MATH & BASEBALL
Baseball statistics can be used to teach statistical concepts.
Batting Averages and More (Grades 7-10)
Objectives
- set up and solve a system of equations
- apply their understanding of percentages and their calculation
- set up a spreadsheet and do calculations using columns
- reason from data
Activity
A. The following data is for eleven of the greatest hitters in the baseball hall of fame.
Player |
Games |
At Bats |
Hits |
Batting Average (BA) |
Runs Batted In (RBI) |
Home Runs |
Hank Aaron |
3298 |
12364 |
3771 |
0.305 |
2297 |
755 |
Rod Carew |
2469 |
9315 |
3053 |
0.328 |
1015 |
92 |
Ty Cobb |
3034 |
11429 |
4191 |
0.367 |
1961 |
118 |
Lou Gehrig |
2164 |
8001 |
2721 |
0.34 |
1990 |
493 |
Rogers Hornsby |
2259 |
8173 |
2930 |
0.358 |
1584 |
301 |
Reggie Jackson |
2820 |
9864 |
2584 |
0.262 |
1702 |
563 |
Mickey Mantle |
2401 |
8102 |
2415 |
0.298 |
1509 |
536 |
Willie Mays |
2992 |
10881 |
3283 |
0.302 |
1903 |
660 |
Stan Musial |
3026 |
10972 |
3630 |
0.331 |
1951 |
475 |
Babe Ruth |
2503 |
8399 |
2873 |
0.342 |
2211 |
714 |
Ted Williams |
2292 |
7706 |
2654 |
0.344 |
1839 |
521 |
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- Place this data in a spreadsheet using either a computer or in a graphing calculator. Determine the percentage of home runs per games, the percentage of home runs per at bat, and the home runs per hit.
Solution:
Player |
Games |
At Bats |
Hits |
Batting Average (BA) |
Runs Batted In (RBI) |
Home Runs |
HR/Games |
HR/At Bats |
HR/Hits |
Hank Aaron |
3298 |
12364 |
3771 |
0.305 |
2297 |
755 |
0.229 |
0.061 |
0.200 |
Rod Carew |
2469 |
9315 |
3053 |
0.328 |
1015 |
92 |
0.037 |
0.010 |
0.030 |
Ty Cobb |
3034 |
11429 |
4191 |
0.367 |
1961 |
118 |
0.039 |
0.010 |
0.028 |
Lou Gehrig |
2164 |
8001 |
2721 |
0.340 |
1990 |
493 |
0.228 |
0.062 |
0.181 |
Rogers Hornsby |
2259 |
8173 |
2930 |
0.358 |
1584 |
301 |
0.133 |
0.037 |
0.103 |
Reggie Jackson |
2820 |
9864 |
2584 |
0.262 |
1702 |
563 |
0.200 |
0.057 |
0.218 |
Mickey Mantle |
2401 |
8102 |
2415 |
0.298 |
1509 |
536 |
0.223 |
0.066 |
0.222 |
Willie Mays |
2992 |
10881 |
3283 |
0.302 |
1903 |
660 |
0.221 |
0.061 |
0.201 |
Stan Musial |
3026 |
10972 |
3630 |
0.331 |
1951 |
475 |
0.157 |
0.043 |
0.131 |
Babe Ruth |
2503 |
8399 |
2873 |
0.342 |
2211 |
714 |
0.285 |
0.085 |
0.249 |
Ted Williams |
2292 |
7706 |
2654 |
0.344 |
1839 |
521 |
0.227 |
0.068 |
0.196 |
- Rank these eleven players in descending order of the ratio of home runs per game.
Solution:
Player HR/Game
Babe Ruth 0.285
Hank Aaron 0.229
Lou Gehrig 0.228
Ted Williams 0.227
Mickey Mantle 0.223
Willie Mays 0.221
Reggie Jackson 0.2
Stan Musial 0.157
Rogers Hornsby 0.133
Ty Cobb 0.039
Rod Carew 0.037
- Rank these eleven players in descending order of the ratio of home runs per at bat.
Solution:
Player HR/At bat
Babe Ruth 0.085
Ted Williams 0.068
Mickey Mantle 0.066
Lou Gehrig 0.062
Willie Mays 0.061
Hank Aaron 0.061
Reggie Jackson 0.057
Stan Musial 0.043
Rogers Hornsby 0.037
Rod Carew 0.01
Ty Cobb 0.01
- Rank these eleven players in descending order of the ratio of home runs per hit.
Solution:
Player HR/Hit
Babe Ruth 0.249
Mickey Mantle 0.222
Reggie Jackson 0.218
Willie Mays 0.201
Hank Aaron 0.2
Ted Williams 0.196
Lou Gehrig 0.181
Stan Musial 0.131
Rogers Hornsnby 0.103
Rod Carew 0.03
Ty Cobb 0.028
- Based on all of the data, make an argument that Babe Ruth is the best hitter in baseball history.
Solution: Student answers will vary.
Sample: Babe Ruth has the highest ratio in all three categories computed. >
- Based on all of the data, make an argument that one of these eleven is the number two greatest hitter in baseball history.
Solution: Student answers will vary.
Sample: This will be a very personal or statistical argument and difficult to provide an example
- Make a statistically based argument that Henry Aaron is the greatest home run hitter of all time.
Solution: Student answers will vary.
Sample: Henry Aaron has the most home runs in a career and ranks second in home runs per game. He ranks sixth in home runs per time at bat and fifth in home runs per hit. No other player, outside of Babe Ruth, ranks that consistently high.
B. The morning edition of the daily newspaper reported that Mark McGuire had a batting average of .350. After the Saturday night game in which McGuire had 3 hits in 5 times at bat, his new batting average was reported to be .352. Determine the number of at bats and hits McGuire had prior to the Saturday night game.
Solution:
Let H = the number of hits prior to Saturday night’s game.
Let AB = the number of times at bat prior to Saturday night’s game.
H/AB = 0.350 and (H + 3)/(AB + 5) = 0.352
H = 0.350AB and H + 3 = 0.352(AB + 5)
H + 3 = 0.352AB + 1.760
H = 0.352AB + 1.760 – 3
H = 0.352AB – 1.24
By substitution: 0.350AB = 0.352AB – 1.24
1.24 = 0.002AB
1.240.002 = AB
620 = AB
0.35(620) = H
217 = H
Prior to that Saturday night’s game Mark McGuire had 217 hits and 620 times at bat.
C. At different times in the season, when Mark McGuire had 3 hits for 5 times at bat, his batting average jumped from .350 to.366, from .350 to .358, and from .350 to .355. Explain why the increase is not the same amount each time. Solution:
Percentage, which is what batting averages measure, is a relative measurement. The measure is relative to the base, which represents 100%. For example, we know that 50% is always half, however, 50% of 100 items is 50 while 50% of 40 items is only 20. Increasing the 50 items to 53 and the base number to 105 produces a percentage of 50.5%, while increasing 20 to 23 and 40 to 45 produces a percentage of 51.1%.
Since the base (number of times at bat) is constantly changing, the increase in percentage relative to the increase of 3 hits and 5 times at bat will also change. Simply, the total number of hits and at bats is never the same. Hence, the ratio of hits to times at bat will have different values because the numerator and denominator of each ratio are different numbers.
Sources: http://www.pbs.org/teachersource/math.htm
Picture: MLB.com
