Excerpt

## Table of Contents

List of Figures

List of Tables

1 Introduction

2 Capacity

2.1 Minimum and maximum capacity in each group

2.2 Meancomparison

2.3 Kruskal-Wallis test of capacity

2.4 Mean capacity in each group

3 Performances

4 Visitors

4.1 Regarding the type of a ticket

4.2 Regarding the type of a performance

4.3 Mean comparison for visitors

4.3.1 Kruskal-Wallis test of opera visitors

4.3.2 Kruskal-Wallis test of puppet theatres

5 Methodology

6 Conclusion

References

## List of Figures

Figure 1: Minimum and maximum capacity in each group

Figure 2: Kruskal-Wallis test results for capacity

Figure 3: Mean capacity in each group

Figure 4: Numbers for each type of performance

Figure 5: Comparing the number of each type of performance between the groups

Figure 6: Numbers for each type of a ticket

Figure 7: Comparing the number of each type of a ticket sold between the groups

Figure 8: Numbers for visitors with regard to the type of performance

Figure 9: Comparing the number of visitors with regard to the type of performance between the groups

Figure 10: Kruskal-Wallis test results for opera visitors

Figure 11: Kruskal-Wallis test results for visitors of puppet theatres

## List ofTables

Table 1: ANOVA results for capacity

Table 2: Summary statistic for opera visitors

Table 3: Summary statistic for visitors of puppet theatres

## 1 Introduction

This essay will review the information surveyed on current data from Theaterstatistik 2008/2009 collected by Deutscher Buhnenverein. The data that was given there consists of 112 German cities, number of inhabitants in these cities, capacity of theatres, i.e. seats in all theatres that are available to visitors per 1000 inhabitants in each city. Moreover, there are 146 theatres given with the number of different types of performances produced in that particular theatre as well as the number of visitors presented in two ways: sold tickets to each type of the performance and regarding the type of a ticket.^{[1]}

The aim of this essay is to summarize data about German theatres as well as to investigate interesting similarities and differences, do mean comparison and discuss the methodology used here.

This paper has been divided into four parts. The first part deals with number of inhabitants of given German cities and capacity of theatres in these cities. The second part is concerned with types of performances presented in these theatres. Third part is about visitors in German theatres, which types of performance are mostly visited and which type of a ticket is usually bought. And 4th part contains discussion about methods used when working with the given data and the way the results of the research were presented.

## 2 Capacity

After exploring data about inhabitants of the given German cities from the first table “Theateruntemehmen”, one could present the summary statistics about inhabitants of each city and capacity of its theatres. So, from 112 given German cities the maximum number of inhabitants is 3 431 675 (Berlin), minimum 11 455 (Dinkelsbuhl) and the mean number is 531 533.1. For easier comparison of further data, the cities, their theatres and their numbers should be divided with regard to their size into three groups - large, middle size and small cities. Throughout this essay the term “large city” will be now used to refer to the group of cities that have more than 500 000 inhabitants, for example Berlin, Stuttgart, Bremen. The term “middle city” will be used for a group of cities that have between 500 000 and 100 000 inhabitants, e.g. Bonn, Halle, Ingolstadt. And last but not least, term “small city” will be used to refer to the group of cities with less than 100 000 inhabitants, for instance Dessau, Weimar, Eisleben.

### 2.1 Minimum and maximum capacity in each group

Taking into consideration previously described division of the cities into three groups, the capacity data could be now summarized for each group. A part of summary statistics could be seen in Figures 1 and 3.

Figure 1: Minimum and maximum capacity in each group

illustration not visible in this excerpt

Source: own illustration

First of all, it must be mentioned that the minimum capacity in the given German cities is 1.1 seats per 1000 inhabitants, which is also the minimum capacity of the group of the middle size cities. In other words, in Muhlheim an der Ruhr there is approximately 1 seat available to 1000 inhabitants.

Second, the maximum capacity is 65, which is at the same time the maximum in the group of small cities. Furthermore, 65 is comparatively large number, because the mean capacity of all given cities is only 8.7839 seats. Namely, there are 65 seats available to 1000 people in Anklam, where the number of inhabitants is only 13 423. On the other hand, in the group of large cities almost 7 seats are available to 1000 inhabitants in Numberg, where the total number ofinhabitants is 503 638.

Moreover, when comparing maximum capacities between the groups, it is visible that the maximum capacity in the group of large cities is much smaller than the maximum capacity of middle size or small cities.

What one could see more from Figure 1 is that the dispersion of the capacity in all three groups is different. So the interval between the minimum and maximum capacity in the group of large cities is comparatively smaller than the interval of middle cities. And distance between minimum and maximum capacity of small cities is very large when comparing to all other groups. This could mean that the capacity of theatres in small cities is not balanced, because there are some cities that have very big number of seats per 1000 inhabitants, for example, Anklam with 65, and some cities which have the minimum capacity number in this group (Aalen 1.8).

### 2.2 Mean comparison

In addition, one could be interested if the means of these three groups are equal. This is a reason to do ANOVA (analysis of variance), because it is a statistical method for determining the existence of differences among several population means.^{[2]} There are three groups given - 15 large cities, 54 middle size and 43 small cities. Each of them has the capacity of all theatres located in this city. In this case, we have a one-way ANOVA. That is, there is one factor we are looking at across these 3 groups.

Step 1: State the null hypothesis and the alternative hypothesis. The null hypothesis in ANOVA is that the means of the groups are equal.^{[3]} In other words, mean of group of large cities is equal to the mean of group of middle and the mean of the group of small cities. And alternative hypothesis is that means of these three groups are not equal.3 In other words, if the hypothesis is true, then the "between group variance" (MSTR) will be equal or close to the "within group variance" (MSE). MSTR (Mean Square Treatments) is the estimate of the population variance based on the differences among the sample means. And MSE (Mean Square Error) is the estimate of the population variance based on the variation within the sample.3 If, on the other hand, the null hypothesis is not true and differences do exist among the population means, then MSTR will tend to be larger than MSE.^{[4]} Under the assumption of ANOVA, the ratio MSTR/MSE possesses an F distribution with r-1 degrees of freedom for the numerator and n-r degrees of freedom for the denominator when the null hypothesis is true.4

Step 2: Select the level of significance. In this case, 0.05 is chosen.

Step 3: Calculate the F statistic using Excel's Data Analysis. Results are presented in Table 1.

Table 1: ANOVA results for capacity

illustration not visible in this excerpt

Source: own illustration

Step 4: Interpret the results. As one could see, the mean level of capacity in small cities (14.3558) is higher than that of either middle (5.6685) or large cities (4.0267). And “between group variance” is not equal to “within group variance”. According to the test result F equals 20.3367. With a significance level of 0.05, the critical F is 3.0796. Therefore, since the F statistic is greater than the critical value, and the p-value is much smaller than 0.05, the null hypothesis is rejected. From above, the null hypothesis was that all 3 of these groups' means were equal. So, it should be rejected that large, middle and small cities have the same level of capacity.

However, an important problem to mention is that variances of the groups are not equal, so ANOVA should not be used and instead one could use a nonparametric technique called Kruskal-Wallis test.^{[5]}

### 2.3 Kruskal-Wallis test of capacity

Taking into consideration the last conclusion, one could use the Kruskal-Wallis which is also known as nonparametric one way ANOVA. The Kruskal-Wallis test is an analysis of variance that uses ranks of the observations rather than the data themselves. Although the hypothesis is stated in terms of the distributions of the populations, the test is most sensitive to differences in the locations of the populations. Therefore, the procedure is actually used to test the ANOVA hypothesis of equality of k population means. The only assumptions required for Kruskal-Wallis test are that the k samples are random and independently drawn.^{[6]} Step 1: Null hypothesis is that all 3 populations have the same distribution. And alternative hypothesis is that not all of them have the same distribution.^{[6]}

Step 2: 0.05 level of significance could be chosen.

Step 3: Calculate Kruskal-Wallis test statistic with the help of Excel Analyse-it. And the results are presented in Figure 2.

Figure 2: Kruskal-Wallis test results for capacity

illustration not visible in this excerpt

Source: own illustration

Step 4: As one could see the Kruskal-Wallis test statistic is 53.66. And the critical value that could be obtained from chi-square distribution table with k - 1 = 2 degrees of freedom and 0.05 significance level is 5.99.^{[7]} That is why it should be concluded that the null hypothesis is rejected, so all three groups do not have the same level of capacity.

### 2.4 Mean capacity in each group

In addition, the mean comparison in dot plot in Figure 3 shows that capacity of theatres in small cities is much bigger than in middle or large cities. Hence, capacity of middle size cities is also bigger than the one in large cities. For example, capacity of theatres in Halle (middle size city) is 11.1, which is much larger than 2.7, the capacity of Berlin theatres. Moreover, the capacity of Eisleben from the group of small cities is 21, which is higher than the capacity of any city from the group of large or middle size.

**[...]**

^{[1]} Theaterstatistik (2010; p. 5f)

^{[2]} Aczel (2006; p. 370)

^{[3]} Lind (2005; p. 395f)

^{[4]} Aczel (2006; p. 383)

^{[5]} Aczel (2006; p. 372)

^{[6]} Aczel (2006; p. 665f)

^{[7]} Aczel (2006; p. 779)

- Quote paper
- Marina Cuvilceva (Author), 2011, Explaining the situation of German theatres, Munich, GRIN Verlag, https://www.grin.com/document/197208

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