Catcher is kind of a weak position, without a ton of offense. Wally Schang had one really good year in the decade, but one bad year too. Carrigan gets the nod here, mostly due to playing more games than any other catcher in the decade, but he had a good year in 1914 when he hit .253/.395/.309. He was also the manager for the 2015 and 2016 World Champions. Carrigan was also a very good defensive catcher. He did not have much power, but he could get on base and was very good behind the plate. In this era, that is good enough.
I did not want to have Jake Stahl here again, even though he was probably a better player, albeit in a shorter period of time. Hoblitzell had some good years though. He was brought over from Cincinnati in the middle of the 1914 season and was terrific, hitting .319/.386/.389 the rest of the year. He had another very good year in 1915, hitting .283/.351/.396 with 12 triples. He declined a bit in 1916, but rebounded in 1917. He was finished in 1918. For his career with Boston, Hoblitzell hit .269/.345/.346.
Second base was really a black hole for this decade. Yerkes was mostly in competition with Jack Barry. Though Barry was the better fielder, Yerkes was a better hitter and played quite a few more games. Yerkes was mostly a shortstop in 1911, his first full season, but switched places with seconnd-baseman Heinie Wagner the following season. Yerkes had a pretty good year in 1913, hitting .267/.338/.358 while hitting 29 doubles and stealing 11 bases. It was more productive than any other second-baseman this decade.
He could not hit a lick, but Everett Scott was one of the greatest defensive shortstops ever and spent a lot more time at the position than his nearest competitor, Heinie Wagner. Scott was also the record holder for longest consecutive games streak before Lou Gehrig, a streak which he started toward the end of this decade. Scott led the league in fielding percentage several seasons in a row, starting in 1916 and continuing through the 1923 season.
An easy choice for third base, Gardner was one of the top hitters for the Red Sox during this time period. He spent more time at second base in his first couple of seasons but gradually settled in at third and solidified the position for several years. In 1912, Gardner hit .315/.383/.449 with 24 doubles, 18 triples, three home runs, 86 RBIs, and 25 stolen bases for a terrific all-around season. It was the best season of his Red Sox career. For his Red Sox career, Gardner hit .282/.350/.377 with 16 home runs and 481 RBIs and stealing 134 bases. Gardner's biggest moment was hitting two home runs in the 1916 World Series, despite only hitting two home runs all season.
Harry Hooper is a Hall of Famer (elected by the Veterans' Committee in 1972) and is, to date, the only Red Sox player to play in four World Series. Hooper was a terrific defensive right-fielder and a very good hitter. Hooper spent the entire decade patrolling right field for Boston. Hooper's biggest moment was hitting two home runs in Game 5 of the 1915 World Series including hitting the eventual game-winner in the 9th inning. Hooper still holds the Red Sox career record for stolen bases with 300 and triples with 130. For his Red Sox career, Hooper hit .272/.362/.367.
Speaker was one of the greatest players of his generation. He got his start in Boston and was a terrific player almost immediately. Speaker was a great hitter who had a lifetime batting average of .345 and also had quite a bit of power for a Deadball Era player. He was also one of the greatest defensive center fielders of all time and revolutionized the way the position was played by playing shallow enough to be a fifth infielder. His tremendous speed allowed him to go back and catch balls that might have gone over his head though. Speaker had his best season for the Red Sox in 1912 when he hit .383/.464/.567 with a league-leading 10 home runs, 53 doubles, 90 RBIs, and 52 stolen bases.
The third outfielder in what is possibly Boston's best outfield of all time was not quite as great as the other two. Duffy Lewis is not a Hall of Famer, though that may be more due to the fact that his career did not last quite as long. Lewis was an expert at playing Boston's tricky left field, which at the time included a sloping hill up to the left field fence. He was so good at it that it became known as "Duffy's Cliff". Lewis was also a tremendous hitter who hit .289/.340/.395 for his Red Sox career. He played on World Champion teams in 1912, 1915, and 1916.
Wood holds the Red Sox single season record of 34 wins and the team's career ERA mark with 1.99. He had an otherworldly season in 1912, going 34-5 with a 1.91 ERA and 258 strikeouts in 344 innings. He also led the league in complete games with 35 and shutouts with 10. It was his second consecutive season with more than 20 wins. And he was just 22 at the time. Unfortunately, injuries shortened his Major League career. Wood was still a very good pitcher through the 1915 season, but mostly gave up pitching after that. He later resurrected his career as an outfielder with the Indians and had some decent seasons. He spent his entire Red Sox career as a pitcher.
As a pitcher, Babe Ruth was almost as good as he would later become as a hitter with the Yankees. He came up in 1914 with Boston and was already a great starting pitcher in his first full season. He went 18-8 with a 2.44 ERA in 1915, but was not used in the World Series that season because Boston had several better options. Ruth was Boston's best pitcher in 1916 and 1917 though, going 23-12 with a 1.75 ERA and 24-13 with a 2.01 ERA respectively. Ruth was also terrific in the World Series. By this point, Ruth's hitting was becoming harder to ignore and he started playing the field on days he was not pitching. He led the league in home runs with 11 in 1918 and then set a new record with 29 in 1919. Unfortunately for the Red Sox, they sold him to the Yankees early in 1920 where he stopped pitching entirely, except for a few relief appearances.
Carl Mays was a borderline Hall of Famer who had some great years with the Red Sox in the late 1910's. He was a submarine pitcher and nicknamed "Sub" for his unusual pitching style. From 1916 through 1918, he was one of the best pitchers in the game. In 1917, Mays was 22-9 with a 1.74 ERA. Unfortunately, he struggled a bit in 1919 and was eventually traded to the Yankees, where he rebounded. He is mostly known now for hitting Ray Chapman with a pitch that eventually killed him. Mays was 72-51 with a 2.21 ERA for the Red Sox.
The answer to the trivia question of lowest ERA in a season for a starting pitcher is Dutch Leonard. The left-hander had a shockingly low 0.96 ERA in his sophomore season in 1914. Leonard was virtually unhittable that season, going 19-5 while striking out 176 and walking just 60. That was no flash in the pan season either, as Leonard was a great pitcher for several years. He played for Boston from 1913 through 1918 and was 90-64 with a 2.13 ERA for them. He pitched one game in each of the 1915 and 1916 World Series and gave up just two runs in 18 innings.
Ernie Shore had a brief four-year career with the Red Sox, but was 58-33 with a 2.12 ERA during that stretch. He was acquired at the same time as Babe Ruth from the minor league Baltimore Orioles, but was a more developed pitcher at the time and stepped immediately into the rotation and was 10-5 with a 2.00 ERA in 20 games. Shore was terrific in 1915, going 19-8 with a 1.64 ERA. Shore pitched four games in the 1915 and 1916 World Series and was 3-1 with a 1.82 ERA. Shore's most famous moment was taking over from Babe Ruth after he was tossed out of a game after walking the first batter. That batter was caught stealing on Shore's first pitch and he proceeded to set down the next 26 batters in a row for a bizarre no-hitter.
Foster had the shortest career of any of the pitchers in this post, pitching from 1913 through 1917, all with the Red Sox. But he had a record of 58-33, identical to Ernie Shore, and a 2.36 ERA. Foster's best year was also 1915, when he went 19-8 with a 2.11 ERA. He pitched in three World Series games and was 2-0 with a 1.71 ERA.