FILMS AND STATISTICS:
Give and Take

 

Lines and Graphs

It would be nice if 'dramatic tension' could be quantified, so that we could get on with the important job of analysing its relation to all the visible features of movies. It IS something that is conceivably possible, but far away at present. In the meantime, we are just messing about with what little can be got from measuring the shot lengths in films. Nowadays, the only measurement of shot lengths that matters is in hours, minutes, seconds and frames. In the Cinemetrics system, the number of frames is reduced to a duration measured in tenths of a second, but this involves an irregular rounding up and down process from the original 24 frames per second at which motion pictures are still shot.
Mostly this does not matter, but sometimes it is better to return to using the original measurement of shot lengths in terms of their length in film frames. There are also some minor problems in measuring the actual length of a shot which depend on the type of transition used by the film-makers between one shot and the next. The majority of shot transitions are straight cuts, which are unproblematic. But most films also include some dissolves from one shot to the next. Here, the obvious transition point is halfway down  the length of the dissolve. That is, one counts frames from the frame in which the first faint part of the image of the incoming shot is visible to the frame in which the last faint part of the image of the outgoing shot is visible, and then halves this count to get the exact transition point. However, some people, including James Cutting's team, estimate this transition by looking for the frame in which the two superimposed images appear equally bright. This will probably be correct to within a frame or so if the incoming and outgoing shots have the same average overall brightness. But quite often one of the shots will be much darker than the other, and if this is so, the estimate of the mid-point of the transition can be out by several frames. Sometimes, the transition from one shot to the next is accomplished by a fade-out followed by a fade-in. After the frame in which the last sign of the outgoing shot is visible at the end of the fade-out, there are a number of frames of black before the beginning of the fade-in of the next shot. This number is usually fairly large, of the order of hundreds of frames, or seconds in time. For their purposes, the James Cutting team count this length of black as a shot, though film-makers and others would not consider it as such. This can create a discrepancy between the Cutting team's results and those of other people working in a frame-accurate way. This is not particularly important in general, and it certainly wouldn't matter if we are comparing it with the approximate length measurements generated by the original Cinemetrics application, where lengths are measured on the fly as the film runs past. Another arguable point in measuring shot lengths is for films in which there are a number of split screen shots. That is, the film frame is divided up into a polyptych with two or more different images visible within it. These subsidiary images can change while the polyptych is visible on the screen, sometimes quite frequently. My practice is to count any such change as a transition to a new shot, but at least one other analyst just counts the whole length of the split-screen sequence as one shot.

Using the Lines

            So having measured the lengths of the shots in a film, they will be represented by a string of numbers in the first place. One can look along this string of numbers and spot any interesting regularities, but it sometimes is easier to see this from a graph in which the lengths of vertical lines are proportional to the size of the shot lengths. It is also easier to get the information onto a page of paper in this form. In modern times, the first visual and non-numerical representation of shot lengths was on the screen of a computer running a non-linear editing program, where the lengths of the shots are represented by horizontal lengths, but more recently we have the Cinemetrics graph, where the shot lengths are represented by the lengths of vertical lines, as in the more orthodox inverted form of the same.
As always, what is of most interest is the relation between the content of the film and its form. Thirty years ago I started investigating how the cutting rate within a film scene varied depending on the type of action within it. This is quite easy to show, and does not require any mathematics beyond counting and averaging. You can see it in all editions of my Film Style and Technology: History and Analysis. Much more recently, I made an attempt at locating the boundaries between scenes by taking repeated rolling averages of the shot lengths in a film (in 'Speeding Up and Slowing Down' in the articles section of the Cinemetrics website.) Unsurprisingly, this approach proves to be only capable of detecting the approximate position of the scene boundaries, and that only in the best cases.
The other thing one can study is the shot length frequency distribution for a film. The most basic way of doing this requires no more mathematics than simply the counting of the number of shots in the film whose lengths fall within a series of fixed intervals -- say from zero to one second, greater than one second to two seconds, and so on -- and then drawing a bar chart where the heights of the columns represent those numbers. When looking at the resulting graph, one is seeing the numbers of shots in each 'class interval' or 'bin' directly, without any mathematical transformation of the data. This procedure is laborious, but it can be automated nowadays using a computer spreadsheet. In the data analysis tools in Excel, there is a function called 'Histogram', which when supplied with a list of shot lengths and a list of the class intervals ('bins') to put them in, produces an array like this one for the film Casablanca (1942).



It so happens that a class interval of one second works quite well for films with an ASL in the region from about 5 seconds to about 12 seconds, which is where the vast majority of films made from the 'thirties into the 'fifties dwell. So for Casablanca I used a class interval of one second.
The first row interval, numbered 1, has the total number of shots (which is 23), with lengths between zero seconds and up to and including 1 second. That is, it includes lengths 0.1 seconds, 0.2 seconds, 0.3 seconds, and so on. The second bin or class interval contains a count of the number of shots greater than one second, and up to and including shots with exactly two seconds length. That is, lengths 1.1 seconds, 1.2 seconds, etc. The third bin or class interval contains the number of shots with lengths from 2.1 seconds to 3 seconds, and so on. The 'More' bin contains the number of shots with lengths greater than 50 seconds. Other spreadsheets can also do this, but the name of the appropriate commands varies. Having got this distribution, it can be graphed with the graph function in a spreadsheet, and the result looks like this after the width of the bars in the chart has been adjusted so that they touch one another to give a true histogram:


(The 'More' column at the right end of the graph represents the total number of shots with lengths greater than 50 seconds.)
This graph is a bit jagged and lumpy, and we can smooth this out by using a class interval of width two seconds, as in the following graph.

For films with a really short ASL, of the kind that have emerged in recent decades, things are slightly different. Take Shoot 'em up (2007), with an ASL of 1.64 seconds.

This distribution has a smooth profile, but it is not very informative, with 1484 shots, almost half the total, in the first interval containing those shots between 0 and 1 second. If we change the time measurement from seconds to frames, and makes the class interval 8 frames (or a third of a second) wide, it looks like this:


As well as providing more precise information about how many shots there are with each length, the shape, though equally smooth, shows the dive from the mode at 16 frames to the origin, which is characteristic of film shot length frequency distributions. Now if we go for broke, and decrease the class interval to one frame, we get:


This graph has a much more jagged shape than the previous one, but it also shows what was not visible before, which is the way the distribution dives towards the origin. This can be made clearer with an enlarged view of the beginning of this graph from zero up to nine frames.

 

I have added the theoretical values derived from the Lognormal distribution that best fits the actual distribution of shot lengths for this film. Although the fit is not perfect in this region, you can see the way the actual values show the same sort of approach to the origin as the theoretical curve. The way the curve approaches the origin asymptotically is particularly characteristic of the Lognormal distribution, and it is gratifying that it appears in the actual experimental data as well. This effect also appears in the distributions for other films with an extremely short ASL, such as Derailed, but not in the distributions for films with longer ASLs, which have in general no shots shorter than eight frames.

            If one wants to compare the shape of two distributions with closely similar ASLs, one can interleave them on the graph, as in this comparison of Shoot 'em up and Derailed (2002).


The resemblance is very close, which is not surprising since the median for both distributions is 1.04 seconds, though there is a small difference in their ASLs (1.59 seconds for Derailed, and 1.64 seconds for Shoot 'em up.)
To actually measure the difference between the two distributions, we can get the Pearson correlation coefficient, which is 0.992, and indicates the closeness of the two distributions.

Going Slower

Looking at the slow end of film cutting rates, this is what we get from graphing Sunset Blvd., which has an ASL of 15.5 seconds, with a one second class interval.


This has a pretty rugged outline, which is not too surprising, given the low number of shots in each interval. So let us try a class interval of two seconds.


That is a little bit better, but more importantly it is starting to get the characteristic shape we expect from film shot length distributions. So let's try an interval of 4 seconds.


I like that shape or profile; it looks like Derailed or Shoot 'em up in their 8 frame class interval incarnation, and also the two second interval graph of Casablanca.
Moving upwards, Panic in the Streets (1950) has an ASL of 24.3 seconds, and in this case its distribution shown with a one second class interval looks like this:


This is a lot more jagged a shape than that for Sunset Blvd., and an obvious idea is to see if the isolated peaks at 17 to 18 seconds, and 34 seconds have any significance. An examination of how the shots of these lengths occur in the film with respect to what is going on in them shows no relation to the lengths of the lines of dialogue being spoken in them, for instance. (Panic in the Streets includes lots of dialogue, and none of the panic in the streets promised by the title.) Nor can I see any relation to any other variable related to the film's content that occurs to me.
Even when the class intervals are widened to 4 seconds, the distribution does not smooth out that much, unlike Sunset Blvd.:


In other words, as we get towards, and past an ASL of 20 seconds, we are in a region where any close resemblance to any standard probability distribution is vanishing, as I have often said. To rub this point in, I recall the distribution for la Signora senza camelie (1953), which has an ASL of 59.4 seconds, and is shown here with a class interval of ten seconds:

Even with such a large interval, the usual characteristic shape is only vaguely there, and the distribution is quite jagged.

Looking For Significance

            My search for a reason for the excessive number of shots with lengths of 17 to 18 seconds in Panic in the Streets failed, but one failure is not enough to stop looking for reasons for such peaks in shot length distributions. Perhaps in musicals there might be specially favoured lengths in the way that musical numbers are edited, related to the regular structure of the pieces of popular music being used on the sound track? Say cuts only at the beginning and end of each chorus of a ballad being sung or played, or every four bars, or something similar.
My first candidate is Good News (1947). The shot length distribution for this film with 2 second class intervals is:


The class of shots with lengths of 39 or 40 seconds sticks out, but when looking down the table of lengths for this film, and seeing where they occur in the films, shows that they nearly all appear separately, and mostly in dialogue scenes. (I find that the best way to study this matter is to use a table of the shot lengths written down in order in a spreadsheet in conjunction with a copy of the film in a non-linear editor, with the length of the shots marked on the time-line.) And there is no sign of groups of shots of nearly equal length in other places down the list of lengths.
Another try with Singin' in the Rain (1952) failed in the same way, and indeed when simply playing a DVD of the film it is apparent that the editor was again not cutting the musical numbers in any rigid way at the end of each chorus, or the end of each phrase, or in even multiples of the bar length. However, a final try with Anchors Aweigh (1945) did just slightly better. (All the previous examples use my own data, but here I am using the data from the James Cutting team in the Cinemetrics database.) 


The column representing shots of length between 14 and 15 seconds looks a bit too big, and indeed from shot 421 to shot 426 there occurs the following sequence of lengths:

 14.1, 13.5, 27.3, 14.2, 13.5, 14.4.

But these shots cover a conversation in the studio cafeteria between Gene Kelly and Kathryn Grayson, not a song. However, after failing to find what I was looking for in about a score of musical numbers in all three films, I got lucky with just one song in Anchors Aweigh, which is the song "The Charm of You" sung by Frank Sinatra to Pamela Britton in a Mexican-type restaurant. This runs from shots 463 to 473. The shot lengths are, in sequence: 

26.7, 6.3, 5.7, 6, 6.8, 6.6, 7.5, 6.7, 12.1, 8.6, 23.9.

The first shot covers the whole 32 bars of the first chorus of the song, then the subsequent shots go to different angles on the pair of lovers for each subsequent 8 bar line over two more choruses, and then the shots lengthen out to cover the last chorus with three shots, which depart from the 8 bar pattern. If you check with the Cutting data in the Cinemetrics database, you will find that he gives the first shot as 31.5 seconds. This is definitely wrong, and there seem to be a number of similar errors in his data for this film. The negative figure of -0.5 seconds for shot 396 is another of them. Shot 396 does not actually exist, so if you remove it from the list you will have something a bit closer to the actual lengths, but still having many shots with slightly wrong lengths. Although there is hardly any cutting on regular numbers of bars in these musicals, there is a visible tendency to cut about a second before the end of a chorus in musical numbers. It is possible that some regularities might emerge in this area in the way that they have in my examination of dialogue cutting in the article Reaction Time: How to edit movies in The New Review of Film and Television Studies (Vol. 9, No. 3 September 2011).
It is quite possible that if one examines musical numbers in recent decades, where the kind of pop music used is very different to that of the 'forties, and the cutting is much faster, then one might find a greater regularity in the cutting of such numbers. The observations about a scene in Wedding Crashers (2005) on page 438 of the paper Attention and the Evolution of Hollywood Film, published in Psychological Science (2010, 21:432) by James Cutting, Jordan DeLong, and Christine Nothelfer is a case in point.

The moral of my story is that you can do plenty, and see what is going on, by staying close to the data, without gussying it up with excessive manipulation.

                                                                                                            Barry Salt, 2013