NON-GAUSSIAN ERROR DISTRIBUTIONS OF LMC DISTANCE MODULI MEASUREMENTS

Transcription

1 The Atrophyical Journal, 85:87 (0pp), 05 December The American Atronomical Society. All right reerved. doi:0.088/ x/85//87 NON-GAUSSIAN ERROR DISTRIBUTIONS OF LMC DISTANCE MODULI MEASUREMENTS Sara Crandall and Bharat Ratra Department of Phyic, Kana State Univerity, 6 Cardwell Hall, Manhattan, KS 66506, USA; ara990@phy.ku.edu, ratra@phy.ku.edu Received 05 July 8; accepted 05 November 9; publihed 05 December ABSTRACT We contruct error ditribution for a compilation of 3 Large Magellanic Cloud (LMC) ditance moduli value from de Grij et al. that give an LMC ditance modulu of (m M) 0 = 8.49 ± 0.3 mag (median and σ ymmetrized error). Central etimate found from weighted mean and median tatitic are ued to contruct the error ditribution. The weighted mean error ditribution i non-gauian flatter and broader than Gauian with more (le) probability in the tail (center) than i predicted by a Gauian ditribution; thi could be the conequence of unaccounted-for ytematic uncertaintie. The median tatitic error ditribution, which doe not make ue of the individual meaurement error, i alo non-gauian more peaked than Gauian with le (more) probability in the tail (center) than i predicted by a Gauian ditribution; thi could be the conequence of publication bia and/or the non-independence of the meaurement. We alo contruct the error ditribution of 47 SMC ditance moduli value from de Grij & Bono. We find a central etimate of ( m - M) 0 = mag (median and σ ymmetrized error), and imilar probabilitie for the error ditribution. Key word: Magellanic Cloud method: tatitical. INTRODUCTION The Large Magellanic Cloud (LMC) i a widely tudied nearby extragalactic etting with a plethora of tellar tracer. The cloene of the LMC and the abundance of tracer ha reulted in a large number of ditance meaurement to thi nearby galaxy. A the LMC ditance provide an important low rung of the comological ditance ladder, it i of great interet to tudy collection of LMC ditance moduli meaurement. Following Schaefer (008), de Grij et al. (04) compiled a lit of 37 LMC ditance moduli publihed during and ued thee data to examine the effect of publication bia and correlation between the meaurement. They conclude that the overall effect of publication bia i not trong, although there are ignificant effect due to meaurement correlation, epecially in ome individual tracer (maller) ubample. In thi paper we extend and complement the analyi of de Grij et al. (04) by contructing and tudying the error ditribution of the full (3 meaurement) ample and two individual tracer ubample of the de Grij et al. (04) compilation. More pecifically, we examine the Gauianity of thee error ditribution. We begin by following Chen et al. (003) and Crandall et al. (05) and contruct an error ditribution, a hitogram of meaurement a a function of N σ, the number of tandard deviation that a meaurement deviate from a central etimate. Thi i imilar to the z core analyi of de Grij et al. (04), however, we ue a central etimate from the data compilation itelf wherea de Grij et al. (04) ue two publihed value that are aumed to well repreent the meaurement. We ue two technique to find the central etimate: weighted mean and median tatitic. Since median Five of the de Grij et al. (04) entrie do not have error bar, o here we only conider the 3 meaurement that do. Conventionally one aume a Gauian ditribution of error. For intance, thi i ued when determining contraint from CMB aniotropy data (ee, e.g., Ganga et al. 997; Ratra et al. 999; Chen et al. 004; Bennett et al. 03) and ha been teted for uch data (ee e.g., Park et al. 00; Ade et al. 05). Schaefer (008) alo aume the LMC ditance moduli meaurement error have a Gauian ditribution. tatitic doe not make ue of individual meaurement error bar, median tatitic contraint are typically weaker than weighted mean one, but are more reliable in the preence of unaccounted-for ytematic error. We find larger probability tail (error) in the weighted mean ditribution. For the median 3 meaurement cae, we find that the ditribution i narrower than a Gauian ditribution at mall (intermediate) N where the probability i higher (lower) than expected for a Gauian ditribution (a imilar effect i een in the maller ub-ample we tudy). We attempt to analytically categorize thee ditribution by fitting to wellknown non-gauian ditribution: Cauchy, Student t, and the double exponential. Uing a Kolmogorov Smirnov (KS) tet, the fit are poor (<0.%) for the Cauchy, and double exponential cae. A Student t cae with an n = 39 give a probability of %, and i the bet fit. Given that the weighted mean error ditribution are ignificantly non-gauian, it i proper to focu more on our median tatitic central etimate reult. In thi cae, for all three data et, the error ditribution are narrower than Gauian. Thi could be the reult of mild publication bia, or more likely, a argued by de Grij et al. (04), the conequence of correlation between meaurement. In Section we ummarize our method of graphically and numerically decribing the error ditribution of the ditance moduli value. Section 3 decribe our finding from analye of the ditribution of all 3 meaurement. Section 4 and 5 ummarize our analye of the two individual tracer ubample. In Section 6 we decribe the reult found uing SMC ditance moduli meaurement from de Grij & Bono (05). We conclude in Section 7.. SUMMARY OF METHODS Of the 37 LMC ditance moduli value collected by de Grij et al. (04), five do not have a quoted error. For our analye here we ue the 3 meaurement with ymmetric tatitical error bar. To determine the error ditribution of the 3 meaurement we mut firt find a central etimate. We do

2 The Atrophyical Journal, 85:87 (0pp), 05 December 0 thi uing two tatitical technique: weighted mean and median tatitic. The weighted mean (Podariu et al. 00) i å N D i i = i Dwm =, ( ) N å i= i where D i i the ditance modulu and i i the one tandard deviation error of i =,, ¼., N meaurement. While de Grij et al. (04) ue only the quoted tatitical error, in our analye i i the quadrature um of the ytematic (if quoted) and tatitical error. Since many do not quote a ytematic error, 3 and if one i tated it i mall, the difference i not large. The weighted mean tandard deviation i N - wm = ( å. i i ) ( ) = We can alo determine a goodne of fit, c,by c N Di - D = å N - i= i ( ) wm. ( 3) The number of tandard deviation that χ deviate from unity i a meaure of good-fit and i given by N = c - ( N - ). ( 4) The median tatitic technique i beneficial becaue it doe not make ue of the individual meaurement error. However, conequently, thi will reult in a larger uncertainty on the central etimate than for the weighted mean cae. To ue median tatitic we aume that all meaurement are tatitically independent and have no ytematic error a a whole. A meaurement then ha a 50% chance of either being below or above the median value. For a detailed decription of median tatitic ee Gott et al. (00). 4 Once a central etimate i found, we can contruct an error ditribution uing N σ defined a N i = Di - D + CE ( i CE), ( 5) where D CE i the central etimate of D i, either D wm or D med,and CE i the error of the central etimate, either wm or med.here D med i the median ditance modulu, with 50% of the meaurement being above it and 50% below, and med i defined a in Gott et al. (00) uch that the range D med med include 68.3% of the probability. de Grij et al. (04) conider aimilarvariable,a z core. Their z core i different in that they ue two reference value for their central etimate (Freedman et al. 00 and Pietrzynśki et al. 03) while we ue the weighted mean and median central etimate. de Grij et al. (04) aume that the reference value are a good repreentation of the ditance moduli meaurement. Therefore they ue the z core auming Gauianity. We do not aume Gauianity a 3 de Grij et al. (04) note that only 49 meaurement have a quoted nonzero ytematic error, and four additional meaurement include ytematic uncertaintie in their error. The ignificance of thi i conidered in Section 7. 4 For application and dicuion of median tatitic ee Chen & Ratra (003), Mamajek & Hillenbrand (008), Chen & Ratra (0), Calabree et al. (0), Croft & Dailey (05), Andreon & Hurn (0), Farooq et al. (03), (04), Ding et al. (05), Colley & Gott (05), and Sereno (05). our central etimate are found directly from the collected de Grij et al. (04) data uing our tatitical technique. To numerically decribe the error ditribution, we ue a nonparametric KS analyi. Thi i ued to tet the compatibility of a ample ditribution to a reference ditribution. Thi can be ued in two way, with binned or un-binned data. 5 The tet compare the LMC ditance moduli meaurement to a well know ditribution function. To et convention, we firt ue the Gauian probability ditribution function P ( x ) = exp( - x ). ( 6) p It will alo be of interet to conider other well-known ditribution. Thee include the Cauchy (or Lorentzian) ditribution P ( x ) =. ( 7) p + x Thi ditribution ha extended tail, and i a popular choice for a widened ditribution compared to the Gauian. The Cauchy ditribution ha large extended tail, with an expected 68.3% and 95.4% of the value falling within x.8 and x 4, repectively. The Student t ditribution i decribed by [( n ) ] P ( x ) = G + png n + x n ( ) ( ) ( n+ ). ( 8) Here n i a poitive parameter, 6 and Γ i the gamma function. When n thi become the Gauian ditribution. When n = it i the Cauchy ditribution. Thu, for n >, it i a function with extended tail, but le o than that of the Cauchy ditribution. The lat ditribution that we conider i the double exponential. Thi i given by P ( x ) = x exp ( - ). ( 9 ) The double exponential fall off le rapidly than a Gauian ditribution, but fater than a Cauchy ditribution. For thi ditribution 68.3% and 95.4% of the value fall within x. and x 3., repectively. The comparion between the ample and aumed ditribution yield a p-value (or probability) that the two are of the ame ditribution. 3. ERROR DISTRIBUTION FOR FULL DATASET When uing weighted mean tatitic, the 3 LMC ditance moduli yield a central etimate of ( m - M) 0 = mag. We alo find c = 3.00 and the number of tandard deviation that χ deviate from unity i N = 5.7. For the median cae we find a central etimate of (m M) 0 = 8.49 mag with a range of 8.3 mag ( m - M) mag. Our central etimate are in good accord with de Grij et al. (04) who quote ( m - M) 0 = mag. 7 5 It i more conventional to ue un-binned data for thi tet, but for completene we have ued both (ee Section 5.3. Feigelon & Babu 0). 6 The incluion of n reduce the number of degree of freedom by. 7 de Grij et al. (04) ue a collection of 33 ditance moduli value for their etimate from year 990 to 03, dropping four 04 meaurement.

3 The Atrophyical Journal, 85:87 (0pp), 05 December 0 Figure. Hitogram of the error ditribution in half tandard deviation bin. The top (bottom) row ue the weighted mean (median) of the 3 meaurement a the central etimate. The left column how the igned deviation, where poitive (negative) N repreent a value that i greater (le) than the central etimate. The right column how the abolute ymmetrized ditribution. The mooth curve in each panel i the bet-fit Gauian. Figure how the error ditribution of the 3 meaurement. Thee are hown a a function of N σ, Equation (5), the number of tandard deviation the meaured value deviate from the central etimate. In Figure we how the error ditribution for the weighted mean and median central etimate. 8 In both cae we alo plot the ymmetrized ditribution a a function of N. For a more detailed perpective of the ditribution, ee Figure (with N = 0. bin ize). Figure how that for the weighted mean cae the ditribution ha a more extended tail than expected for a Gauian ditribution. In fact, for a et of 3 value, a Gauian ditribution hould yield value with N, one value with N 3, and none with N 4. However, we find 4 value with N, 3 with N 3, and 9 with N 4 for the weighted mean cae. We alo note that 68.3% of the oberved weighted mean N σ error ditribution fall within -.37 N.6 while 95.4% lie within N The oberved weighted mean N error ditribution ha limit of N.33 and N 3.63 repectively, and 56.5% and 8.9% of the value fall within N 8 The larger σ CE for the median cae reult in a narrower ditribution, ee Equation (5). and N, repectively. Thee reult clearly indicate that the weighted mean error ditribution i non-gauian, o the weighted mean technique i inappropriate for an analyi of thee data. The median cae i narrower than Gauian, with even value of N and none with N % of the data fall within N 0.63, while 95.4% lie within -.97 N.7. The N error ditribution ha limit of N 0.7 and N.66, repectively, and 80.6% and 97.0% of the value fall within N and N, repectively. The median technique i more appropriate becaue of the non-gauianity of the ditribution, however, de Grij et al. (04) note that there are correlation between meaurement (epecially among meaurement of the ame tracer type). Thee correlation mean that the meaurement are not tatitically independent, and the error aociated with the median will need to be lightly adjuted to account for thi. Regardle, the narrowne of the median ditribution i clearly conitent with the preence of uch correlation. Since the ditribution for the weighted mean cae i broader than Gauian while the median ditribution i narrower than a Gauian, it i of interet to try to fit thee oberved ditribution to well-known non-gauian ditribution. 3

4 The Atrophyical Journal, 85:87 (0pp), 05 December 0 Figure. Hitogram of the error ditribution in N = 0. bin (with the exception of the lat, truncated, bin with 5 N 6 that contain the number of meaurement to N = 8). The olid black line repreent the expected Gauian probabilitie for 3 meaurement and the dotted blue (dahed red) line i the number of N value in each bin for the weighted mean (median) cae. To et convention, we firt conider a Gauian probability ditribution function. In thi cae 68.3% of the value have N. The Gauianity of the ditribution can be etablihed by taking a quantitative look at the pread of value. However, the probability given by the KS tet i 0.% for the data et (See Table ). Our firt non-gauian ditribution, the Cauchy (or Lorentzian) ditribution, alo ha a probability of <0.%. Next we conider a ditribution with extended tail, but le o than the Cauchy ditribution, the Student t ditribution. Fitting to thi function yield a probability of % (correponding to a Student t ditribution with n = 39) for a binned KS tet. Thi may appear odd, a we have argued that for the median cae the ditribution i narrower than a Gauian ditribution, while the Student t ditribution i known for extended tail. To explain thi we examine the kurtoi of the N ditribution. We ue the common definition of kurtoi (ee Equation (37.8b) of Olive et al. 04) where the fourth and econd moment are and m m 4 m4 k = m, ( 0 ) n 4 = å Ni - N ( ) n i= n ( ) = å Ni - N. ( ) n i= ( ) Here N i the mean of the n N i value. For a detailed dicuion of kurtoi ee Balanda & MacGillivray (988). The kurtoi can be defined a a meaurement of the peakedne, or that of the tail width of a ditribution. For example, a large kurtoi would repreent a ditribution with more probability in the peak and tail than in the houlder (Balanda & MacGillivray 988). A Gauian ditribution ha k = 3, and k 3 repreent a leptokurtic ditribution with a high peak and wide tail. 9 For the median cae, we find k = Thi may explain why thi cae favor a Student t fit, a a Student t ditribution alo ha a large kurtoi, i.e., wider tail and a higher peak. The median tatitic ditribution appear to favor thi fit becaue it kurtoi i greater than that for a Gauian ditribution, even though it i narrower than a Gauian. The final non-gauian ditribution function we conider i the double exponential, or Laplace ditribution. Again, we do not find a probability of greater than 0.%. To viually clarify the difference between the weighted mean and median tatitic N hitogram, we plot them in bin of N = 0., ee Figure. We ee that the weighted mean cae i cloer to Gauian near the peak, but ha an extended tail. Thi ugget the exitence of unaccounted-for ytematic error. For the median cae the peak i much higher than expected for a Gauian, and the ditribution drop off with increaing N more rapidly than expected for a Gauian. Thi may be a ign of correlation between meaurement or poibly publication bia. Table i a more compact way of diplaying ome of thi information. For a Gauian ditribution of 3 value, there hould be zero meaurement with N 4 while the oberved weighted mean cae ha nine. For illutrative purpoe, we truncate thi ditribution by removing all value with 0 N 4. Thi leave u with 3 value and an unchanged central etimate of ( m - M) = Often an exce kurtoi i ued to decribe the peakedne of a ditribution. Thi i imply three ubtracted from the tandard kurtoi, and i ued to compare to a normal ditribution (which would have an exce kurtoi of zero). 0 For completene, we alo did a median tatitic analyi of thi truncated data. A expected, we found that removing thee nine meaurement doe not increae probabilitie or change the median tatitic reult, which how the robutne of median tatitic. The 3 value alo give a c =.90 and N=8.0 (the number of tandard deviation that χ deviate from unity). 0 4

5 The Atrophyical Journal, 85:87 (0pp), 05 December 0 Function a Table K S Tet Probabilitie Data Set Un-binned Binned Probability Probability (%) b (%) b Gauian Whole (3) <0. <0. Truncated (3) <0. <0. Cepheid (8).5 <0. Truncated Cepheid (75) RR Lyrae (63).5 <0. Truncated RR 0.8 <0. Lyrae (58) Cauchy Whole (3) <0. <0. Truncated (3) <0. <0. Cepheid (8).0 <0. Truncated.9 <0. Cepheid (75) RR Lyrae (63).6 <0. Truncated RR 0.7 <0. Lyrae (58) Double Whole (3) <0. <0. Exponential Truncated (3) <0. <0. Cepheid (8).5 <0. Truncated 3.7 <0. Cepheid (75) RR Lyrae (63).3 <0. Truncated RR 0.6 <0. Lyrae (58) n = 39 Student t Whole (3) <0. n = 3 Student t Truncated (3) <0. 8 n = 3 Student t Cepheid (8) n = Student t Truncated.7 5 Cepheid (75) n = 59 Student t RR Lyrae (63).6 34 n = 94 Student t Truncated RR Lyrae (58) Note. a For the Student t cae, the n correponding with the bet probability i diplayed. b The probability that the data et i compatible with the aumed ditribution. The pread of the value can be een in Figure 3. We alo find that 68.3% of the value fall within -.55 N.05 and 95.4% fall within N.06. For the abolute cae, N.3 and N 3.03 for 68.3% and 95.4% of the value, repectively. In term of percentage, 6.0% and 85.% of the meaurement fall within N and N, repectively. We note that when truncated, the normal tandard deviation become σ = 0.5 while the ymmetrized error for the median cae i σ = 0.6. It would appear that after eliminating N > 4, the median and weighted mean cae converge. However, we do utilize a weighted mean rather than the tandard mean, a the error for the meaurement are not the ame, and the weighted mean and median tatitic error till do not converge even in the truncated cae. It i alo of interet to determine the probabilitie of the four well-known ditribution for the new truncated weighted mean cae. The ditribution have a probability of <0.% for the Gauian, Cauchy, and double exponential ditribution. The truncation to N < 4 improve the probability for the Student t cae, lightly increaing it to 8% compared to Table Expected Gauian and Oberved Number of N Oberved Tracer Value a N Expected b (WM) c the % for the non-truncated cae. Since the probability doe not improve for the Gauian fit, thi till indicate non- Gauianity in the meaurement ditribution, becaue of the larger than expected N > and 3 tail. 4. ERROR DISTRIBUTIONS FOR 8 CEPHEID VALUES It i of interet to alo invetigate the pread of individual tracer meaurement. We firt conider the 8 Cepheid ditance moduli value tabulated by de Grij et al. (04). For the weighted mean cae we find a central etimate of ( m - M) 0 = mag. For igned N, 68.3% of the value fall within -.04 N.73 and 95.4% fall within -.78 N For abolute N σ, 68.3% and 95.4% of the value fall within N.3 and N 4.3, repectively, while 56.8% of the value fall within N and 87.7% fall within N. For the median cae we find a central etimate of ( m - M) 0 = 8.50 mag with a σ range of 8.37 mag ( m - M) mag. For igned N σ, 68.3% of the value fall within N 0.73 and 95.4% fall within -.8 N.76. For abolute N σ, 68.3% and 95.4% of the value fall within N 0.7 and N.63, repectively, while 79.0% of the value fall within N and 97.5% fall within N. We note that for the median cae, the error ditribution i tighter when we ue only the 8 Cepheid value compared to the ditribution from all 3 meaurement. Oberved (Med) c All Type Cepheid RR Lyrae Note. a The number of ditance moduli meaurement ued in our analyi. b The number of value expected to fall outide of the correponding N Gauian ditribution of total number lited in Col (). c The oberved number of value outide of the correponding N. for a We alo find a c =.66 and N = 7.96, which i the number of tandard deviation that χ deviate from unity. 5

6 The Atrophyical Journal, 85:87 (0pp), 05 December 0 Figure 3. Hitogram of the error ditribution in half tandard deviation bin for the truncated weighted mean ditribution (N 4). The left plot ue the weighted mean of the 3 meaurement a the central etimate to how the igned deviation. The right plot how the ymmetrized abolute N σ. The mooth curve in each panel i the bet-fit Gauian. Figure 4. Hitogram of the error ditribution in half tandard deviation bin for Cepheid. The top (bottom) row ue the weighted mean (median) of the 8 meaurement a the central etimate. The left column how the igned deviation, where poitive (negative) N σ repreent a value that i greater (le) than the central etimate. The right column how the abolute ymmetrized ditribution. The mooth curve in each panel i the bet-fit Gauian. The igned and abolute N σ ditribution for the Cepheid tracer can be een in Figure 4. One can ee from the top two plot that there i an extended tail in the ditribution for the weighted mean cae, and from the lower two plot, a narrower ditribution for the median cae. We alo plot N in bin of 0.; ee Figure 5. Thi figure again illutrate the higher than expected peak and rapid dropoff of N for the median cae, and the extended tail for the weighted mean cae. To 6

7 The Atrophyical Journal, 85:87 (0pp), 05 December 0 Figure 5. Hitogram of the error ditribution uing Cepheid tracer in N = 0. bin (with the exception of the lat bin with 5 N 6). The olid black line repreent the expected Gauian probabilitie for 8 meaurement and the dotted blue (dahed red) line i the number of N value in each bin for the weighted mean (median) cae. numerically decribe thee feature, we can again ue the four well-known ditribution. The bet probability come from the Student t ditribution for the median cae with a probability of 6% (ee Table ). It i of interet to alo truncate the Cepheid ub-ample, and we do o by truncating all N > 3, a there hould be none for a normally ditributed et of 8 meaurement (ee Table ). For the weighted mean cae, with a new central etimate of ( m - M) 0 = mag, the ditribution lightly tighten. For igned N, 68.3% of the value fall within N.48 and 95.4% fall within -.43 N.38. For abolute N, 68.3% and 95.4% of the value fall within N. and N.3, repectively, while 65.3% of the value fall within N and 94.7% fall within N. A for the median cae, the ditribution doe not ignificantly change (a expected with median tatitic). Table how the probabilitie for the new truncated cepheid et. The probability for the unbinned KS tet only lightly increae to.7% while the binned probability doe not ignificantly change. 5. ERROR DISTRIBUTION FOR 63 RR LYRAE VALUES de Grij et al. (04) alo tabulate 63 RR Lyrae ditance moduli, 3 whoe error ditribution we tudy here. We find a weighted mean central etimate of ( m - M) 0 = mag. For igned N, 68.3% of the value fall within N.5 and 95.4% fall within -.75 N 3.. For abolute N, 68.3% and 95.4% of the value fall within N.00 and N 3., repectively, while 68.3% of the value fall within N and 88.9% fall within N. For the median cae we find a central etimate of ( m - M) 0 = 8.47 mag, with a σ range of 8.9 mag ( m - M) mag. For igned N σ, 68.3% of the value fall within N 0.48 and 95.4% fall within 3 Three RR Lyrae value in de Grij et al. (04) quote a zero error and were not ued here N.03. For abolute N σ, 68.3% and 95.4% of the value fall within N 0.50 and N.56, repectively, while 8.5% of the value fall within N and 98.4% fall within N. We plot N in bin of 0., ee Figure 6, and the pread of value can be een in Figure 7. In thi cae, the non-gauianity i not a viually triking. We alo fit the RR Lyrae meaurement to the four ditribution. The Student t ditribution give the larget probability of 34% (ee Table ). We alo truncated the RR Lyrae ub-ample by only including value with N <.5, a there hould be none greater than thi for a et of 63 normally ditributed meaurement (ee Table ). In doing o the weighted mean error ditribution wa lightly tightened. 4 We find a lightly changed central etimate of ( m - M) 0 = mag. For igned N σ, 68.3% of the value fall within N 0.79 and 95.4% fall within -.49 N.8. For abolute N σ, 68.3% and 95.4% of the value fall within N 0.75 and N.87, repectively, while 75.9% of the value fall within N and 98.3% fall within N. When fitting the ub-ample error ditribution to well-known ditribution, we find a light increae in probabilitie given by the KS tet (ee Table ). Wefind that the highet probability of 37% i given by an n = 94 Student t ditribution, which lightly increaed from 34%. 6. SMC DISTANCE MODULI We have alo analyzed 47 SMC ditance moduli meaurement 5 compiled by de Grij & Bono (05) and find imilar reult to thoe given by LMC ditance Moduli meaurement. 6 For the weighted mean cae, which give a central etimate of ( m - M) 0 = mag, we find extended tail in the error ditribution. For igned N σ, 4 The median cae did not ignificantly change. 5 de Grij & Bono (05) collected 304 etimate, but we have only included meaurement with non-zero error. 6 We thank Jacob Peyton for helping with thi analyi. 7

8 The Atrophyical Journal, 85:87 (0pp), 05 December 0 Figure 6. Hitogram of the error ditribution uing RR Lyrae tracer in N = 0. bin. The olid black line repreent the expected Gauian probabilitie for 63 meaurement and the dotted blue (dahed red) line i the number of N value in each bin for the weighted mean (median) cae. Figure 7. Hitogram of the error ditribution in half tandard deviation bin for RR Lyrae. The top (bottom) row ue the weighted mean (median) of the 63 meaurement a the central etimate. The left column how the igned deviation, where poitive (negative) N repreent a value that i greater (le) than the central etimate. The right column how the abolute ymmetrized ditribution. The mooth curve in each panel i the bet-fit Gauian. 8

9 The Atrophyical Journal, 85:87 (0pp), 05 December 0 we find that 68.3% and 95.4% of the meaurement fall within -.0 N.9 and N 4.76, repectively. For the unigned N we find that 68.3% and 95.4% of the meaurement are within N.9 and N 5.6, repectively. Converely, 45.8% of the meaurement fall within N and 70.5% fall within N. Thee wider tail ugget unaccounted-for ytematic error. A for the median cae, which give a central etimate of ( m - M) 0 = 8.94 mag with a σ range of 8.8 mag ( m - M) mag, the ditribution i narrower than expected for a Gauian. For igned N,wefind that 68.3% and 95.4% of the meaurement fall within N 0.78 and -.60 N.9, repectively. For the unigned N we find that 68.3% and 95.4% of the meaurement are within N 0.79 and N.68, repectively. Converely, 78.5% of the meaurement fall within N and 96.8% fall within N. Thi narrow ditribution indicate the preence of correlation between meaurement (epecially within imilar tracer type), a uggeted by de Grij & Bono (05). We alo examine the ditribution given by the two tracer type with a greater number of meaurement: Cepheid (0 meaurement) and RR Lyrae (30). For the Cepheid weighted mean cae, we find a central etimate of ( m - M) 0 = mag. 68.3% and 95.4% of the meaurement are within -.55 N 0.98 and N.76 for igned N σ. For the abolute cae N.6 and N 4.0 for 68.3% and 95.4% of the meaurement, repectively. Alternatively, 56.5% and 86.% of the meaurement fall within N and N, repectively. For the median cae, we find a central etimate of ( m - M) 0 = 8.98 mag, with a range of 8.83 mag ( m - M) mag. The ditribution how that 68.3% and 95.4% of the meaurement are within N 0.68 and -.9 N.46 for igned N σ. For the abolute cae N 0.8 and N.54 for 68.3% and 95.4% of the meaurement, repectively. Alternatively, 8.% and 98.0% of the meaurement fall within N and N, repectively. Again, we ee a wider (narrower) than Gauian ditribution for the weighted mean (median) cae. For the ub-ample of RR Lyrae tracer type, we notice imilar ditribution. For the weighted mean cae, we find a central etimate of ( m - M) 0 = mag. 68.3% and 95.4% of the meaurement are within -.40 N 0.88 and -.40 N.47 for igned N σ. 7 For the abolute cae N.49 and N 3.6 for 68.3% and 95.4% of the meaurement, repectively. Alternatively, 50.0% and 80.0% of the meaurement fall within N and N, repectively. For the median cae, we find a central etimate of ( m - M) 0 = 8.90 mag, with a range of 8.74 mag ( m - M) mag. The ditribution how that 68.3% and 95.4% of the meaurement are within -0.5 N 0.86 and -.3 N.4 for igned N. For the abolute cae N 0.65 and N.8 for 68.3% and 95.4% of the meaurement, repectively. Alternatively, 83.3% and 00% of the meaurement fall within N and N, repectively. We alo attempt to fit the error ditribution to four wellknown ditribution. The probabilitie, found by uing the KS tet, are given in Table 3. Wefind that all ditribution are fit 7 The two lower bound are the ame due to the ditribution being weighted toward the poitive N σ ide (there are more value with N > 0). Symmetrizing thi ditribution give a clearer undertanding of the error. Table 3 K S Tet Probabilitie Un-binned Binned Function a Data Set Probability(%) b Probability(%) b Gauian Whole (47) < 0. < 0. Cepheid (0) <0. <0. RR Lyrae (30) Cauchy Whole (47) <0. <0. Cepheid (0) <0. <0. RR Lyrae (30) 33 5 Double Exponential Whole(47) <0. <0. Cepheid (0) <0. <0. RR Lyrae (30) 36 0 n = Student t Whole (47) 59 <0. n = Student t Cepheid (0) 74 <0. n = Student t RR Lyrae (30) 3 Note. a For the Student t cae, the n correponding with the bet probability i diplayed. b The probability that the data et i compatible with the aumed ditribution. bet by a Student t ditribution. The whole (47) ditribution i bet fit byann = Student t with a probability of 74%. 7. CONCLUSION We have tudied the error ditribution of LMC ditance moduli compiled by de Grij et al. (04). Wefind that the error ditribution are non-gauian with extended tail when uing a weighted mean central etimate, probably a a conequence of unaccounted-for ytematic error. In fact, only 53 of the 37 value tabulated by de Grij et al. (04) have a non-zero ytematic error. Becaue the weighted mean error ditribution are non-gauian, it i more appropriate to ue the median tatitic error ditribution. The median tatitic error ditribution are narrower than Gauian, upporting the concluion of de Grij et al. (04), who argue that thi i a conequence of correlation between ome of the meaurement, with publication bia poibly alo contributing mildly. We thank R. de Grij, J. Wicker, and G. Bono for providing u with the data. In addition, we thank R. de Grij, J. Wicker, G. Horton-Smith, and A. Ivanov for valuable comment and advice. Finally, we thank Jacob Peyton for the SMC ditance moduli analyi. Thi work wa upported in part by DOE grant DEFG 03-99EP4093 and NSF grant AST REFERENCES Ade, P. A. R., Aghanim, N., Arnaud, M., et al. 05, arxiv: Andreon, S., & Hurn, M. A. 0, arxiv:0.63 Balanda, K. P., & MacGillivray, H. L. 988, American Statitician, 4, Bennett, C. L., Laron, D., Weiland, J. L., et al. 03, ApJS, 08, 0 Calabree, E., Archidiacono, M., Melchiorri, A., & Ratra, B. 0, PhRvD, 86, Chen, G., Gott, R., & Ratra, B. 003, PASP, 5, 83 Chen, G., Mukherjee, P., Kahniahvili, T., Ratra, B., & Wang, Y. 004, ApJ, 6, 655 Chen, G., & Ratra, B. 003, PASP, 5, 43 Chen, G., & Ratra, B. 0, PASP, 3, 7 Colley, W., & Gott, J. R. 05, MNRAS, 447, 034 Crandall, S., Houton, S., & Ratra, B. 05, MPLA, 30, 5 Crandall, S., & Ratra, B. 04, PhLB, 73, 330 Croft, R. A. C. 05, Rev. Quarterly Phy,, 9

10 The Atrophyical Journal, 85:87 (0pp), 05 December 0 de Grij, R., & Bono, G. 05, AJ, 49, 79 de Grij, R., Wicker, J. E., & Bono, G. 04, AJ, 47, Ding, X., Bieiada, M., Cao, S., Li, Z., & Zhu, Z.-H. 05, ApJL, 803, L Farooq, O., Crandall, S., & Ratra, B. 03, PhLB, 76, 7 Feigelon, E. D., & Babu, G. J. 0, Modern Statitical Method for Atronomy with R Application (Cambridge: Cambridge Univ. Pre) Freedman, W. L., Madore, B. F., Gibon, B. K., et al. 00, ApJ, 553, 47 Ganga, K., Ratra, B., Gunderen, J. O., & Sugiyama, N. 997, ApJ, 484, 7 Gott, J. R., Vogeley, M. S., Podariu, S., & Ratra, B. 00, ApJ, 549, Mamajek, E. E., & Hillenbrand, L. A. 008, ApJ, 687, 64 Olive, K. A. & (Particle Data Group) et al. 04, ChPhC, 38, Park, C. G., Park, C., Ratra, B., & Tegmark, M. 00, ApJ, 556, 58 Pietrzynśki, G., Graczyk, D., Gieren, W., et al. 03, Natur, 495, 76 Podariu, S., Souradeep, T., Gott, J. R., Ratra, B., & Vogeley, M. S. 00, ApJ, 559, 9 Ratra, B., Stompor, R., Ganga, K., et al. 999, ApJ, 57, 549 Schaefer, B. E. 008, AJ, 35, Sereno, M. 05, MNRAS, 450,