Active galactic nuclei (AGN) display an extreme range in the narrow emission‐line equivalent widths. Specifically, in the Palomar–Green (PG) quasar sample, the equivalent width of the narrow [O iii]λ5007 line has a range of >300 (<0.5 to 157 Å), while the broad Hβ line, for example, has a range of 10 only (23 to 230 Å). The strength of [O iii]λ5007 is modulated by the covering factor (CF) of the narrow‐line region (NLR) gas, its density (ne) and ionization parameter (U). To explore which of these factors produces the observed large range in [O iii]λ5007 strength, we measure the strength of the matching narrow Hβ and [O iii]λ4363 lines, detected in 40 out of the 87 PG quasars with z < 0.5 in the Boroson & Green sample. The photoionization code cloudy is then used to infer CF, ne and U in each object, assuming a single uniform emitting zone. We find that the range of CF (∼0.02–0.2) contributes about twice as much as the range in both ne and U towards modulating the strength of the [O iii]λ5007 line. The CF is inversely correlated with luminosity, but it is not correlated with L/LEdd as previously speculated. The single‐zone [O iii]λ5007 emitting region is rather compact, having RNLR= 40L0.4544 pc. These emission lines can also be fitted with an extreme two‐zone model, where [O iii]λ4363 is mostly emitted by a dense (ne= 107 cm−3) inner zone at RinNLR=L0.544 pc, and [O iii]λ5007 by a low‐density (ne= 103 cm−3) extended outer zone at RoutNLR= 750L0.3444 pc. Such an extended [O iii]λ5007 emission should be well resolved by Hubble Space Telescope imaging of luminous AGN. Further constraints on the radial gas distribution in the NLR can be obtained from the spectral shape of the infrared continuum emitted by the associated dust.