From a sample of 84 local barred, moderately inclined disc galaxies, we determine the fraction which host boxy or peanut-shaped (B/P) bulges (the vertically thickened inner parts of bars). We find that the frequency of B/P bulges in barred galaxies is a very strong function of stellar mass: 79% of the bars in galaxies with log (M_{star}/M_{sun}) >~ 10.4 have B/P bulges, while only 12% of those in lower-mass galaxies do. (We find a similar dependence in data published by Yoshino & Yamauchi 2015 for edge-on galaxies.) There are also strong trends with other galaxy parameters -- e.g., Hubble type: 77% of S0-Sbc bars, but only 15% of Sc-Sd bars, have B/P bulges -- but these appear to be side effects of the correlations of these parameters with stellar mass. In particular, despite indications from models that a high gas content can suppress bar buckling, we find no evidence that the (atomic) gas mass ratio M_{atomic}/M_{star} affects the presence of B/P bulges, once the stellar-mass dependence is controlled for. The semi-major axes of B/P bulges range from one-quarter to three-quarters of the full bar size, with a mean of R_{box}/L_{bar} = 0.42 +/- 0.09 and R_{box}/a_{max} = 0.53 +/- 0.12 (where R_{box} is the size of the B/P bulge and a_{max} and L_{bar} are lower and upper limits on the size of the bar).
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