15 Model Parameter Shift due to Systematic Uncertainty Absolute E Had$_{.}$ Relative E Had$_{.}$ Normalization CC Background ND Selection Total m 4 = m 1 Δ θ 23 0 . 3 0 . 6 0 . 3 0 . 1 0 . 1 0 . 7 Δ θ 34 3 . 6 9 . 9 12 . 6 9 . 9 9 . 9 21 . 6 m 4 ≫ m 3 Δ θ 23 0 . 2 0 . 6 0 . 1 0 . 2 0 . 2 0 . 7 Δ θ 24 1 . 5 2 . 1 5 . 1 0 . 3 0 . 3 5 . 7 Δ θ 34 4 . 5 9 . 9 6 . 3 9 . 9 9 . 9 18 . 8 Oscillations Δ α (GeV/km) 2 . 54 × 10 − 4 0 . 70 × 10 − 4 6 . 25 × 10 − 4 1 . 23 × 10 − 4 1 . 15 × 10 − 4 6 . 99 × 10 − 4 with decay Δ θ 2 . 6 3 . 7 0 . 9 4 . 0 3 . 9 7 . 2 TABLE III: Summary of mixing-angle deviations introduced by the major systematic uncertainties from best-fit results in which systematic shifts have been neglected. Angular deviations, shown in degrees, are displayed for each mixing angle fitted, for each of the neutrino models analyzed in this work. FIG. 13: Contours representing 90% confidence level for the m 4 = m 1 model. The solid line and best-fit point (solid sym- bol) are obtained assuming null ν$_{e}$ appearance, whereas the dashed line and corresponding best-fit point (open symbol) are obtained with ν$_{e}$ appearance set at the CHOOZ limit. FIG. 12: Projections of Δ χ 2 as a function of the mixing angles for the m 4 = m 1 model. The solid line contours are obtained with null ν$_{e}$ appearance, whereas the dashed line contours include ν$_{e}$ appearance at the CHOOZ limit. The ranges of values allowed at 68% and 90% confidence levels lie within contours below the horizontal dashed lines.  (deg) 34 0 20 40 60 80 100  (deg) 23 30 40 50 60 =0 13 Best fit point 90% C.L. =12 13 Best fit point 90% C.L.  (deg) 23 30 40 50 60 2 0 1 2 3 =0 13 =12 13  (deg) 34 20 40 60 68% 90% f$_{s}$ ≡ P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{s}$ 1 − P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{µ}$ . (15) between the best-fit point and the remaining points in the space are shown in Fig. 12 for the m$_{4}$ = m$_{1}$ model. The two-dimensional 90% confidence level contours for that model are shown in Fig. 13. The projections and contours for the m$_{4}$ ≫ m$_{3}$ model are shown in Figs. 14 and 15. As seen in these figures, θ$_{34}$ < 38 $^{◦}$(56$^{◦}$) at 90% con- fidence level for the m$_{4}$ = m$_{1}$ model. The number in parentheses represents the 90% confidence level limit ob- tained when maximal ν$_{e}$ appearance is allowed. For the m$_{4}$ ≫ m$_{3}$ model, θ$_{24}$ < 10 $^{◦}$(11$^{◦}$) and θ$_{34}$ < 38 $^{◦}$(56$^{◦}$) at the 90% confidence level. These limits indicate that any coupling between the active neutrinos and a sterile neu- trino is submaximal. Furthermore, the χ 2 values indicate that these four-flavor models fit the data no better than oscillations among only the active neutrinos. A straightforward method to quantify the coupling be- tween the active and sterile neutrinos is to determine the fraction of disappearing ν$_{µ}$ that transition to ν$_{s}$ . That fraction is expressed as For the m$_{4}$ = m$_{1}$ model, the disappearance fraction f$_{s}$ is energy independent, as can be seen upon inserting the expressions from Eq. (12) into Eq. (15). The 90% confi- dence level limit for f$_{s}$ is determined by selecting a large number of test values of θ$_{23}$ and θ$_{34}$ from Gaussian distri- butions with mean and σ given in Table II. The value of f$_{s}$ that is larger than 90% of the test cases represents the limit. The value corresponding to the 90% confidence level is f$_{s}$ < 0 . 51 (0 . 55), with the value in parentheses indicating the value obtained for maximally-allowed ν$_{e}$ appearance in the beam. This new limit on the value of f$_{s}$ represents a reduction of 33% compared to the previ- ous MINOS result without ν$_{e}$ appearance [17]. For the m$_{4}$ ≫ m$_{3}$ model, inserting the expressions from Eq. (13) into (15) shows that f$_{s}$ is energy depen- dent because of the constant terms in Eq. (13). For this reason the 90% confidence level value of f$_{s}$ for this model is presented at E$_{ν}$ = 1 . 4 GeV, the energy where the ν$_{µ}$