FILMS AND STATISTICS:
Give and Take

 

The average shot length as a statistic of film style

A shot length distribution is a description of the data set created for a film by recording the length of each shot in seconds. Since a film typically comprises several hundred (if not thousands) of shots we need summary statistics that accurately reflect the style of a film in order to simplify our analysis and to communicate our results. The average shot length (ASL) is the most commonly cited statistic of film style, and is used to describe how quickly a film is edited with a low ASL representing a fast editing style and a high ASL indicating a slow cutting rate. By comparing the ASLs of two films we can determine if they have similar styles or if one film is edited more quickly than the other.

The ASL commonly referred to in film studies is the arithmetic mean and is equal to the sum of the data values (i.e. the total running time) divided by the number of shots. The mean is the point at which a data set is balanced, and as a 'centre of gravity' is a representative statistic of central tendency when the distribution of the data is symmetrical. However, the distribution of shot lengths in a motion picture is characterised by its lack of symmetry so that the majority of shot lengths are less than the mean due to the influence of a number of shots that are of exceptionally long duration relative to the rest of the shots in a film [1]. In statistical terms, the mean shot length is not a robust statistic of film style because it does not provide a stable description of a data set when underlying assumptions (e.g. a symmetrical normal distribution) are not met [2]. It has the worst possible breakdown point of 1/n so that just a single outlying data point can lead to the mean becoming an arbitrarily bad estimate of the centre of a data set (Wilcox 2011: 19-21) [3]. Consequently, the mean shot length does not give an accurate or reliable description of a film's style and use of this statistic to compare shot length distributions inevitably leads to flawed inferences.

This lack of robustness in the presence of outliers has led several researchers - typically from outside film studies - to prefer the median shot length as a statistic of film style in place of the mean. Adams, Dorai, and Venkatesh (2002: 72) preferred the median as a measure of location because it 'provides a better estimate of the average shot length in the presence of outliers.' Similarly, Vasconcelos and Lippman (2000: 17) reject the use of the mean because it is 'well known in the statistics literature [...] that the sample mean is very sensitive to the presence of outliers in the data,' and that '[m]ore robust estimates can be achieved by replacing the sample mean by the sample median.' Kang (2003: 245) used the median shot length 'because it shows a better estimate than the average [mean] shot length in the presence of outliers' when analysing the relationship between emotion and film style. Finally, in television studies Schaefer and Martinez (2009) used the median shot length in order to study changing editing patterns in news bulletins because it provided better indicators of shot length than the means and because the means are inordinately influenced by a few outlier values from the longest shot.

The median is the middle value when shot length data is ranked by order of magnitude, so that for any film 50 per cent of shots will be less than or equal to the median and 50 per cent will be greater than or equal to the median shot length. If the data set contains an odd number of observations the median is the centre value of the order statistics. If the data set contains an even number of values the median is equal to the mean of the two middle values. Since the median is based on the ranked data rather than the data values themselves so that it locates the centre of a distribution irrespective of its shape. It has the highest possible breakdown point of 0.5, which means that half the data can take on extreme values before the median is heavily influenced (Wilcox 2011: 22-24). Consequently, the median shot length is a robust statistic of film style resistant to the influence of outlying data points and accurately describes the style of a film without requiring assumptions about the underlying probability distribution of the data.

The duration of a shot in a motion picture carries information about the stylistic decisions of filmmakers and it is desirable that we retain all this information in our data set. It is often the unusual deployment of style that is of interest to the film analyst and so including this data is important for the analysis of film style. For this reason the removal of outlying data points or the trimming or winsorising of the data is undesirable. At the same time, it is necessary to ensure these unusual events do not distort our understanding of a film's style so that our description of shot length data and the conclusions we draw accurately reflect the style of a film and not the influence of a small proportion of atypically long takes.

The average shot lengths of Lights of New York and Scarlet Empress

To illustrate the difference between using the different ASLs we compare the mean and the median shot lengths of two early Hollywood sound films: Lights of New York (Bryan Foy, 1928) and Scarlet Empress (Josef von Sternberg, 1934), using data from the Cinemetrics database (O'Brien 2007, Salt 2007). Table 1 presents the descriptive statistics for these films.

Table 1 Descriptive statistics for Lights of New York (1928) and Scarlet Empress (1934)

 

 Lights of New York 

 Scarlet Empress 

Length (s)

3329.7

5948.4

Shots

338

601

Mean Shot Length (s)

9.9

9.9

Standard Deviation (s)

14.5

9.6

Skew

3.7

2.3

Minimum Shot Length (s)

0.9

0.3

Lower Quartile (s)

3.0

3.6

Median Shot Length (s)

5.1

6.5

Upper Quartile (s)

10.2

12.9

Maximum Shot Length (s)

95.6

64.2

Looking at the average shot lengths, we see both films have a mean shot length of 9.9 seconds; whereas the median shot length for Scarlet Empress is 1.4 seconds greater than that of Lights of New York. Therefore we can conclude that either

  • Lights of New York and Scarlet Empress are cut equally quickly,

or that

  • Lights of New York is cut more quickly than Scarlet Empress.

These statements are contradictory and cannot be true at the same time since they purport to describe the same thing.

From the descriptive statistics in Table 1 we know that the distribution of shot lengths in these films is asymmetrical: both films exhibit positive skewness with a long right-tail. The distance between the median and the maximum shot length is much greater than the distance between the minimum shot length and the median; while the distance from the upper quartile (Q3) to maximum is much greater than the distance between the minimum and the lower quartile (Q1). The maximum shot lengths are much greater than both the median and the mean. It is in precisely these circumstances that the mean provides a misleading description of a data set, and being aware of this should lead us to prefer the median to the mean and, therefore, to conclude that Lights of New York is cut more quickly than Scarlet Empress.

This can be more easily appreciated if we compare the shot length distributions of these films graphically. Numerical descriptions are valuable but it is often simpler and more informative to use graphical representations of shot length data to aid us in analysing film style. Box-plots are an excellent method for conveying a large amount of information about a data set quickly and clearly. The box plot provides a graphical representation of the five-number summary, which includes the minimum value, the lower quartile, the median, the upper quartile, and the maximum value of a data set. The core of the data is defined by the box, which covers the interquartile range (IQR) and is equal to the distance between the lower and upper quartiles, and the horizontal line within the box represents the median value of the data. The inner fences are marked by error bars extending from the box, and data points beyond these limits are classed as outliers. An outlier is defined as greater than Q3 + (IQR 1.5) and the error bars extend to largest value within this limit; while an extreme outlier has a value greater than Q3 + (IQR 3). Typically, there are no outliers at the low end of a shot length distribution, and the error bar descends to the value of the shortest shot in a film.

Description: F1.png
Figure 1 The distribution of shot lengths in Lights of New York (1928) and Scarlet Empress (1934)

Based on the criteria stated above, Lights of New York has 33 outliers, with 22 classed as 'extreme,' covering a range of 21.2 seconds to 95.6 seconds; and Scarlet Empress has 39 outliers, including nine classed as 'extreme,' covering a range of 27.3 seconds to 64.2 seconds. These outliers comprise only a small proportion of the shots in each film: 10 per cent in the case of Lights of New York and 6 per cent in Scarlet Empress. Lights of New York has a small number of shots that exceed the maximum shot length of Scarlet Empress: specifically, there are six shots in excess of 70 seconds and the longest shot is more than 30 seconds longer than maximum shot of von Sternberg's film. Lights of New York therefore not only has more outliers but they tend to lie relatively further away from the mass of the data. The influence of outliers on the mean is obvious, and from Figure 1 we also see that, rather than locating the centre of the distribution for these films, the mean is actually greater than the majority of shot lengths. In Lights of New York the mean shot length is greater than or equal to 74 per cent of shots, and in Scarlet Empress the mean is greater than or equal to 66 per cent of the shots. It is also immediately apparent from Figure 1 that the median shot length of Lights of New York lays to the left of the median of Scarlet Empress.

Once we understand the nature of the data sets we are dealing with and how the statistics behave in these circumstances we can come to a conclusion regarding how the styles of these films differ. Clearly, using the mean shot length to compare the editing style of Lights of New York and Scarlet Empress leads the researcher to draw the wrong conclusion that both films have similar editing style. This is because of the influence of a small number of takes of long duration on the mean shot length making it an unrepresentative statistic of film style. The five-number summary, the interquartile range, and the box plot provide accurate and reliable descriptions of film style leading us to the correct conclusion that the duration of takes in Scarlet Empress tends to be longer than those in Lights of New York.

Conclusion

The mean shot length has been widely used as a statistic of film style even though it is an obviously inappropriate statistic of film style. The mean shot length does not locate the centre of a shot length distribution and is not resistant to the influence of outliers. Unfortunately, this means film scholars have been laboring under a series of misconceptions about the nature of film style due to the use of this statistic. Specifically, use of the mean shot length leads to the researcher (i) identifying differences in film style when they do not in fact exist (Type I error), (ii) failing to identify changes in film style when they do occur (Type II error), and (iii) incorrectly estimating the size of any change in style correctly identified.

These problems may be overcome by using the median shot length which is resistant to the influence of outlying data points whilst retaining the complete set of shot length for a film. The median has a clearly defined meaning that is easy to understand, and fulfils precisely the role film scholars erroneously believe the mean performs.

  • There is no purpose in retaining the mean shot length as a statistic of film style since it has no defined meaning once we accept that it does not describe the cutting rate of a motion picture. If we state the 'mean shot length of Lights of New York is 9.9 seconds and the mean shot length of Scarlet Empress is 9.9 seconds,' and we know that this does not accurately describe the difference in editing (since Lights of New York is edited more quickly), what should we conclude about the respective styles of these films? Unless it is possible to state what the mean means then it is of no use. Citing both the median and mean shot lengths of a film is unnecessary since only the former provides useful information and only creates the opportunity for further confusion.

Finally, it should be noted that the style of a motion picture cannot be adequately described using the average shot length alone and it is also necessary to use statistics that describe the variation in shot lengths. Again, it is necessary to be concerned with the nature of the data and the nature of the statistics so that we do not employ statistics that give a misleading description of film style. IN fact, the case of Lights of New York and Scarlet Empress is a perfect demonstration of the necessity of robust measures of scale. The standard deviation of the shot lengths in Lights of New York is greater than that of Scarlet Empress (14.5 seconds compared to 9.6 seconds) indicating that there is greater variation in the former than the latter. However, the interquartile ranges lead us to the opposite conclusion: the IQR of Scarlet Empress is 9.3 seconds while the IQR of Lights of New York is 7.2 seconds, indicating the latter film exhibits less variation in its shot lengths. Like the mean, the standard deviation is not robust and has a breakdown point of 1/n. The IQR is resistant to the influence of outliers since it is based on the middle 50 per cent of the data and has a breakdown point of 0.25. Use of the IQR leads us to the correct conclusion regarding the differences in styles of these films that Lights of New York exhibits less variability in its shot lengths that Scarlet Empress.

See Redfern (2010) for a discussion of robust measures of scale from which we may choose.

Notes

1. An outlier is an observation (or subset of observations) well separated from the bulk of the data, or that in some way deviates from the general pattern of the data (Maronna, Martin, & Yohai 2006: 1). Since it is possible to accurately record the length of each and every shot in a film we are concerned with true outliers (correctly observed data values distant from the mass of the data) and not with gross errors (atypical values arising through human or technological error, faulty sampling, inappropriate assumptions about populations, etc.), though obviously film scholars are not immune to the latter.
2. The robustness of a statistic refers to 'the ability of a procedure or an estimator to produce results that are insensitive to departures from ideal assumptions' (Hella 2003: 17).
3. The breakdown point of an estimator is the smallest proportion of outliers a data set can contain before it becomes unreliable.

References

Adams B, Dorai C, and Venkatesh S 2002 Formulating film tempo: the computational media aesthetics methodology in practice in C Dorai and S Venkatesh (eds.) Media Computing: Computational Media Aesthetics. Norwell, MA: Kluwer Academic Publishers: 57-84.
Hella H 2003 On robust ESACF identification of mixed ARIMA models, PhD thesis, Bank of Finland Studies.
Kang H-B 2003 Affective Contents retrieval from video with relevance feedback, in TMT Sembok, H Zaman, H Chen, S Urs, and S-H Myaeng (eds.) Digital Libraries: Technology and Management of Indigenous Knowledge for Global Access. Berlin: Springer: 243-252.
Maronna R, Martin D, and Yohai V 2006 Robust Statistics: Theory and Method. Chichester: John Wiley & Sons.
O'Brien C (2007) Lights of New York, Cinemetrics Database, http://www.cinemetrics.lv/movie.php?movie_ID=690, accessed 24 January 2011.
Redfern N 2010 Robust measures of scale for shot length distributions, Research into Film [blog], http://nickredfern.wordpress.com/2010/07/15/robust-measures-of-scale/, accessed 10 July 2012.
Salt B (2007) Scarlet Empress, Cinemetrics Database, http://www.cinemetrics.lv/movie.php?movie_ID=615, accessed 24 January 2011.
Shaefer RJ and Martinez TJ 2009 Trends in network news editing strategies from 1969 through 2005, Journal of Broadcasting and Electronic Media 53 (3): 347-364.
Vasconcelos N and Lippman A 2000 Statistical models of video structure for content analysis and characterization IEEE Transactions on Image Processing 9 (1): 3-19.
Wilcox RR 2011 Modern Statistics for the Social and Behavioural Sciences: A Practical Introduction. Boca Raton, FL: CRC Press.