# The relative dispersion of shot lengths

Studies comparing the change in shot length distributions in Hollywood films with the coming of synchronous sound have focused on measures of central location – the mean or median shot length of a film. The change in the mean shot length from the silent to sound era has been put at approximately six seconds, although this figure is suspect due to the asymmetrical nature of shot length distributions; while the change in the median shot length has been estimated at 2.9 seconds. Similar attention has not been paid to the change in the dispersion of shot lengths that also occurred in the shift from silent to sound cinema. In fact, it is common for mean shot lengths to be presented with no measures of dispersion at all and this severely hampers any useful interpretation of the results.

In my study of the impact of sound on shot length distributions I noted that the interquartile range of sound films was greater than those of silent films, indicating that there is greater variation in the shot length distributions of the sound films. While this method of comparing the variation of shot length distributions is perfectly fine, it is not perhaps the simplest method and using measures of relative dispersion may prove easier to interpret.

## Measures of Relative Dispersion

In order to compare the relative dispersion of shot length distributions, three measures of relative dispersion were calculated for each film from a sample of Hollywood silent films produced from 1920 to 1928 inclusive (n = 20) and from a sample of sound films produced in Hollywood from 1929 to 1931 inclusive (n = 30) (see my earlier study for the descriptive statistics of these films). The mean values of each coefficient for the two samples were compared using a t-test assuming unequal variances. Calculations were conducted using Microsoft Excel 2007 and GraphPad Instat v3.10 (2009).

The three measures of dispersion considered are the coefficient of variation (CV), the quartile coefficient of dispersion (QCD), and the coefficient of median deviation (MD). The relative measures of dispersion for the silent films are presented in Table 1 and for the sound films in Table 2.

TABLE 1 Relative measures of dispersion for Hollywood silent films, 1920 to 1928

TABLE 2 Relative measures of dispersion for Hollywood sound films, 1929 to 1931

## Coefficient of variation

The coefficient of variation is the ratio of the standard deviation to the mean:

CV = SD/M

The coefficient of variation for the sound films (M = 1.1912, SD = 0.2319) is greater than those silent films (M = 0.9015, SD = 0.1393), t (47) = 5.5217, p = <0.0001. On this measure of dispersion, the shot lengths of a Hollywood sound film are more dispersed by almost a third (32.14%) than the silent films.

## Quartile coefficient of dispersion

The quartile coefficient of dispersion is calculated using the lower (Q1) and upper (Q3) quartiles of the shot length distribution:

QCD = Q3-Q1/Q3+Q1

The quartile coefficient of dispersion for the sound films (M = 0.5748, SD = 0.0617) is greater than those silent films (M = 0.4833, SD = 0.0522), t (45) = 5.6409, p = <0.0001. On this measure of dispersion, the shot lengths of a Hollywood sound film are more dispersed by almost a fifth (18.83%) than the silent films.

## Coefficient of median deviation

The coefficient of median deviation is the ratio of the median absolute deviation from the median shot length (MAD) to the median shot length [1]:

The coefficient of median deviation for the sound films (M = 0.5825, SD = 0.0680) is greater than those silent films (M = 0.4735, SD = 0.0473), t (47) = 6.6813, p = <0.0001. On this measure of dispersion, the shot lengths of a Hollywood sound film are more dispersed by almost a quarter (23.01%) than the silent films.

## Discussion

All three measures of relative dispersion provide similar results, but the coefficient of median deviation is the most reliable.

While the coefficient of variation makes complete use of the data and is the best understood of measures of relative dispersion, it relies on the mean shot length. As the distribution of shot lengths in a motion picture is typically positively-skewed with a number of outlying data points, the mean shot length is an unreliable statistic of film style. Consequently, the coefficient of variation can be expected to overestimate the dispersion of shot lengths in a film as the mean value is pulled towards the higher end of the distribution.

The quartile coefficient of dispersion is not dependent upon the mean shot length and so provides a more robust estimation of relative dispersion than the coefficient of variation. A drawback is that it uses only a limited amount of information in calculating the coefficient, and as a film may feature shot lengths that are much greater than the upper quartile it may underestimate the actual dispersion of shot lengths.

Like the quartile coefficient of dispersion, the median deviation does not use the mean shot length and can be relied upon as a more robust measure of relative dispersion. The median deviation has an advantage over the quartile coefficient of dispersion in that it uses more of the data by calculating the absolute deviation of each shot length from the median rather than relying on just two positional values. The quartile coefficient of dispersion can be regarded as an estimator of the coefficient of median deviation for the films looked at here.

In conclusion, we can say that with the introduction of synchronous sound to Hollywood in the late-1920s we not only see an increase in the median of the shot lengths of a motion picture, but also an increase in the variation shot lengths of sound films relative to silent films. Using the coefficient of median deviation we can estimate that increase to be of the order of 23%.

## Notes

1. The coefficient of median deviation is based on the coefficient of mean deviation, but replaces the average absolute deviation with the median absolute deviation in order to prevent extra weight being given to shots of duration that are unusually long.