# Blog Archives

## Power functions and the mean relative frequency of shot scales in motion pictures

UPDATE (26 October 2009): This post has been getting a lot of traffic recently, and I think it is important to point the reader in the direction of a follow up post (here) where I point out that while the use of rank-frequency plots is useful for analysing film style (see here, for example) the power laws approach is not. This is not to say that films or groups of films will not be described by a non-linear power model, but other non-linear models (logarithmic, exponential) are also evident and there does not seem to be any general rule for which model can be applied to specific types of films (e.g. genres, eras, etc.). In general, a power laws approach to the distribution of shot scales is not going to get you anywhere – certainly not to the extent that I suggest in this post. I’ll leave this post here because there is still some useful information, and it’s also nice to see how wrong you can be.

Power functions describe a wide range of social phenomena, from the distributions of city size to the popularity of websites, the citations of academic papers, and the frequency of words in the corpus of a language (Schroeder 1991: 33-38, 103-119; Newman 2005). While power functions have been used for over half a century in analysing language and communication (e.g. Zipf’s law) they have yet to be applied to the empirical analysis of film style. This brief survey looks at the applicability of power functions in describing the distribution of the mean relative frequency of shot scales in the films of two directors – Alfred Hitchcock and Fritz Lang – whose careers encompass both European and Hollywood filmmaking.

Data on the frequency of shot scales was collected from the Cinemetrics database. Seven shot scales were used – big close-up, close-up, medium close-up, medium shot, medium-long shot, long shot, and very long shot (see Salt 2006). The relative frequency of shot scales in a motion picture was calculated by dividing the frequency with which each scale occurred by its normalising value (~500), and this data was then ranked from the event of the highest frequency to the lowest. The average value of the seven ordered relative frequencies was taken to give the mean relative frequency of each shot scale, thereby removing the problem of films that have zero frequency for a particular shot size. This data was then used to calculate predicted values for linear regression (*f(x) = ax+b*, where *a* is the slope and *b* is the intercept of the regression line) and power regression (*f(x) = cx ^{α}*, where

*c*is a constant equal to the frequency of the most frequently occurring scale and

*α*is the exponent of the distribution such that Σ

*f(x)*= 1). The coefficient of determination (

*R*) was used as a measure of goodness-of-fit for the predicted mean relative frequencies to the empirically observed values.

^{2}Data on shot scales was taken from the Cinemetrics database for 43 films directed by Alfred Hitchcock between 1925 and 1963, of which 23 were produced in the UK between 1925 and 1939; and from 21 films directed by Fritz Lang between 1919 and 1955, of which 11 were produced in Germany between 1919 and 1933. The results are presented in Table 1, and show that power regression provides the better model only for Lang’s German films, while for the other classes the linear model is superior. These results can be seen clearly in Figures 1-4, in which the observed values and the linear and power regression lines are plotted on linear axes. (The power regression line is straight when the log rank is plotted against the log frequency).

**Table 1** Linear and power regression for the films of Alfred Hitchcock and Fritz Lang

Table 1 also shows that while the distribution of the mean relative frequencies of shot scales in the films of Alfred Hitchcock are consistent for his British and Hollywood films, there is a change in Lang’s style in his shift from Germany to Hollywood. It is also worth noting the similarity in the figures of *R ^{2}* for both linear and power regression for Hitchcock’s and Lang’s Hollywood films, which suggests that both filmmakers are working within a consistent institutional style (such as classical Hollywood cinema) rather than auteurist idiosyncrasies. The value

*c*is the mean relative frequency with a rank of 1, and this too is similar for Hitchcock and Lang’s Hollywood films, reinforcing the idea of an institutional style. This does not, however, account for why Hitchcock’s British films are so similar to his Hollywood movies. It is possible that German cinema in the 1920s was different from British and Hollywood cinema in general, and that British films style was influenced by Hollywood, but a larger scale study is needed to resolve these questions.

**Figure 1** Linear and power regression for the mean relative frequency (MRF) of shot scales in Alfred Hitchcock British films, 1925-1939

**Figure 2** Linear and power regression for the mean relative frequency (MRF) of shot scales in Alfred Hitchcock’s Hollywood films, 1940-1963

**Figure 3** Linear and power regression for the mean relative frequency (MRF) of shot scales in Fritz Lang’s German films, 1919-1933

** **

**Figure 4** Linear and power regression for the mean relative frequency (MRF) of shot scales in Fritz Lang’s Hollywood films, 1936-1956

Table 1 also shows that while the distribution of the mean relative frequencies of shot scales in the films of Alfred Hitchcock are consistent for his British and Hollywood films, there is a change in Lang’s style in his shift from Germany to Hollywood. It is also worth noting the similarity in the figures of *R ^{2}* for both linear and power regression for Hitchcock’s and Lang’s Hollywood films, which suggests that both filmmakers are working within a consistent institutional style (such as classical Hollywood cinema) rather than auteurist idiosyncrasies. The value

*c*is the mean relative frequency with a rank of 1, and this too is similar for Hitchcock and Lang’s Hollywood films, reinforcing the idea of an institutional style. This does not, however, account for why Hitchcock’s British films are so similar to his Hollywood movies. It is possible that German cinema in the 1920s was different from British and Hollywood cinema in general, and that British films style was influenced by Hollywood, but a larger scale study is needed to resolve these questions.

This survey has shown that while power functions can be used to describe the distribution of mean relative frequencies of shot scales in motion pictures, they cannot be applied universally and linear functions may provide a better means of modelling film style. These distributions may be used as a measure of film style in order to distinguish between different groups of films. However, this approach cannot tell us what changes in the use of shot scales have occurred. It is necessary, then, to look more closely at where the continuities and discontinuities lie. As I have shown elsewhere, in there is a shift in the use of particular shot scales in the films in the films of Hitchcock and Lang when they arrive in Hollywood (Redfern 2009, unpublished). For both directors we find that there is a shift from distant shots to closer framing. Armed with the knowledge that different regression models explain the distribution of the mean relative frequencies of shot scales for Hitchcock and Lang prior to their arrival in Hollywood, we can extend this argument to state that: (1) Hitchcock’s Hollywood films feature closer framing than his British films, but there is no change in the distribution of the mean relative frequencies of the scale overall; and, (2) Lang’s Hollywood films feature a large change in the distribution of the mean relative frequencies as well as a shift to closer framing. In Lang’s German films it is the long shot that dominates, while in his Hollywood films there is no single shot scale that determines the films’ style.

## References

Newman, M.E.J. (2005) Power laws, Pareto distributions, and Zipf’s law, *Contemporary Physics* 46 (5): 323-351.

Redfern, N. (2009) Shot scales in the films of Fritz Lang.

Redfern, N. (unpublished) Cinemetric analysis of shot types in the films of Alfred Hitchcock.

Salt, B. (2006) *Moving into Pictures: More on Film History, Style, and Analysis*. London: Starwood.

Schroeder, M. (1991) *Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise*. New York: W.H. Freeman & Co.