App to quickly analyze audio loudness of several video files?
ffmpeg, a free multi-platform command-line tool, can do this. 

Use the volumedetect and EBUR filters on each of the files

ffmpeg -i input.mov -af volumedetect,ebur128 -f null - 2&gt; input.log


This will produce a log file with the initial lines looking like this:

[Parsed_ebur128_1 @ 0000000002d42400] t: 12.7       M: -32.7 S: -32.7     I: -30.7 LUFS     LRA:   3.1 LU
[Parsed_ebur128_1 @ 0000000002d42400] t: 12.8       M: -33.2 S: -32.7     I: -30.8 LUFS     LRA:   3.1 LU
[Parsed_ebur128_1 @ 0000000002d42400] t: 12.9       M: -33.2 S: -32.7     I: -30.8 LUFS     LRA:   3.1 LU
[Parsed_ebur128_1 @ 0000000002d42400] t: 13         M: -34.3 S: -32.9     I: -30.8 LUFS     LRA:   3.1 LU
[Parsed_ebur128_1 @ 0000000002d42400] t: 13.1       M: -34.3 S: -32.9     I: -30.9 LUFS     LRA:   3.1 LU


and the final lines:

[Parsed_volumedetect_0 @ 00000000005db400] n_samples: 18700288
[Parsed_volumedetect_0 @ 00000000005db400] mean_volume: -31.5 dB
[Parsed_volumedetect_0 @ 00000000005db400] max_volume: -5.3 dB
[Parsed_volumedetect_0 @ 00000000005db400] histogram_5db: 22
[Parsed_volumedetect_0 @ 00000000005db400] histogram_6db: 32
[Parsed_volumedetect_0 @ 00000000005db400] histogram_7db: 250
[Parsed_volumedetect_0 @ 00000000005db400] histogram_8db: 470
[Parsed_volumedetect_0 @ 00000000005db400] histogram_9db: 816
[Parsed_volumedetect_0 @ 00000000005db400] histogram_10db: 936
[Parsed_volumedetect_0 @ 00000000005db400] histogram_11db: 1796
[Parsed_volumedetect_0 @ 00000000005db400] histogram_12db: 2632
[Parsed_volumedetect_0 @ 00000000005db400] histogram_13db: 4516
[Parsed_volumedetect_0 @ 00000000005db400] histogram_14db: 6028
[Parsed_volumedetect_0 @ 00000000005db400] histogram_15db: 8846
[Parsed_ebur128_1 @ 0000000002d42400] Summary:

  Integrated loudness:
    I:         -28.6 LUFS
    Threshold: -38.9 LUFS

  Loudness range:
    LRA:         9.0 LU
    Threshold: -49.0 LUFS
    LRA low:   -33.6 LUFS
    LRA high:  -24.6 LUFS


The mean_volume is the RMS value.

(Download the latest nightly/snapshot binary for your platform)