Introduction & RPubs link

The modern Summer Olympic Games has been held every four years from 1896, except in times of war and pandemic.

Nations bid to host the Olympics, and while it requires significant investment and planning, rewards include retained infrastructure, an opportunity to invoke national pride on a global stage, and a potential increase in tourism. It has been said that host nations tend to do well at their home Games.⁵

This investigation aims to determine if the medal tally of a country’s host Games is typical of their performance in non-hosting years prior.

Problem Statement

• Using data from the Summer Olympics from 1896 to 2012, this analysis will focus on on seven nations who hosted the games from 1984 to 2012 - Great Britain, China, Greece, Australia, USA, Spain and South Korea.

• The sample of these countries’ medal tallies from years prior to hosting will be compared to the medal tally of their host Games.

• After visual analysis, data will be compared via t-testing and a critical value approach to see if a country’s sample medal tally is statistically significant, compared to their host year.

Data

The summer.csv open source data was collected from Kaggle, and details the following variables:

• Year (Year the Olympics were hosted): numeric
• City (Host city of the Olympics): character
• Sport (Activity governed by an International Federation⁶): character
• Discipline (Branch of a sport with one or more events⁶): character
• Athlete (Name of the athlete who won the medal: surname, given name): character
• Country (Abbreviation of the country competing at the Olympics): character
• Gender (Men’s or Women’s): character
• Event (A competition within a discipline⁶): character
• Medal (Gold, Silver or Bronze for first, second and third place): character.

The data was provided by the IOC Research and Reference Service and published by The Guardian’s Datablog. It can be found at: https://www.kaggle.com/the-guardian/olympic-games?select=summer.csv ⁷

Data: Preprocessing

Medal and Gender were factored into their levels. The following required transformation:

When a duo, trio, team, etc wins a medal, the source dataset lists each athlete as a medalist. In Olympic medal tallies, countries are only awarded one medal per event, not per person. Therefore, if the data were grouped into medals per country, it would count multiple medals for team events.

This was addressed by grouping the number of medals (“n”) per Year, Event and Country:

• If n=1, that value remained a 1;

• If n>2, n was imputed with a 1, with the assumption that it was unlikely that there was a three-way tie between compatriots in an individual event;

• If n=2, events were filtered as either individual (ie. there was a tie between two compatriots), or a team event. If it was a team event, n was imputed with a 1. If it was a tie in an individual event, both medals were awarded and it remained n=2;

• There was only one case of a three-way tie between three compatriots in an individual event, which was the men’s Pommel Horse in the 1948 gymnastics, when three Finnish men tied for Gold.⁸ As the 3 medals had been imputed with a 1, 2 medals were manually added.

Data: Preprocessing

The dataset was subsetted to include four variables: Year, City, Country, and the new variable:

• n (Number of medals earned): numeric.

As the data were grouped from the original source listing each medalist, countries that won no medals in an Olympiad had no observation for that year:

• Each host country’s “Year” variable was checked against official records to determine whether they attended those Olympics and earned no medals, or if they did not attend.

• Greece, Australia, Spain and South Korea all had years where they earned no medals,⁹ and these were manually added into the data.

• Two additional years for Australia were also imputed under country code “ANZ”, under which they competed in 1904 and 1908.⁹

After preprocessing, the data were filtered to include only the seven host countries from 1984 to 2012:

• Great Britain, China, Greece, Australia, USA, Spain and South Korea.

Descriptive Statistics

• Descriptive statistics summarise the min, max, mean, median, SD, IQR, number of years & missing.
• This summary indicates that each country has a wide range and IQR.
• The mean and median rarely have similar values, and at times SD > mean, suggesting non-normality.

Host_countries <- Olympics6 %>% filter(Country %in%
c("USA", "AUS", "CHN", "GRE", "KOR", "ESP", "GBR"))
Host_countries %>% group_by(Country)  %>% 
summarise (Min = min(n, na.rm = TRUE),
Q1 = quantile (n,probs = .25, na.rm = TRUE), 
Median = median(n, na.rm = TRUE),
Q3 = quantile(n,probs = .75,na.rm = TRUE),
Max = max(n,na.rm = TRUE),
Mean = mean(n, na.rm = TRUE), 
SD = sd(n, na.rm = TRUE), 
n = n(), Missing = sum(is.na(n))) -> table1

kbl(table1, booktabs = T) %>%
  kable_styling(font_size = 13,
                latex_options = "striped")

Visualisation

A coloured line graph was used to visualise the medal tallies of seven recent host nations:

Host_countries_plot <- Host_countries %>% ggplot(mapping = aes(x=Year, y = n, color = Country)) + labs(y = 
"Total medals") + expand_limits(x= max(Host_countries$Year), y = 0:max(Host_countries$n)) + 
scale_x_continuous(breaks = seq(1896, 2012, by = 4)) + scale_y_continuous(breaks = seq(0, 230, by = 25)) +
geom_line() + theme(axis.text.x=element_text(angle=45,hjust=1)) + ggtitle("Medal tallies of
7 Olympic host nations, 1896-2012") + theme(plot.title = element_text(hjust = 0.5))
Host_countries_plot

Visualisation

• An initial observation is that some countries tend to perform better than others. Hence, the analysis will compare a country’s sample with its own medal tally.
• Since we are comparing a country’s medal tally with its previous performances, significant spikes should be noted. Upon first glance, noted spikes include USA in 1908, 1924, 1932 and 1984; Australia in 1956 and 2000; Great Britain in 1984 and 2012, and China in 2008.
• Each country’s sample need to be investigated individually to determine if these spikes are hosting years or anomalies, and whether they are statistically significant.

The following code was used to plot each country’s sample with host years and most medals:

GBR_historic_plot <- GBR_historic %>% ggplot(mapping = aes(x=Year, y = n)) + labs(y = "Total medals")+ 
  expand_limits(x= max(GBR_historic$Year)+4, y = 0:max(GBR_historic$n)+10) + 
  scale_x_continuous(breaks = seq(1896, 2012, by = 4))+ scale_y_continuous(breaks = seq(0, 160, by = 10)) + 
  geom_line() + geom_point(data = GBR_historic, aes(x=Year, y=n), color = "black", size=2) + 
  geom_text(data=GBR_historic, aes(x=Year, y=n, label = n), size = 3.5, color = "blue", hjust = -0.2, vjust =1) + 
  geom_point(data=subset(GBR_historic, Most_medals == "Yes"), color = "red", size = 4) + 
  geom_point(data=subset(GBR_historic, Host_city == "Yes"), color = "darkgreen", size = 2) + 
  geom_text(data=subset(GBR_historic, Most_medals == "Yes"), 
            aes(label = "Most medals", hjust=0.5,vjust=-2.5), color = "red") + 
  geom_text(data=subset(GBR_historic, Host_city == "Yes"), aes(label = "Host city", hjust=.5,vjust=-1), color = "darkgreen") + 
  theme(axis.text.x=element_text(angle=45,hjust=1)) + ggtitle("Medals won by Great Britain at the Summer Olympics, 1896-2012") + 
  theme(plot.title = element_text(hjust = 0.5), legend.position = "none")

Hypothesis Testing: Methods and assumptions

• We test the hypothesis \(H_0\) that the sample mean of years prior to hosting, \(\mu_1\), is the same as the host year, \(\mu_2\).

\[H_0: \mu_1 = \mu_2 \]

• The alternative hypothesis \(H_A\) is that the sample mean of years prior to hosting, \(\mu_1\), is lower than the host year, \(\mu_1\).

\[H_A: \mu_1\ < \mu_2\] The same hypotheses will be used for the seven host countries from 1984 to 2012.

• As there have been fewer than 30 Olympiads, no country’s sample can be assumed to be normal via Central Limit Theorem (n>30).
• Samples will be tested for normality using boxplots, Q-Q Plots and a histogram with a normal curve overlay.
• If sample is proven to be normally distributed, it will be tested using a lower, one-tailed, one-sample t-test and a critical value approach.

Great Britain

GBR_mean = mean(GBR_clean$n)
GBR_sd = sd(GBR_clean$n)
hist(GBR_clean$n, density=80, breaks=25,
     prob=TRUE)
lines(density(GBR_clean$n),
      lwd = 2, col = "orange")
curve(dnorm(x, mean=GBR_mean, sd=GBR_sd), 
col="darkblue", lwd=2, add=TRUE, yaxt="n")

Results: Great Britain

A lower, one-tailed, one-sample t-test to test Great Britain’s 1896-2008 medal tally against their 68 medals in 2012. A critical value approach was also applied for the data with 24 degrees of freedom.

Decision: Reject \(H_0\): \(\mu\) = 68 as \(p\)-value < .001 and 95% CI [0, 27.387] did not capture \(H_0\).

Conclusion: The estimated mean medal tally, based on the pre-host year sample, was \(\bar{x}\) = 23.32, 95% CI [0, 27.387]. The results of the one-sample t-test found that the medal tallies of previous years were significantly lower than the 2012 medal tally of 68, \(t\)(\(df\)=24) = -18.793, \(p\) < .001.

t.test(GBR_clean$n, mu = 68, 
       conf.level = .95, alternative = "less")

## 
##  One Sample t-test
## 
## data:  GBR_clean$n
## t = -18.793, df = 24, p-value = 3.659e-16
## alternative hypothesis: true mean is less than 68
## 95 percent confidence interval:
##      -Inf 27.38758
## sample estimates:
## mean of x 
##     23.32

qt(0.025, df = 25-1,
   lower.tail = FALSE)

## [1] 2.063899

Australia

Aus_mean_clean_1948_2000 = mean(Aus_clean_1948_2000$n)
Aus_sd_clean_1948_2000 = sd(Aus_clean_1948_2000$n)
hist(Aus_clean_1948_2000$n, density=80, 
     breaks=20, prob=TRUE)
lines(density(Aus_clean_1948_2000$n),
      lwd = 2, col = "orange")
curve(dnorm(x, mean=Aus_mean_clean_1948_2000, 
            sd=Aus_sd_clean_1948_2000), 
      col="darkblue", lwd=2, add=TRUE, yaxt="n")

Results: Australia

A lower, one-tailed, one-sample \(t\)-test to test Australia’s 1948-1996 sample against their 58 medals in 2000. A critical value approach was also applied for the sample with 11 degrees of freedom.

Decision: Reject \(H_0\): \(\mu\) = 58 as \(p\)-value < .001 and 95% CI [0, 22.121] did not capture \(H_0\).

Conclusion: The estimated mean medal tally, based on the pre-host year sample was \(\bar{x}\) = 17.75, 95% CI [0, 22.121]. The results of the one-sample \(t\)-test found that the medal tallies of previous years were significantly lower than the 2000 medal tally of 58, \(t\)(\(df\)=11) = -16.444, \(p\) < .001.

t.test(Aus_clean_1948_2000$n, mu = 58,
       conf.level = .95, alternative = "less")

## 
##  One Sample t-test
## 
## data:  Aus_clean_1948_2000$n
## t = -16.534, df = 11, p-value = 2.036e-09
## alternative hypothesis: true mean is less than 58
## 95 percent confidence interval:
##      -Inf 22.12184
## sample estimates:
## mean of x 
##     17.75

qt(0.025, df = 12-1,
   lower.tail = FALSE)

## [1] 2.200985

China & Greece

Both sample plots show an upward trend in the lead-up to their peak medal tally at their host games.

Spain & South Korea

Spain_mean = mean(Spain_clean$n)
Spain_sd = sd(Spain_clean$n)
hist(Spain_clean$n, density=80, breaks=20, prob=TRUE)
lines(density(Spain_clean$n), lwd = 2, col = "orange")
curve(dnorm(x, mean=Spain_mean, sd=Spain_sd), 
      col="darkblue", lwd=2, add=TRUE, yaxt="n")

## [1] 9 4

USA

USA samples from pre-1996, 1984 and 1932 were tested for normality via boxplot, Q-Q Plot and normal curve overlay, yielding non-normal results. A BoxCox transformation was also attempted, which resulting in non-normality.

boxcox_USA <- BoxCox(USA_pre_1996$n,lambda = "auto")

The USA achieved their highest medal tallies in 1904 and 1984, with a local maximum in 1932. However, it did not achieve a maximum in Atlanta 1996, suggesting this sample does not support the research hypothesis. A larger sample or a normal distribution is required to prove statistical significance.

Discussion: Results

• Only two samples were normally distributed and could be tested for statistical significance. Both samples - Great Britain pre-2012 and Australia 1948-2012 - were statistically significant, with host year tallies outside the 95% confidence interval of the sample.
• Samples with a non-normal distribution were analysed visually, with many showing clear maxima or local maxima in host years. However, these cannot be proven to be statistically significant.
• While statistical and visual analyses tend to support the research hypothesis, two samples cannot conclusively suggest the same for all samples of Olympic host nations in the past 120 years.
• Having small, non-normal samples and many factors that affect a host nation’s performance means that this project should be revisited in years to come, when all samples move towards a normal distribution and countries have hosted multiple Games.

Discussion: Context

• The number of medals given out in each Olympiad changes depending on which events are included. One limitation of this study is that it compares the number of medals won by each nation, rather than the percentage of available medals won. It is possible that the sample distribution and results may differ if future research compared each country’s medal tally as a percentage of total medals awarded that year.
• In the late 1800s and early 1900s, large host country tallies could be attributed to difficulty and cost of international travel for those countries not hosting the Games. This was evident in Athens 1896, St Louis 1904 and London 1908, when host nation medal tallies far outweighed non-host nations.
• More recently, a “home ground advantage” can be attributed to funding levels, cost of travel (and therefore size of team), performance psychology, a home crowd and the ability of host nations to remove or include certain sports.¹⁰ The commodification of sports and the Olympic Games has created an intricate web of factors that contribute to a team’s success in any one year.
• Further study may determine if a “home ground advantage” truly exists, and provide insight into the social, psychological, cultural and economic factors that may contribute to such an advantage.

References

1 Encyclopaedia Britannica 2020, Athens 1896, viewed 13 Oct 2020, https://www.britannica.com/event/Athens-1896-Olympic-Games.

10 Council of Foreign Relations 2018, The economics of hosting the Olympic Games, viewed 11 October 2020, https://www.cfr.org/backgrounder/economics-hosting-olympic-games.