Barry Salt: The Metrics in Cinemetrics

The original impulse for the creation of Cinemetrics could be said to have been to find some kind of order in the succession of shot lengths that occur over the duration of a motion picture. The most common graphical representation of the shot lengths in a film is through the successive lengths marked down the timeline of a computer non-linear editing program (an NLE), but Yuri Tsivian and Gunars Civjans invented their own graphical representation, which is the Cinemetrics graph. This shows the lengths of the shots represented by the lengths of vertical bars on the negative y-axis of a graph. The x-axis intrinsically shows the ordinal shot number in equal divisions, but it is actually calibrated with a non-linear scale that gives the cumulative time-lapse from the beginning of the film to the end of each successive shot.

At a quick glance, these Cinemetrics shot length graphs are very complex and very varied, but the initial method that was tried to find structures and regularities in them was to plot a polynomial curve of best fit through the shot length values. This is referred to as a “trendline”, because that is what the makers of computer spreadsheet programs call such a line, though “trendline” had long meant something

Y = Ax + Bx2 + Cx3 + Dx4 + Ex5 + Fx6 + Gx7 + Hx8 + Ix9 + Jx10 + Kx11 + Lx12 … etc.

Here “y” stands for the length on the y-axis, and “x” for the ordinal number of the shot in the sequence of shots. The coefficients or parameters A, B, C etc. represent numbers determined to give the curve of best fit through the varied values using the standard “least squares” algorithm. In the case of Cinemetrics, the degree or order of the trendline curve can vary from the first degree, which

The hope was that frequent patterns might be observed in the trendlines over the results for a large number of films, but this hope has not been properly realised so far, though there do seem to be a few shapes that recur sometimes. A more systematic search for favoured shapes in the Cinemetrics trendlines might involve the use of scatter diagrams to see if there is any clumping of the values

Another more mathematically sophisticated approach to finding recurring patterns in the sequence of shot lengths in a film is that of James Cutting and his colleagues, in Attention and Hollywood Films, by James Cutting, Jordan DeLong, and Christine Nothelfer. That paper is discussed in the articles

on the Cinemetrics website.

Tsivian and Civjans initially considered another means of finding a structure in the series of shot lengths, which was to use a moving or rolling average, but this was not implemented until recently. Although it does not detect simple recurring patterns in the shot lengths, it has a use at the next level of investigation, which is relating the patterns in shot lengths to the nature of what is going on

The same sort of idea, but using the Cinemetrics trendlines to relate to the dramatic content of the film dates back to the very beginning of Cinemetrics, with Yuri Tsivian’s initial investigation of Intolerance using trendlines, which can be found on the Cinemetrics website also in the “Movie Measurement Articles” section, and there are other examples of this approach scattered throughout the comments attached to individual Cinemetrics graphs.

A caution about this approach is sounded by taking a Cinemetrics graph for Darby O’Gill and the Little People with a 12th. degree trendline added, as below:

Figure 1

and then comparing it with a similar Cinemetrics graph created by using a random selection of shot lengths which follow the same frequency distribution.

Figure 2

Although you can see that the distribution of long and short shots is more even in the random graph, the trendline for this latter still has a pronounced set of wiggles that suggests a degree of structure equal to that in the real film, even though it actually has no meaning in relation to any film content.

Enter the Metrics

A more general and more abstract method of investigating the shot lengths in a film is to ignore the order in which they occur, and see what mass features can be found in them. The most obvious of these measures or metrics is the arithmetic mean of all the shot lengths in a film, popularly called the average. Less obvious is the median value of the shot lengths, which is that

Statistics and Shot Lengths

Much of the elaborate apparatus of present-day statistics is fortunately not necessary for investigating shot lengths. The reason for this is that most of the methodological superstructure of modern statistics is directed towards estimating how reliable a sample taken from a population is in representing the characteristics of the entire population. But in Cinemetrics we are dealing with all the shot lengths for the entire film, and this is the

One way to investigate the global properties of the collection of shots making up a film is to group the shots purely according to their length. That is, one groups and counts how many shots in a film have lengths between, say, 0 and 1 second, 1 and 2 seconds, 2 and 3 seconds, and so on. These groups of lengths are correctly called “class intervals”, but the popular name for

In this sort of treatment the collection of shots whose lengths are being investigated is called, in general, a population, and their lengths are the variable describing the population being studied. As the text-books tell you, the first thing you should do when investigating how a characteristic variable describing a population varies is to plot it on a graph. So here is a graph illustrating the distribution of shot lengths

Figure 3

This shows, for instance, that there were 246 shots with lengths between two and three seconds out of the 1160 shots making up Darby O’Gill.

Examination of many such shot length distributions shows that they have similar-looking asymmetric profiles, as you can see in the article “The Numbers Speak” elsewhere on the Cinemetrics website. This resemblance suggests looking for a standard theoretical statistical distribution to which they conform. The best-known distribution is the Normal distribution, which is also referred to as the Gaussian distribution in Europe, and as “the bell curve” by the innumerate.

Figure 4

where m is the median value of the distribution of shot lengths, and is what is called the “shape factor”. The probability density has to be multiplied by the total number of shots in the distribution to get the expected number, according to the equation, within each class interval.

The Lognormal distribution results when the quantity under consideration, in our case shot length, is determined as a result of the probabilities associated with a large number of independent causative factors being multiplied together. The Lognormal distribution turns up in various diverse areas; for instance the separation between cars on a motorway. A current hot topic is its use to describe cancer survival rates. One would perhaps not expect

There are various methods of finding the values of the parameters m and to get the closest fit of the theoretical equation, but I still use a standard pencil and graph-paper method that dates back to the years B.C. (Before Computers), when I started this enterprise. You can find this method described in the original, and still standard text, The Lognormal Distribution by J. Aitchinson and J.A.C. Brown (Cambridge Univerity

Figure 5

As you can see, there is not an absolutely exact correspondence, but there is quite enough similarity to warrant seeing where this idea leads us. Since computers became widely available, programs have been developed that perform such a distribution fitting automatically, and here is the result of applying one of these to Darby O’Gill.

Figure 6

At first glance it might look as though it is just as good as my manual result, perhaps even better, so we need a precise way of comparing the accuracy of the two fits. This is provided by the Pearson correlation coefficient, r, which is an efficient and accurate way of comparing two matching sets of data, and it can have values between 1 and -1. If its value is zero,

In the case of the manually derived Lognormal distribution above, r = 0.975407, and for the automatic computer fitting result, r = 0.970011. But from this point onwards, I will use the coefficient R2 , which is the square of r, to measure goodness of fit of the theoretical distribution with the actual observed distribution, because this exaggerates the discrepancy between the two distributions being compared, making the difference more

Nick Redfern has checked 40 shot length records from the Cinemetrics database in his piece "Some brief notes on cinemetrics II" on his website for how well they conform to the Lognormal distribution, and he has found that 20 of them pass the strict test derived from random sampling theory that he is using. He had already done three Chaplin films in his piece "Testing normality in cinemetrics" in the same

The films that seriously fail to conform to the Lognormal distribution must do so for a reason. That is, the makers must either have been forced into it, or have made at least a semi-conscious effort to break away from the norm of the Lognormal distribution, so identifying the reasons that this happens is important.

To return to the Lognormal distribution, its shape is ordinarily defined by two parameters, the median and the shape factor. So the median is a good measure to have in this case, as in others. However, in the case of the Lognormal distribution, these two factors can be derived from the mean (that is, the ASL) and the standard deviation by inverting the two relations:

Figure 7

In other words, all you need in theory to characterize a Lognormal distribution is the standard deviation and the ASL.

So for Lognormal distributions the median IS related to the ASL, and hence the ASL is useful, too. Because actual shot length distributions of feature films are not exactly Lognormal, these relationships cannot be used to derive the value of the shape factor of the Lognormal distribution that is the best fit to any particular actual shot length distribution. In any case, the mean exists as a basic characteristic of

Figure 8

Figure 9

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

The Coefficient of Variation

As Yuri Tsivian has noticed that for feature films, the ASL and the Standard Deviation (STD) are usually of a rather similar size. This is remarkable, and needs explanation, because for distributions in general this is not true. This ratio of (STD)/(ASL) is called the “coefficient of variation”, with the standard abbreviation Cv, and in the case of the Lognormal distribution it depends solely on the Shape Factor through a

Nick Redfern has observed that this feature indicates that there is greater variation in the shot lengths of the sound films than the silent films, and in fact for the 181 silent features (1913-1929) in the Cinemetrics database the Coefficient of Variation is 0.97, and for the 1607 sound features in the database, the Coefficient of Variation = 1.14.

The distribution of values of the Coefficient of Variation for film shot lengths is fairly close to being a Normal distribution, as shown here.

Figure 10

As mentioned, the mean value of Cv for sound features in the Cinemetrics database is 1.14. The correlation between the actual experimental values from the Cinemetrics database and the theoretical Normal distribution given by R2 is 0.97, which is very good.

Now, the general shape of the distribution of Cv suggests to me that mostly film-makers are unconsciously working for some sort of standard mix of long and short shots in their scene dissection, but don’t always quite hit it. However, there are also some who want to put in extra long takes beyond the normal mixture of lengths, and they are represented in the vestigial right tail of the graph above,

The Front Page (1931)

Rain (1932)

Citizen Kane (1941)

Touch of Evil (1958)

Lady From Shanghai (1947)

Macbeth (1948)

Forty Guns (1957)

Ride Lonesome (1959)

Verboten! (1959)

Who’s Afraid of Virginia Woolf? (1966)

Week End (1967)

En Passion (1969)

Tout va bien (1972)

Electra Glide in Blue (1973)

1900 (1976)

Paris, Texas (1984)

Wild at Heart (1990)

Amores Perros(2000)

Code Unknown (2000)

Yo (Me) (2007)

You will notice that after two early sound films, there are none from the rest of the ‘thirties. Although these films nearly all have long ASLs, Amore Perros (ASL = 4.9) shows that the association of large Cv with large ASL is not necessary. Conversely, long ASLs do not necessarily produce large values of Cv, as shown by Werckmeister Harmonies (not in the list) with an ASL of 219 seconds, but

The distribution histogram for Ride Lonesome shows the sort of thing that is going on in its cutting, which is somewhat like that in the earliest sound films such as Lights of New York.

Figure 11

Again the film contains a number of very long takes, with 16 shots greater than 50 seconds in length. Partly because of that, the fit of the theoretical Lognormal distribution is not particularly good. These long take shots could reasonably be referred to as “outliers” in this particular case, but to disregard their existence in an investigation is to shut your eyes to the very thing that makes this film

There is also another larger group of sound films in which the director has set out to do the whole thing in long takes, and such films typically have ASLs longer than 15 seconds. For example, here is the shot length distribution for On the Beach (1959), which has an ASL of 18.4 seconds.

Figure 12

The shape here is flattened, and there are many more of the shots in the long tail, which extends beyond the right end of the graph, with 36 shots having lengths greater than 50 seconds. In fact, 75 shots of the 407 shots making up the film have lengths greater than 30 seconds.

Median/ASL Ratio

I have noticed that there is also a relatively fixed ratio between the ASL and the Median for movies, and now, with a bit of help from Gunars Civjans, I have the proof of this. Considering all the 1520 sound fiction feature films in the Cinemetrics database, I find that the average ratio of their Median to their ASL is 0.620. The values are clustered quite close to this average value,

Figure 13

This distribution has the common Normal (or Gaussian) shape, and its Standard Deviation is 0.124 . This is the reason that you can roughly predict what the Median value for any film will be, just from its ASL. Putting this another way, 82% of films have a Median/ASL coefficient in the range 0.5 to 0.7. This is of course another remarkable fact, and again demands explanation.

I had predicted to myself that removing films with an ASL of 15 seconds and higher, which is the region where the fit to the Lognormal distribution begins to break down, would sharpen the above distribution, but in fact doing this makes hardly any change to the shape of the distribution, or to its mean value. However, removing non-American films from the population does sharpen the relation slightly, without changing

When we turn to all the 186 silent fictional features in the Cinemetrics database, the Median/ASL ratio changes appreciably, to a mean value of 0.711, and a standard deviation of 0.082 , as in the graph here:

Figure 14

But concentrating just on the 92 American silent fiction features in the Cinemetrics database has very little effect on the result we get. (Median/ASL ratio = 0.714, with a standard deviation of 0.090).

Since the Median/ASL ratio relates approximately to the shape factor for Lognormal distributions, this means that all the sound films have fairly similar shape factors, and the silent films mostly have a different shape factor. Why there is an appreciable difference between sound and silent films in the shape of their shot length distribution is yet another interesting question.

Actually, it turns out that the difference in the Median/ASL ratio between American sound and silent is more complicated than I thought at first. Closer inspection of the American sound feature film corpus on the Cinemetrics database shows that the ratio varies a bit with the magnitude of the ASL. To take an extreme case, the mean Median/ASL ratio for the 25 feature films with ASL less than 3 seconds is

So what we have here is a correlation between the Median/ASL ratio and the ASL for American sound films. It is not a strong correlation, because when we calculate the correlation coefficient (r) for this relation for all the American sound films, it comes out at about 0.3.

Turning back to the 84 American silent films in the database, we find that only 17 of them have ASLs greater than 7 seconds, so their Median/ASL ratio of 0.73 may reasonably be compared with that of 0.68 for the group of sound films with ASL less than 7 seconds. However, the remaining difference between silent films and the group of faster cut sound films remains to be addressed.

Experimental Science

Up to this point, we have just been groping around with descriptive statistics. It is time to start being real scientists, and look for the cause of the phenomenon. The most obvious difference between sound and silent films in this context is that the silent films have dialogue intertitles. The unexamined convention in film analysis is that a dialogue intertitle should be counted as a shot. But up to the

So the first simple thing to do is look at any silent films that have no dialogue titles. There are very few of these, with the much the best known being Der letzte Mann (1925). This has a Median/ASL ratio of 0.63, just like the average sound film. This is encouraging, but it could be a lucky fluke. How to bring more silent films into the enquiry? The fairly obvious thing

So here is the shot length distribution for It (1927), counting the dialogue titles as shots in the usual way, and with the theoretical Lognormal distribution corresponding to the shape factor and median shot length determined from the actual distribution of shot lengths imposed on the histogram as well.

Figure 15

I next removed the dialogue titles in a non-linear editing program to create a new version of the film without them. In American silent films of the ‘twenties, most dialogue titles occur between shots of the scene taken from different camera positions, so removing them does not disturb the length of the remaining shots. But in the case where the dialogue title has been cut into the middle of a continuous

Figure 16

Although it is not obvious from the graph, the theoretical distribution is actually not quite so good a fit (with R2 = 0.945) as it was for the original film with the titles in it, which has R2 = 0.968. But our real concern is with the Median/ASL ratio, which was 0.76, but is now reduced to 0.68, and so the distinction between the shapes of the shot length distributions for

I next treated two more silent films in the same way. Here are the median/ASL ratios (which determine the shape factor for Lognormal distributions) for three American silent films. They are given for the original form of the film, with dialogue titles treated as shots, and then for the film modified by omitting the dialogue titles.

(MOST IMPORTANT: When this has been done, if the dialogue title (or titles) are cut into the middle of what was obviously one continuous take, then the divided parts of this take are joined together again to make one continuous shot. This does NOT give the same result as leaving these fragments as separate shots in the analysis.)

Figure 17

Now for American silent films made between 1920 and 1928 in the Cinemetrics database, the ASLs cover the range from 3.3 seconds to 7.6 seconds, and the mean value of the Median/ASL ratio is 0.72 for this group. So what we need for a comparable group of sound films are those that cover the same range of ASLs. When this comparable group has been selected, they are found to have a

Although it is not necessary for accuracy of my demonstration above, to actually see the difference between the distributions with, and without, dialogue titles, I add the distribution graphs for Seventh Heaven and Little Annie Rooney to that for It shown above. Here is the shot length distribution for Seventh Heaven (1927), counting the dialogue titles as shots in the usual way, and with the theoretical Lognormal distribution corresponding

Figure 18

Then we have the shot length distribution with the dialogue titles cut out, as previously described.

Figure 19

The correlation coefficient, which indicates the goodness of fit of the actual distribution to the theoretical Lognormal distribution using the median and shape factor derived from the actual values goes from R2 = 0.974 to R2 = 0.962 on removal of the titles. Both these values imply a fairly good fit to the Lognormal distribution for this data, though a whisker better for original distribution.

The similar results found for Little Annie Rooney are as follows:

Figure 20

Figure 21

In this case the fit between the actual and theoretical distributions is given by R2 = 0.876 when including the dialogue titles, going up to R2 = 0.964 when they are removed. This indicates that there is an appreciably better fit to the Lognormal distribution when the dialogue titles are cut out in this particular case.

If we look at the distribution of the lengths of dialogue titles on their own for Seventh Heaven, we get the following graph. (The results are similar for Little Annie Rooney and It.)

Figure 22

You can see immediately that this distribution is a quite different shape to that of a Lognormal distribution, as it lacks the extended right tail, and also that it has broader shoulders.

Removing these dialogue titles has the appreciable effect on the original distributions that we have observed because in the typical American silent film they make up a substantial proportion of the number of shots in the film. The proportion of dialogue titles in an American silent film is usually around 15% of the shots in the film, though this is not true for slapstick comedy, which uses much less titling

Incidentally, this last part of my investigation represents the first appearance of a sub-discipline that might be called “experimental film history”, as a kind of analogue to the “experimental archaeology” that has come to the fore in archaeology in recent times.

Barry Salt, 2011