | 1,500 LV | ±3.0% |
| 62 | IBD/TIPP Tracking | October 10–15, 2012 | 47% | 45% | 2 | 931 LV | ±3.5% |
| 63 | Gallup Tracking | October 9–15, 2012 | 46% | 50% | 4 | 2,700 LV | ±2.0% |
| 64 | Washington Post-ABC News | October 10–13, 2012 | 49% | 46% | 3 | 923 LV | ±3.5% |
| 65 | Washington Post-ABC News | October 10–13, 2012 | 50% | 43 | 7 | 1,063 RV | ±3.5% |
| 66 | Angus Reid Public Opinion[permanent dead link] | October 10–11, 2012 | 47% | 47% | Tie | 906 RV | ±3.5% |
| 67 | Politico/George Washington University/Battleground | October 7–11, 2012 | 49% | 48% | 1 | 1,000 LV | ±3.1% |
| 68 | Reuters/Ipsos | October 7–11, 2012 | 44% | 47% | 3 | 1,092 LV | ±3.4% |
| 69 | Fox News | October 7–9, 2012 | 46% | 44% | 2 | 1,204 RV | ±3% |
| 70 | Fox News | October 7–9, 2012 | 45% | 46% | 1 | 1,109 LV | ±3% |
| 71 | Zogby / JZAnalytics | October 5–7, 2012 | 45% | 45% | Tie | 800 LV | ±3.5% |
| 72 | Investor’s Business Daily/TIPP | October 4–9, 2012 | 46% | 47% | 1 | 812 LV | ±3.5% |
| 73 | Pew Research Center | October 4–7, 2012 | 46% | 46% | Tie | 1,201 RV | ±3.3% |
| 74 | Pew Research Center | October 4–7, 2012 | 45% | 49% | 4 | 1,112 LV | ±3.4% |
| 75 | Gallup Tracking | October 4–6, 2012 | 47% | 47% | Tie | 1,387 RV | ±3.0% |
| 76 | PPP/Daily Kos & SEIU | October 4–7, 2012 | 47% | 49% | 2 | 1,300 LV | ±2.7% |
| 77 | Rasmussen Reports | October 4–7, 2012 | 47% | 49% | 2 | 1,500 LV | ±3.0% |
| 78 | CNN/Opinion Research | September 28–30, 2012 | 50% | 46% | 4 | 883 RV | ±3.5% |
| 79 | CNN/Opinion Research | September 28–30, 2012 | 50% | 47% | 3 | 783 LV | ±3.5% |
| 80 | PPP/Daily Kos & SEIU | September 27–30, 2012 | 49% | 45% | 4 | 1,100 LV | ±2.8% |
| 81 | 47% | 47% | Tie | 1,005 LV | ±3.7% | ||
| 82 | NBC/Wall Street Journal | September 26–30, 2012 | 51% | 44% | 7 | 1,000 RV | ±3.10% |
| 83 | NBC/Wall Street Journal | September 26–30, 2012 | 49% | 46% | 3 | 832 LV | ±3.40% |
| 84 | Quinnipiac University | September 25–30, 2012 | 49% | 45% | 4 | 1,912 LV | ±2.2% |
| 85 | ABC News/Wash Post | September 26–29, 2012 | 49% | 44% | 5 | 929 RV | ±3.5% |
| 86 | ABC News/Wash Post | September 26–29, 2012 | 49% | 47% | 2 | 813 LV | ±4% |
| 87 | Politico/George Washington University/Battleground | September 24–27, 2012 | 50% | 47% | 3 | 1,000 LV | ±3.1% |
| 88 | Fox News | September 26–29, 2012 | 49% | 41% | 8 | 1,208 RV | ±3% |
| 89 | Fox News | September 26–29, 2012 | 48% | 43% | 5 | 1,092 LV | ±3% |
| 90 | Rasmussen Tracking | September 23–25, 2012 | 46% | 46% | Tie | 1,500 LV | ±3.0% |
| 91 | Gallup Tracking | September 19–25, 2012 | 50% | 44% | 6 | 3,050 RV | ±2.0% |
| 92 | Bloomberg | September 21–24, 2012 | 49% | 43% | 6 | 789 LV | ±3.5% |
| 93 | Reuters/Ipsos | September 20–24, 2012 | 49% | 43% | 6 | 1,313 RV | ±3.1% |
| 94 | PPP/Daily Kos & SEIU | September 20–23, 2012 | 50% | 45% | 5 | 1,200 LV | ±2.8% |
| 95 | Zogby / JZAnalytics | September 21–22, 2012 | 49% | 41% | 8 | 860 LV | ±3.4% |
| 96 | Politico/George Washington University | September 16–20, 2012 | 50% | 46% | 4 | 1,000 LV | ±3.1% |
| 97 | Angus Reid Public Opinion[permanent dead link] | September 19–20, 2012 | 48% | 46% | 2 | 865 RV | ±3.5% |
| 98 | Allstate/National Journal | September 15–19, 2012 | 50% | 43% | 7 | 1,055 LV | ±3% |
| 99 | UConn/Hartford Courant | September 11–18, 2012 | 46% | 43% | 3 | 1,148 LV | ±3% |
| 100 | Reason-Rupe | September 13–17, 2012 | 52% | 45% | 7 | 787 LV | ±4.3% |
| 101 | Associated Press/GfK Roper | September 13–17, 2012 | 52% | 37% | 15 | 1,512 adults | ±3.2% |
| 102 | Associated Press/GfK Roper | September 13–17, 2012 | 50% | 40% | 10 | 1,282 RV | ±3.4% |
| 103 | Associated Press/GfK Roper | September 13–17, 2012 | 47% | 46% | 1 | 807 LV | ±4.3% |
| 104 | Ipsos/Reuters | September 12–17, 2012 | 46% | 41% | 5 | 737 RV | ±4.1% |
| 105 | Ipsos/Reuters | September 12–17, 2012 | 48% | 43% | 5 | 591 LV | ±4.1% |
| 106 | Monmouth University | September 13–16, 2012 | 48% | 41% | 7 | 1,571 RV | ±2.5% |
| 107 | Monmouth University | September 13–16, 2012 | 48% | 45% | 3 | 1,344 LV | ±2.5% |
| 108 | Pew Research Center | September 12–16, 2012 | 51% | 42% | 9 | 2,424 RV | ±2.3% |
| 109 | Pew Research Center | September 12–16, 2012 | 51% | 43% | 8 | 2,192 LV | ±2.4% |
| 110 | NBC/Wall Street Journal | September 12–16, 2012 | 50% | 44% | 6 | 900 RV | ±3.27% |
| 111 | NBC/Wall Street Journal | September 12–16, 2012 | 50% | 45% | 5 | 736 LV | ±3.61% |
| 112 | Zogby / JZAnalytics | September 11–12, 2012 | 47.7% | 42% | 5.7 | 1,014 LV | ±3.1% |
| 113 | UPI/CVOTER | September 8–14, 2012 | 49% | 45% | 4 | 3,000 LV | ±3% |
| 114 | CBS/New York Times | September 8–12, 2012 | 51% | 43% | 8 | 1,170 RV | ±3% |
| 115 | CBS/New York Times | September 8–12, 2012 | 49% | 46% | 3 | 1,162 LV | ±3% |
| 116 | Democracy Corps | September 8–12, 2012 | 50% | 45% | 5 | 1,000 LV | ±3.1% |
| 117 | Fox News | September 9–11, 2012 | 46% | 42% | 4 | 1,224 RV | ±3% |
| 118 | Fox News | September 9–11, 2012 | 48% | 43% | 5 | 1,056 LV | ±3% |
| 119 | Rasmussen Tracking | September 8–11, 2012 | 46% | 45% | 1 | 1,500 LV | ±3.0% |
| 120 | Gallup Tracking | September 6–12, 2012 | 50% | 44% | 6 | 3,050 RV | ±2.0% |
| 121 | YouGov/The Economist | September 8–10, 2012 | 49% | 45% | 4 | ≤1,000 RV | ±3.9% |
| 122 | Esquire/Yahoo! News | September 7–10, 2012 | 52% | 41% | 11 | ≤1,002 RV | ±4.5% |
| 123 | Esquire/Yahoo! News | September 7–10, 2012 | 50% | 46% | 4 | ≤1,002 LV | ±5% |
| 124 | Daily Kos/PPP | September 7–9, 2012 | 50% | 44% | 6 | 1,000 LV | ±3.1% |
| 125 | ABC/Washington Post | September 7–9, 2012 | 50% | 44% | 6 | 826 RV | ±4% |
| 126 | ABC/Washington Post | September 7–9, 2012 | 49% | 48% | 1 | 710 LV | ±4.5% |
| 127 | CNN/Opinion Research | September 7–9, 2012 | 53% | 45% | 8 | 875 RV | ±3.5% |
| 128 | CNN/Opinion Research | September 7–9, 2012 | 52% | 46% | 6 | 709 LV | ±3.5% |
| 129 | Ipsos/Reuters Tracking | September 6–10, 2012 | 45% | 41% | 4 | 1,434 RV | ±3.0% |
| 130 | Ipsos/Reuters Tracking | September 6–10, 2012 | 48% | 43% | 5 | 1,182 LV | ±3.0% |
Above we see a list of polls conducted, and we see how although Obama leads most of the polls leading up to the election, his lead is slight and usually within the margin of error. However, Nate Silver cleverly used aggregate polling, a method that involves combining many polls and treating them as one poll, to improve the certainty of his prediction. The increased sample size led to a smaller margin of error and a more confident prediction.
names(table)[3] <- 'Obama'
names(table)[4] <- 'Romney'
names(table)[6] <- 'Sample_Size'
clean_table <- table[3:ncol(table)] %>% apply(2, str_replace_all, 'Tie', '0') %>% apply(2, str_replace_all, '[^(0-9\\.)]', '') %>% as.data.frame() %>% mutate(Obama_prop = as.numeric(as.character(Obama))/100) %>% mutate(Romney_prop = as.numeric(as.character(Romney))/100) %>% mutate(Obama_num = Obama_prop * as.numeric(as.character(Sample_Size))) %>% mutate(Romney_num = Romney_prop * as.numeric(as.character(Sample_Size)))## Warning: NAs introduced by coercion
## Warning: NAs introduced by coercionknitr::kable(clean_table)| Obama | Romney | Leading by % | Sample_Size | Margin of Error (MoE) | Obama_prop | Romney_prop | Obama_num | Romney_num |
|---|---|---|---|---|---|---|---|---|
| 48 | 49 | 1 | 1500 | 3.0 | 0.480 | 0.49 | 720.000 | 735.00 |
| 50 | 48 | 2 | 918 | 3 | 0.500 | 0.48 | 459.000 | 440.64 |
| 49 | 49 | 0 | 693 | 3.5 | 0.490 | 0.49 | 339.570 | 339.57 |
| 49 | 46 | 3 | 2854 | 2 | 0.490 | 0.46 | 1398.460 | 1312.84 |
| 49 | 50 | 1 | 2551 | 2 | 0.490 | 0.50 | 1249.990 | 1275.50 |
| 49 | 45 | 4 | 1080 | 3.0 | 0.490 | 0.45 | 529.200 | 486.00 |
| 48 | 42 | 6 | 5158 | 3.0 | 0.480 | 0.42 | 2475.840 | 2166.36 |
| 48 | 47 | 1 | 3805 | 3.4 | 0.480 | 0.47 | 1826.400 | 1788.35 |
| 51 | 48 | 3 | 1050 | 3.1 | 0.510 | 0.48 | 535.500 | 504.00 |
| 50 | 47 | 3 | 1200 | 2.8 | 0.500 | 0.47 | 600.000 | 564.00 |
| 48 | 47 | 1 | 1475 | 2.6 | 0.480 | 0.47 | 708.000 | 693.25 |
| 49 | 49 | 0 | 3000() | 2.5 | 0.490 | 0.49 | NA | NA |
| 49 | 42 | 7 | 3151 | 2.0 | 0.490 | 0.42 | 1543.990 | 1323.42 |
| 48 | 45 | 3 | 2709 | 2.2 | 0.480 | 0.45 | 1300.320 | 1219.05 |
| 49 | 47 | 2 | 36472 | 0.490 | 0.47 | 17871.280 | 17141.84 | |
| 49 | 48 | 1 | 2069 | 2.5 | 0.490 | 0.48 | 1013.810 | 993.12 |
| 49 | 48 | 1 | 1200 | 2.8 | 0.490 | 0.48 | 588.000 | 576.00 |
| 48 | 49 | 1 | 1000 | 3.1 | 0.480 | 0.49 | 480.000 | 490.00 |
| 49 | 49 | 0 | 800 | 3.5 | 0.490 | 0.49 | 392.000 | 392.00 |
| 49 | 48 | 1 | 1293 | 3 | 0.490 | 0.48 | 633.570 | 620.64 |
| 49 | 48 | 1 | 3633 | 3.5() | 0.490 | 0.48 | 1780.170 | 1743.84 |
| 46 | 45 | 1 | 1230 | 3 | 0.460 | 0.45 | 565.800 | 553.50 |
| 46 | 46 | 0 | 1128 | 3 | 0.460 | 0.46 | 518.880 | 518.88 |
| 45 | 48 | 3 | 1015 | 3.1 | 0.450 | 0.48 | 456.750 | 487.20 |
| 46 | 43 | 3 | 805 | 3.5 | 0.460 | 0.43 | 370.300 | 346.15 |
| 47 | 49 | 2 | 1500 | 3.0 | 0.470 | 0.49 | 705.000 | 735.00 |
| 48 | 46 | 2 | 757 | 4.6 | 0.480 | 0.46 | 363.360 | 348.22 |
| 48 | 47 | 1 | 688 | 4.6 | 0.480 | 0.47 | 330.240 | 323.36 |
| 50 | 45 | 5 | 713 | 4.4 | 0.500 | 0.45 | 356.500 | 320.85 |
| 48 | 47 | 1 | 563 | 4.0 | 0.480 | 0.47 | 270.240 | 264.61 |
| 49 | 49 | 0 | 1259 | 3.5 | 0.490 | 0.49 | 616.910 | 616.91 |
| 49 | 49 | 0 | 1400 | 2.6 | 0.490 | 0.49 | 686.000 | 686.00 |
| 47 | 45 | 2 | 1678 | 2.8 | 0.470 | 0.45 | 788.660 | 755.10 |
| 47 | 47 | 0 | 1495 | 2.9 | 0.470 | 0.47 | 702.650 | 702.65 |
| 51 | 39 | 12 | 1133 | 3.3 | 0.510 | 0.39 | 577.830 | 441.87 |
| 49 | 46 | 3 | 795 | 4.0 | 0.490 | 0.46 | 389.550 | 365.70 |
| 48 | 48 | 0 | 3050 | 2 | 0.480 | 0.48 | 1464.000 | 1464.00 |
| 46 | 51 | 5 | 2700 | 2 | 0.460 | 0.51 | 1242.000 | 1377.00 |
| 45 | 44 | 1 | 942 | 3.5 | 0.450 | 0.44 | 423.900 | 414.48 |
| 47 | 48 | 1 | 1000 | 3.1 | 0.470 | 0.48 | 470.000 | 480.00 |
| 49 | 48 | 1 | 1000 | 3.1 | 0.490 | 0.48 | 490.000 | 480.00 |
| 49 | 48 | 1 | 1200 | 2.8 | 0.490 | 0.48 | 588.000 | 576.00 |
| 47 | 50 | 3 | 1386 | 3.0 | 0.470 | 0.50 | 651.420 | 693.00 |
| 48 | 40 | 8 | 1186 | 3.5 | 0.480 | 0.40 | 569.280 | 474.40 |
| 45 | 44 | 1 | 1041 | 3.5 | 0.450 | 0.44 | 468.450 | 458.04 |
| 45 | 47 | 2 | 839 | 4.2 | 0.450 | 0.47 | 377.550 | 394.33 |
| 47 | 45 | 2 | 938 | 3.5 | 0.470 | 0.45 | 440.860 | 422.10 |
| 46 | 50 | 4 | 1500 | 3 | 0.460 | 0.50 | 690.000 | 750.00 |
| 49 | 48 | 1 | 1376 | 3 | 0.490 | 0.48 | 674.240 | 660.48 |
| 45 | 48 | 3 | 1402 | 2.6 | 0.450 | 0.48 | 630.900 | 672.96 |
| 45 | 52 | 7 | 2700 | 2 | 0.450 | 0.52 | 1215.000 | 1404.00 |
| 46 | 49 | 3 | 3050 | 2 | 0.460 | 0.49 | 1403.000 | 1494.50 |
| 50 | 47 | 3 | 800 | 3.5 | 0.500 | 0.47 | 400.000 | 376.00 |
| 48 | 46 | 2 | 790 | 4 | 0.480 | 0.46 | 379.200 | 363.40 |
| 47 | 47 | 0 | 816 | 3.4 | 0.470 | 0.47 | 383.520 | 383.52 |
| 49 | 44 | 5 | 1000 | 3.1 | 0.490 | 0.44 | 490.000 | 440.00 |
| 49 | 47 | 2 | 1200 | 2.8 | 0.490 | 0.47 | 588.000 | 564.00 |
| 47 | 49 | 2 | 1000 | 3.1 | 0.470 | 0.49 | 470.000 | 490.00 |
| 48 | 45 | 3 | 1023 | 3 | 0.480 | 0.45 | 491.040 | 460.35 |
| 47 | 49 | 2 | 1500 | 3.0 | 0.470 | 0.49 | 705.000 | 735.00 |
| 47 | 45 | 2 | 931 | 3.5 | 0.470 | 0.45 | 437.570 | 418.95 |
| 46 | 50 | 4 | 2700 | 2.0 | 0.460 | 0.50 | 1242.000 | 1350.00 |
| 49 | 46 | 3 | 923 | 3.5 | 0.490 | 0.46 | 452.270 | 424.58 |
| 50 | 43 | 7 | 1063 | 3.5 | 0.500 | 0.43 | 531.500 | 457.09 |
| 47 | 47 | 0 | 906 | 3.5 | 0.470 | 0.47 | 425.820 | 425.82 |
| 49 | 48 | 1 | 1000 | 3.1 | 0.490 | 0.48 | 490.000 | 480.00 |
| 44 | 47 | 3 | 1092 | 3.4 | 0.440 | 0.47 | 480.480 | 513.24 |
| 46 | 44 | 2 | 1204 | 3 | 0.460 | 0.44 | 553.840 | 529.76 |
| 45 | 46 | 1 | 1109 | 3 | 0.450 | 0.46 | 499.050 | 510.14 |
| 45 | 45 | 0 | 800 | 3.5 | 0.450 | 0.45 | 360.000 | 360.00 |
| 46 | 47 | 1 | 812 | 3.5 | 0.460 | 0.47 | 373.520 | 381.64 |
| 46 | 46 | 0 | 1201 | 3.3 | 0.460 | 0.46 | 552.460 | 552.46 |
| 45 | 49 | 4 | 1112 | 3.4 | 0.450 | 0.49 | 500.400 | 544.88 |
| 47 | 47 | 0 | 1387 | 3.0 | 0.470 | 0.47 | 651.890 | 651.89 |
| 47 | 49 | 2 | 1300 | 2.7 | 0.470 | 0.49 | 611.000 | 637.00 |
| 47 | 49 | 2 | 1500 | 3.0 | 0.470 | 0.49 | 705.000 | 735.00 |
| 50 | 46 | 4 | 883 | 3.5 | 0.500 | 0.46 | 441.500 | 406.18 |
| 50 | 47 | 3 | 783 | 3.5 | 0.500 | 0.47 | 391.500 | 368.01 |
| 49 | 45 | 4 | 1100 | 2.8 | 0.490 | 0.45 | 539.000 | 495.00 |
| 47 | 47 | 0 | 1005 | 3.7 | 0.470 | 0.47 | 472.350 | 472.35 |
| 51 | 44 | 7 | 1000 | 3.10 | 0.510 | 0.44 | 510.000 | 440.00 |
| 49 | 46 | 3 | 832 | 3.40 | 0.490 | 0.46 | 407.680 | 382.72 |
| 49 | 45 | 4 | 1912 | 2.2 | 0.490 | 0.45 | 936.880 | 860.40 |
| 49 | 44 | 5 | 929 | 3.5 | 0.490 | 0.44 | 455.210 | 408.76 |
| 49 | 47 | 2 | 813 | 4 | 0.490 | 0.47 | 398.370 | 382.11 |
| 50 | 47 | 3 | 1000 | 3.1 | 0.500 | 0.47 | 500.000 | 470.00 |
| 49 | 41 | 8 | 1208 | 3 | 0.490 | 0.41 | 591.920 | 495.28 |
| 48 | 43 | 5 | 1092 | 3 | 0.480 | 0.43 | 524.160 | 469.56 |
| 46 | 46 | 0 | 1500 | 3.0 | 0.460 | 0.46 | 690.000 | 690.00 |
| 50 | 44 | 6 | 3050 | 2.0 | 0.500 | 0.44 | 1525.000 | 1342.00 |
| 49 | 43 | 6 | 789 | 3.5 | 0.490 | 0.43 | 386.610 | 339.27 |
| 49 | 43 | 6 | 1313 | 3.1 | 0.490 | 0.43 | 643.370 | 564.59 |
| 50 | 45 | 5 | 1200 | 2.8 | 0.500 | 0.45 | 600.000 | 540.00 |
| 49 | 41 | 8 | 860 | 3.4 | 0.490 | 0.41 | 421.400 | 352.60 |
| 50 | 46 | 4 | 1000 | 3.1 | 0.500 | 0.46 | 500.000 | 460.00 |
| 48 | 46 | 2 | 865 | 3.5 | 0.480 | 0.46 | 415.200 | 397.90 |
| 50 | 43 | 7 | 1055 | 3 | 0.500 | 0.43 | 527.500 | 453.65 |
| 46 | 43 | 3 | 1148 | 3 | 0.460 | 0.43 | 528.080 | 493.64 |
| 52 | 45 | 7 | 787 | 4.3 | 0.520 | 0.45 | 409.240 | 354.15 |
| 52 | 37 | 15 | 1512 | 3.2 | 0.520 | 0.37 | 786.240 | 559.44 |
| 50 | 40 | 10 | 1282 | 3.4 | 0.500 | 0.40 | 641.000 | 512.80 |
| 47 | 46 | 1 | 807 | 4.3 | 0.470 | 0.46 | 379.290 | 371.22 |
| 46 | 41 | 5 | 737 | 4.1 | 0.460 | 0.41 | 339.020 | 302.17 |
| 48 | 43 | 5 | 591 | 4.1 | 0.480 | 0.43 | 283.680 | 254.13 |
| 48 | 41 | 7 | 1571 | 2.5 | 0.480 | 0.41 | 754.080 | 644.11 |
| 48 | 45 | 3 | 1344 | 2.5 | 0.480 | 0.45 | 645.120 | 604.80 |
| 51 | 42 | 9 | 2424 | 2.3 | 0.510 | 0.42 | 1236.240 | 1018.08 |
| 51 | 43 | 8 | 2192 | 2.4 | 0.510 | 0.43 | 1117.920 | 942.56 |
| 50 | 44 | 6 | 900 | 3.27 | 0.500 | 0.44 | 450.000 | 396.00 |
| 50 | 45 | 5 | 736 | 3.61 | 0.500 | 0.45 | 368.000 | 331.20 |
| 47.7 | 42 | 5.7 | 1014 | 3.1 | 0.477 | 0.42 | 483.678 | 425.88 |
| 49 | 45 | 4 | 3000 | 3 | 0.490 | 0.45 | 1470.000 | 1350.00 |
| 51 | 43 | 8 | 1170 | 3 | 0.510 | 0.43 | 596.700 | 503.10 |
| 49 | 46 | 3 | 1162 | 3 | 0.490 | 0.46 | 569.380 | 534.52 |
| 50 | 45 | 5 | 1000 | 3.1 | 0.500 | 0.45 | 500.000 | 450.00 |
| 46 | 42 | 4 | 1224 | 3 | 0.460 | 0.42 | 563.040 | 514.08 |
| 48 | 43 | 5 | 1056 | 3 | 0.480 | 0.43 | 506.880 | 454.08 |
| 46 | 45 | 1 | 1500 | 3.0 | 0.460 | 0.45 | 690.000 | 675.00 |
| 50 | 44 | 6 | 3050 | 2.0 | 0.500 | 0.44 | 1525.000 | 1342.00 |
| 49 | 45 | 4 | 1000 | 3.9 | 0.490 | 0.45 | 490.000 | 450.00 |
| 52 | 41 | 11 | 1002 | 4.5 | 0.520 | 0.41 | 521.040 | 410.82 |
| 50 | 46 | 4 | 1002 | 5 | 0.500 | 0.46 | 501.000 | 460.92 |
| 50 | 44 | 6 | 1000 | 3.1 | 0.500 | 0.44 | 500.000 | 440.00 |
| 50 | 44 | 6 | 826 | 4 | 0.500 | 0.44 | 413.000 | 363.44 |
| 49 | 48 | 1 | 710 | 4.5 | 0.490 | 0.48 | 347.900 | 340.80 |
| 53 | 45 | 8 | 875 | 3.5 | 0.530 | 0.45 | 463.750 | 393.75 |
| 52 | 46 | 6 | 709 | 3.5 | 0.520 | 0.46 | 368.680 | 326.14 |
| 45 | 41 | 4 | 1434 | 3.0 | 0.450 | 0.41 | 645.300 | 587.94 |
| 48 | 43 | 5 | 1182 | 3.0 | 0.480 | 0.43 | 567.360 | 508.26 |
Given that for each poll we know the proportion of voters for a given candidate and the sample size, we can deduce the number of voters for each candidate and create an aggregate poll containing all voters.
clean_table$Sample_Size <- as.numeric(as.character(clean_table$Sample_Size))## Warning: NAs introduced by coercionobama_voters = round(sum(clean_table$Obama_num, na.rm = T))
romney_voters = round(sum(clean_table$Romney_num, na.rm = T))
all_voters = sum(clean_table$Sample_Size, na.rm = T)
obama_prop = obama_voters/all_voters
romney_prop = romney_voters/all_voters
print(c(obama_prop, romney_prop))## [1] 0.4834197 0.4599031We see that in the aggregate poll, Obama captures 48% of the voters and Romney captures 46%, similar to many of the other polls. The strength of using an aggregate poll however is that the certainty of the estimate is much stronger due to the larger sample size. We can calculate the margin of error as 2 * 1.96 * sqrt(p(1-p)/n)
moe = 2 * 1.96 * sqrt(obama_prop*(1-obama_prop)/all_voters)
print(moe)## [1] 0.004301585We see that due to the large sample size, the margin of error for the spread between Obama and Romney is .4%, well below the actual spread of approximately 2.4%. This margin of error is likely overconfident as it fails to account for various sampling biases that are often present in election polling, but it allowed Silver to make confident predictions with a statistical basis.