Fogging the Measure: Runs and Wins from 10 to 9

It is often asserted that 10 runs equals one win. What does that mean? Is it even true? Does it matter? Many good internet primers exist for this sort of thing, and I will not offer a full one here. But I think it’s worth a bit of non-technical discussion, and I will also offer a historical table showing how many runs “equal” a win over the years.

Hint: it might be time to start dividing by nine.

It’s obvious that the less frequently runs are scored, the more a single run is worth. Sometimes it’s worth taking a step back to assess our commonly held beliefs. While scoring runs and preventing the other team from scoring is how one wins baseball games, we can’t forget that the actual goal is to win.

Of course, for various reasons (e.g., not getting mired in pointless discussion of “clutchness”), runs do fine as a basic unit of value for offensive and defensive metrics. However, since runs have different values in different seasonal, league, and park environments, we want to be able to evaluate that difference using a set value — the “win.”

These are not actual team wins from actual records. Rather, these “wins” are to actual wins as linear weights runs created are to actual runs. Just as runs created models (in general usage) measure offensive value by estimating the average value of accumulated events, runs-to-wins conversion attempts to estimate how many wins a number of runs (or the size of a run differential) would be worth on an average team.

The mention of run differential may remind you of the Pythagorean formula for expected win percentage, and it should, since the general principle is the same. In fact, getting some sort of idea of the relative value of runs and wins is really the best use the Pythagorean formula, rather than the common mistake of simplistically using it for predictive purposes.

Putting aside the abstractions, in concrete terms, converting from runs to wins gives us a method for understanding the value of players during different eras. For example, a player who created 40 runs above average in 2011 was actually more valuable than a player who created 40 runs above average in 1998 relative to their respective runs environments (4.3 runs per game per team in 2011 versus 4.8 in 1998).

Nevertheless, it’s assumed that ten runs equals a win. In modern baseball, this is, for the most part, a correct assumption, no matter what method you’re using to calculate the run-per-win conversion. Even in recent years where the run environment has become more stingy, I have found this to still be basically true. But what about right now?

Let’s check according to the three methods graphed above.

The classic Pete Palmer runs to win formula:

10*SQRT(RPG) where RPG is runs scored by both teams

Tom Tango’s simple formula:

(RPG*1.5)+3 where RPG is runs scored by one average team (I’m too lazy to convert these all to the same scale, sorry!).

And Patriot’s adaption of his PythagenPat formula to runs-per-win (a modified, more accurate version of the Pythagorean theorem which adjusts for run environment):

(RPG^(1-z)) * 2, where z is between .27 and .29. I have used .28 here. RPG is runs scored by both teams on average per nine innings in my calculations.

PythagenPat has been found to be the most accurate formula for “predicting” team wins from their runs scored and allowed, for more technical discussion, see Patriot’s post on the topic.

However, Tango’s formula, as you can see in the graph above and the chart below, gives the same results, and tracks the movements of PythagenPat easily from year-to-year. It is also easy enough to do in your head (if you have so bizarre need to do so).

Palmer is not bad, but it doesn’t change as much, and is quite different from the other two. This is most evident in the crazy years of early baseball.

As for practical usage, here is one example: last I checked, Baseball-Reference’s WAR uses a version of Palmer’s formula, while FanGraphs uses Tango’s. People make a big deal about stuff like the differences in fielding metrics and stuff like that, but park factors and runs-to-win conversion tend to be overlooked, and they make a drastic difference.

In 2011, as you can see, PythagenPat and Tango’s formula had the MLB conversion closer to 9 than 10 for the first time in a long time, while Palmer’s formula still lingers around 10. I do not mean to make a mountain out of a molehill, but things like this do add up, as it were.

Take WAR calculations, for example. We know that due to the various moving parts, that these are not terribly precise. One should not make a big deal out of a 0.5 win difference. However, if we see a player at, say, 63 runs above replacement, our habit might be to say he had a six win season, while in truth it might be closer to seven in recent seasons. To make things a bit more concrete: when I wrote about Alex Gordon’s 2011 season, just looking at his runs created above average, he did not have as good a season as the best of Carlos Beltran or Mike Sweeney. However, those players played in a more offensively prolific era. Gordon’s runs created were more valuable, so when we convert to wins he ended up having the best offensive season by a Royal since Danny Tartabull in 1991.

Another implication would be on the high end of the free agent market. Take the 63 runs above replacement player mentioned above — the difference between seeing him as a six or seven win player is about \$5 million a year given the current free agent market. That matters on more than just a speculative level. The more scarce marginal runs are, the more wins they generate, but the more they generally cost to buy, too.

Here is the chart for your reference. At a later date I might add one separating the leagues. “RPG” is just runs per team per nine innings. My thanks to David Appelman for helping me finally get access to the Lahman database from which the basic data for my calculations is drawn.

Season Runs IP RPG Palmer Tango PythagenPat
2011 20808 43527.3348 4.3024 9.7779796583957 9.4536 9.41983690119779
2010 21308 43305.3353 4.4284 9.92009060442494 9.64255 9.61761102885033
2009 22419 43271.9994 4.6629 10.179340342085 9.99428 9.98161782682078
2008 22585 43357.6662 4.6881 10.2068585568724 10.03215 10.0204964545269
2007 23322 43425.6664 4.8335 10.3639342047313 10.25025 10.243305914278
2006 23599 43257.9992 4.9099 10.445484373642 10.3648 10.3595705835384
2005 22325 43232.3343 4.6476 10.1626364197486 9.97134 9.95803972371378
2004 23376 43394.0009 4.8482 10.3797107185123 10.27234 10.265766751373
2003 22978 43334.9994 4.7722 10.2979721401837 10.15826 10.1495566337381
2002 22408 43268.9993 4.6609 10.1771955665596 9.99133 9.97858821885409
2001 23199 43287.3334 4.8234 10.3530715055968 10.23506 10.227848288839
2000 24971 43244.3343 5.197 10.7465325105357 10.79544 10.7922281297507
1999 24691 43211.3333 5.1426 10.6901919627292 10.71391 10.7108446171999
1998 23268 43434.3334 4.8213 10.350895999864 10.23202 10.2247528772813
1997 21604 40453.6666 4.8064 10.3348252234859 10.20958 10.20190304716
1996 22831 40559.9997 5.0661 10.6103202684933 10.59908 10.5957969732738
1995 19554 36032.0002 4.8842 10.4181022936041 10.32624 10.3204893064785
1994 15752 28586.3323 4.9593 10.4979289767077 10.43894 10.4345509927635
1993 20864 40507.0007 4.6356 10.1495956569708 9.95346 9.93964380304715
1992 17341 37829.6671 4.1256 9.57493439142013 9.18836 9.13945318719117
1991 18127 37769.6664 4.3194 9.79729963816561 9.47913 9.44664924294364
1990 17919 37563.6661 4.2933 9.76760068798884 9.43991 9.40544200074417
1989 17405 37715.0001 4.1534 9.60715847688587 9.23008 9.18377680182631
1988 17380 37667.6668 4.1526 9.60628624391341 9.22895 9.18257639455519
1987 19883 37574.667 4.7624 10.2874653049233 10.14366 10.1346476157142
1986 18545 37674.3329 4.4302 9.92214640085501 9.64531 9.62047948729599
1985 18216 37658.6666 4.3534 9.83578522538999 9.53013 9.50013179419523
1984 17921 37704.6666 4.2777 9.74986416315632 9.41654 9.38085664129554
1983 18170 37741.9998 4.3328 9.81250763057028 9.49926 9.46777205391551
1982 18110 37878.3348 4.303 9.77864732976908 9.45448 9.42076348375011
1981 11147 25095.0002 3.9977 9.42541315805307 8.99659 8.93464342250054
1980 18053 37861.3334 4.2914 9.76543820829357 9.43706 9.40244273716757
1979 18713 37478.3335 4.4937 9.99301572099234 9.74057 9.71958316921755
1978 17251 37496.6669 4.1406 9.59236736160579 9.21091 9.16342338157079
1977 18803 37738.999 4.4841 9.98236354777765 9.72621 9.70466647698983
1976 15492 34925.6667 3.9921 9.41881765403705 8.9882 8.9256408124602
1975 16295 34724.0006 4.2234 9.68784769698616 9.33517 9.29505573920666
1974 16046 34942.3332 4.1329 9.58346234927649 9.19939 9.15117632523197
1973 16376 34878.6661 4.2256 9.69033918910995 9.33843 9.29849675318943
1972 13706 33450.333 3.6877 9.05253345754657 8.53151 8.43011466098406
1971 15073 34766.9997 3.9019 9.31174855760184 8.85283 8.77990148085191
1970 16880 34858.667 4.3582 9.84114953651249 9.53726 9.50759305917084
1969 15850 34890.6668 4.0885 9.53180164501969 9.13273 9.08022431938545
1968 11109 29234.3338 3.42 8.71777949939089 8.12998 7.98489121965357
1967 12210 29184.3329 3.7654 9.1474057633845 8.64806 8.55763280886768
1966 12900 29043.9998 3.9974 9.42500610079378 8.99607 8.93408711675766
1965 12946 29182.3326 3.9926 9.41939040490413 8.98893 8.9264219466641
1964 13124 29138.3323 4.0536 9.49108280440119 9.08044 9.02442038860894
1963 12780 29105.3325 3.9519 9.37117702319191 8.92778 8.86070202031848
1962 14461 29004.9999 4.4871 9.98568187957137 9.73068 9.70931397035227
1961 12942 25453.3331 4.5761 10.0842444536019 9.86421 9.84761526094061
1960 10664 22203.3338 4.3226 9.80090030558418 9.48389 9.45164961380715
1959 10853 22100.3337 4.4197 9.91038401879564 9.62956 9.60406141757242
1958 10578 22049.6667 4.3176 9.79525540248951 9.47642 9.44381265547895
1957 10636 22344.3332 4.284 9.7570933991635 9.42606 9.39087597900963
1956 11031 22094.9999 4.4933 9.99252946955875 9.73992 9.71890036909876
1955 11069 22011.0009 4.526 10.0288075063788 9.78895 9.76975357701165
1954 10827 22126.667 4.4039 9.89261405291847 9.60581 9.57927203686986
1953 11426 22032.0008 4.6675 10.1843927261276 10.00123 9.98874983252862
1952 10349 22206 4.1944 9.65448303121405 9.29161 9.24899382045125
1951 11268 22208.3334 4.5664 10.0735009703678 9.84959 9.83251294442715
1950 12013 21972.6664 4.9205 10.4568133291171 10.38078 10.3757562571209
1949 11422 22038.6664 4.6644 10.1810699241288 9.99666 9.98405877473876
1948 11312 21932 4.642 10.1565351867652 9.96298 9.94943131300255
1947 10830 22043.3338 4.4217 9.91266860134041 9.63262 9.60725245680252
1946 9955 22146.6664 4.0455 9.48159602598634 9.06829 9.01143628069039
1945 10275 21987.9988 4.2057 9.66747506849643 9.30856 9.26692067657104
1944 10366 22325.6663 4.1788 9.63647912881048 9.26817 9.22416742300205
1943 9669 22402.6666 3.8844 9.29086007859337 8.82661 8.75155370565804
1942 9996 21951.9998 4.0982 9.54313519761719 9.14732 9.09577723090918
1941 11168 22199.3325 4.5277 10.0307346590367 9.79156 9.77245890850329
1940 11570 22051.666 4.7221 10.2437972549246 10.08314 10.0727596388052
1939 11870 21826.0003 4.8946 10.4292548151821 10.34193 10.3364014538821
1938 11966 21658.3327 4.9724 10.5117975151731 10.45861 10.4544078467964
1937 12069 21907.3335 4.9582 10.4967748380157 10.4373 10.4329005545706
1936 12861 22043.6667 5.2509 10.8021560995942 10.87634 10.8727574547668
1935 12029 21880.0003 4.9479 10.4859093358659 10.42191 10.4173522936368
1934 11993 21711 4.9715 10.5108781364832 10.4573 10.4530900183684
1933 10981 21900.333 4.5127 10.014069103017 9.76901 9.74908509241731
1932 12105 22126.3336 4.9238 10.4602645090839 10.38566 10.3806866589093
1931 11893 21990.6663 4.8674 10.4001957385426 10.30107 10.2949550285859
1930 13689 21862.0004 5.6354 11.1906653957662 11.45309 11.4402923007345
1929 12754 21853.3337 5.2526 10.8038701399082 10.87884 10.8752402497588
1928 11612 22008.6664 4.7485 10.2723934990829 10.12274 10.1132736650782
1927 11745 22011.0004 4.8024 10.3305065509877 10.20356 10.1957645320132
1926 11452 21913.3331 4.7034 10.223543446379 10.05516 10.0440944480433
1925 12589 21833.3334 5.1894 10.738672338795 10.78404 10.7808641675869
1924 11686 21940 4.7937 10.3211865306272 10.19057 10.1825218936871
1923 11876 22088.3325 4.8389 10.3697581263981 10.2584 10.2515960322698
1922 12041 22097.3343 4.9042 10.4394190930339 10.35625 10.3509120788595
1921 11922 22002.6665 4.8766 10.4100280210958 10.31489 10.3089733258803
1920 10769 22263.0002 4.3535 9.83582608630307 9.53018 9.50019021291908
1919 8680 20112.3336 3.8842 9.29059712827975 8.82628 8.75119783698163
1918 7380 18393.3332 3.6111 8.95803921625709 8.41664 8.30369170692734
1917 8944 22503.6666 3.577 8.91567544272446 8.36553 8.24720112250869
1916 8885 22511.3335 3.5522 8.88470684941265 8.32832 8.20598569080101
1915 14215 33325.3334 3.839 9.23636693727572 8.75846 8.67773385962787
1914 14521 33508.999 3.9001 9.30963240949931 8.85017 8.77702819090859
1913 9977 22066.9998 4.0691 9.50918590626979 9.10366 9.04921785423807
1912 11135 21880.0005 4.5802 10.0887287207061 9.87032 9.85392251357394
1911 11150 22044.0006 4.5523 10.0578979314765 9.82839 9.81058871700482
1910 9593 22375.6658 3.8585 9.25985606799587 8.78778 8.70953065563699
1909 8807 22214.0001 3.5682 8.90462451763128 8.35223 8.2324861902887
1908 8427 22312.6669 3.3991 8.69112014644833 8.09865 7.94975306471788
1907 8716 21823.3329 3.5945 8.93743857041826 8.39175 8.27620890519981
1906 8883 21706.0004 3.6832 9.04700697468505 8.52476 8.42270709582975
1905 9643 21980.667 3.9483 9.3670027436742 8.9225 8.85502090237447
1904 9306 21998.667 3.8072 9.19810448951304 8.71085 8.62601511766847
1903 9908 19643.0004 4.5396 10.0439394064281 9.80945 9.79098721440456
1902 9883 19692.6665 4.5168 10.0186021879302 9.77514 9.75543946244253
1901 11069 19542.6664 5.0976 10.6433239826663 10.64642 10.6432883461552
1900 5932 9914 5.3851 10.9393397698399 11.07767 11.0721478359607
1899 9647 15846.3334 5.4791 11.0343496681952 11.21859 11.2108849566565
1898 9129 15954.6666 5.1497 10.6975108226166 10.72448 10.7214039220591
1897 9520 13968.3334 6.1339 11.6751152285534 12.20081 12.1601968078069
1896 9555 13762 6.2487 11.7839140611259 12.37309 12.3237143067079
1895 10482 13753.3335 6.8593 12.346193753542 13.28892 13.1792975568621
1894 11680 13731.3335 7.6555 13.0430770985991 14.48323 14.2636948498495
1893 10315 13864.3334 6.696 12.1983226715807 13.04394 12.9525938806909
1892 9388 16106.6665 5.2458 10.7968907005675 10.86867 10.8651253150709
1891 12667 19485.6667 5.8506 11.4023468549242 11.77591 11.7532043664959
1890 19331 28113.9999 6.1883 11.7268356516155 12.28251 12.237846652336
1889 12989 19021.3336 6.1458 11.6864437961255 12.21868 12.1771923159584
1888 10622 19261.6667 4.9631 10.5019806608087 10.44468 10.4403511498493
1887 13417 18369.9998 6.5734 12.0861542270484 12.86007 12.7814319084249
1886 11519 18282.0004 5.6707 11.2256249892823 11.50599 11.4917917342392
1885 9292 15720.3332 5.3197 10.8727330418805 10.9796 10.9751993681967
1884 16670 26945.0003 5.568 11.1235574255721 11.35201 11.3416299199246
1883 9030 13875.6666 5.857 11.4085892642342 11.78552 11.7624738430515
1882 6095 10150.6667 5.4041 10.9585872264631 11.10612 11.1002093417977
1881 3426 5987.3333 5.1499 10.6977380506348 10.72481 10.7217299036593
1880 3191 6031.3334 4.7616 10.2865971438567 10.14245 10.1334160894249
1879 3409 5797 5.2926 10.8449323741552 10.93885 10.9348119280868
1878 1904 3324 5.1552 10.7033064984611 10.73285 10.7297657921825
1877 2040 3241 5.6649 11.2199408108956 11.49738 11.483411832414
1876 3062 4739.3332 5.8147 11.3673431196564 11.72211 11.7012852006048
1875 4234 6192.3334 6.1537 11.6940045664434 12.23061 12.1885420861259
1874 3470 4172.6667 7.4844 12.8965304248856 14.22663 14.0334821853779
1873 3580 3584.6667 8.9883 14.1329271490375 16.48243 16.0110146846853
1872 3389 3285.3333 9.284 14.3635259807611 16.92598 16.3885480231973
1871 2659 2250 10.636 15.3738594829015 18.954 18.0739047487799