Barry Salt: Graphs and Numbers

Back in 1974, when I initiated the systematic study of film style using statistics, which I called ‘Statistical Style Analysis’ (as I still do), Sight and Sound rejected an article showing my first results, which of course contained graphs and tables of numbers, although they had just published my piece putting forward a general theoretical framework for film analysis, ‘Let a Hundred Flowers Bloom’. Discussing this quite some time later with Ray Durgnat, I said to him that most people’s minds just freeze up when they see a graph. He riposted that it was worse than that, for most peoples minds freeze up when they see a decimal point. Fortunately, ‘The Statistical Style Analysis of Motion Pictures’ was published later that year by Film Quarterly, because the editor had a bit of a scientific education, before he got into writing about movies.

I was already well aware of the problem that most people have with mathematics, so although I briefly indicated in ‘Statistical Style Analysis of Motion Pictures’ what I thought was the nature of shot length distributions in feature films, I did not go further into the matter until much later, but restricted myself to only the most basic use of statistics when dealing with other areas of film style, because I was seeking the widest audience I could get. That audience might be just able to cope with the notion of the arithmetic mean of a collection of numbers, as long as it was referred to as an average, though it is doubtful if many of them could actually calculate it. Obviously I was already using the median of the shot lengths of a film back then in fitting theoretical distributions to the empirical distributions with which I was working. So I was never in any doubt that the median was a necessary measure for this purpose. But how many people can calculate it if given a set of numbers, and pencil and paper?

Nick Redfern seems to be suggesting banning the use of the concept of the Average Shot Length, but he surely can’t be serious. Such an idea seems reminiscent of the Catholic church continuing its ban on the discussion of the idea of the earth going round the sun, even after the concept was in wide use. As I have shown, in ‘The Metrics in Cinemetrics’ and elsewhere, the combination of the standard statistical measures, including the ASL and the median, can reveal major new features of the shot length distributions of feature films. The most important of these is that most sound feature films with an ASL of less than 15 seconds have a shot length distribution of roughly the same shape. Which raises a big question, why should this be so?

Another concept being put up for discussion is the idea of naming some members of a distribution as ‘outliers’, and then excluding them from consideration. I have already indicated why this is a bad idea in ‘The Metrics in Cinemetrics’. As another example, I will use the shot lengths for The Grapes of Wrath, as recorded by myself and placed on the Cinemetrics database.

Figure 1

That little bar on the end of the graph represents the three shots in the film longer than 100 seconds. They are actually 100 seconds, 104 seconds, and 160 seconds in length respectively. You might think they are far detached from the rest of the shots in the film, which are all shorter than 70 seconds, but actually the theoretical distribution that best approximates the actual distribution of shots predicts a certain probability of shots occurring beyond 70 seconds. The probability of any particular length of shot beyond 70 seconds is very low, but if you sum these probabilities from 70 seconds to infinity, then they say that there is likely to be between one or two shots of lengths greater than 70 seconds out there. So having three shots is more than expected, but not a lot more. Turning to the idea of leaving these three shots out of consideration, this would of course remove the uniqueness of this film. To be more specific, the 104 second long shot is the second last shot in the film, in which Ma Joad having the last word in the film by delivering her long speech ending with ‘We’re the people’ in a close two-shot showing her and Pa Joad sitting in the front seat of their truck heading towards another fruit-picking job. Her speech is actually interrupted by two short responses from Pa Joad within the shot, and these short speeches by Pa Joad could have been used to break the scene up into more shots. Another more ordinary director might well have done so, but the way John Ford chose to shoot it is one of the many details that make this film special, and indeed unique.

The series of shot lengths making up any film is also unique, and so is the distribution of lengths resulting from them. They make up the whole population with which our statistical analysis of a film deals, and are not a random sample from some larger population. So any test or method which assumes that they are part of a larger population is being misapplied. However, the shape of the distributions of shot lengths of most feature films do resemble each other, without being identical. How close they can get in shape can be illustrated by taking three films very different in ASL (and median shot length), from very different periods, and then superimposing their shot length distribution graphs. The films are Ace Ventura: Pet Detective (1995), with ASL = 4.7 sec. and Median = 2.7 sec., The 39 Steps (1935) with ASL = 9 sec and Median = 4.3, and On the Beach (1959) with ASL = 19.1 sec. and Median = 8.8 seconds. The values of shot lengths have been normalized by dividing the number of shots in each film into the frequencies for each length of shot, and the shot durations on the y-axis have been adjusted in the cases of Ace Ventura and The 39 Steps by dividing by 2.7/8.8 and 4.3/8.8 respectively. (That is, the ratio of the Median values for each of the first two films to that of the last film mentioned.) Incidentally, I could have used the ASL instead of the Median in this normalization, and got almost identical results.

Figure 2

The coincidence of the three graphs shows that they do indeed have almost the same shape, and the small discrepancies correspond to the general way the shape of shot length distributions changes slightly as we come up to more recent times and faster cutting. In the case of the Lognormal distribution, to which most film shot length distributions approximate if the ASL is less than somewhere around 15 seconds, the shape parameter sigma is fairly constant, and it is this that produces the coincidence of the distributions illustrated above.

I am very pleased to see that Mike Baxter’s detailed paper endorses the results and positions I have put forward in ‘The Metrics of Cinemetrics’ and elsewhere. The one part of his work I have some small doubts about is his analysis of what he calls ‘lumpy’ distributions. Of the twelve distributions he discusses in this context, not all of them look lumpy to me.

I would agree that when looking at the shot length distribution for Pursuit to Algiers directly one can see lumps:-

Figure 3

but it does not look at all like a bimodal distribution to me. I see no second modal peak standing out from amongst the many small lumps. The only one amongst Mike Baxter’s twelve examples quoted that does have a suggestion of a real second maximum peak when we look at the actual distribution is Harvey:

Figure 4

One could take it, perhaps, that there is a second distribution having its mode at 16 seconds, and the cross-over between the two is around 12 seconds, but then how do you tell which shot in the film is in which distribution? As far as I remember the film, the scene dissection was rather clumsy, and in particular the handling of the long takes.

Anyway, kernel density estimates are done by putting the distribution into a very small number of class intervals, so they could be creating something that is not really there on the finer scale of the actual distributions. Look at the shot length distribution of Foreign Correspondent:

Figure 5

That looks fairly smooth to me as shot length distributions go. Maybe the small deficiency in shots of length between seven and eight seconds freaked out Mike Baxter’s KDE calculation for some reason.

Finally, I repeat the knock-down counter example I gave in ‘The Metrics in Cinemetrics’ to Nick Redfern’s comparison of The Scarlet Empress (1935) and The Lights of New York (1928) with my comparison of the distributions for The Lights of New York and The New World (2005). Both these films have median shot lengths of 5.1 seconds, so on this ground alone you might think they have similar distributions, but when you look at their other features it turns out they are very different.

Figure 6

Figure 7

The crucial feature is that in The Lights of New York there are a substantial number of shots with length greater than 50 seconds, in fact 12 of them, represented by the tall bar at the right end of the graph, whereas there is only one for The New World. The reason for this substantial number of long takes in The Lights of New York is that is subject to the technical constraints on shooting and editing synchronized sound that I describe in Film Style and Technology: History and Analysis. Like many films made at the very beginning of the use of synchronized sound, The Lights of New York is a mixture of scenes done in long takes shot with direct sound, and action scenes that were shot wild and post-synchronized. The use of normal fast cutting in the latter was of course unconstrained by technical factors. Another well-known film that has the same hybrid form is Hitchcock’s Blackmail. However, if we take into account the ASLs of the two films, with 9.9 seconds for The Lights of New York, and 6.8 seconds for The New World, this significant difference is highlighted. So just using the median alone as a statistic can also be very misleading.

Barry Salt, 2012