Fresh statistical findings from the World Cup have shown Scotland underperformed at the tournament finals in the United States.
Scotland were eliminated at the group stage of the competition, finishing as one of the bottom four third-placed sides following losses to Brazil and Morocco.
They had started the tournament with a 1-0 victory over Haiti but failed to progress.
Now, Dr Guy Wilkinson, a lecturer in sport at the University of Stirling, Scotland's University for Sporting Excellence, has studied unexpected stories at the World Cup.
Dr Wilkinson created a simulation model that used each squad’s estimated market value to forecast a range of plausible outcomes for every nation.
The statistical model based on historical World Cup data predicted the number of goals in each match, before simulating the tournament 10,000 times to generate a distribution of possible outcomes for every nation.
Comparing the real outcome to the simulation distribution produced a Standardised Performance Index (SPI).
An SPI close to zero indicates a broadly expected tournament, while positive and negative values represent increasingly unusual over or underperformances, respectively.
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In 10,000 simulations, Scotland progressed beyond the group stage around 60 per cent of the time, showing elimination at the group stage is underperformance.
However, crucially, Scotland were also knocked out early in four out of every ten simulations.
The data suggests Scotland underperformed compared to their expectation, but the outcome should be classed as "realistic" rather and a "remarkable upset".
Dr Wilkinson explains: "Scotland provide a useful illustration of how the SPI should be interpreted.
"Their group-stage exit resulted in an SPI of -0.91, indicating a performance below their average simulated expectation.
"However, this should not be viewed as a major shock.
"Although Scotland progressed beyond the group stage in around 60% of simulations, they were still eliminated early in four out of every ten.
"In other words, Scotland underperformed relative to their average expectation, but their elimination remained a realistic outcome rather than a remarkable upset."
All four semi-finalists overperformed according to the fresh data, with Spain against Argentina only predicted 151 times out of the 10,000 simulations.
Outside of the semi-finalists, Morocco, Paraguay, Switzerland, Egypt, Norway and Cape Verde surpassed expectations.
Turkey and Uruguay recorded the largest underperformances at -1.35 and -1.30 as both nations were eliminated in the group stage. They both made the knockout rounds in almost 88 per cent of simulations.
