[Continued] Regression Prediction of Minimum Passing Scores Based on Difficulty Evaluations by the 3 Major Prep Schools for the University of Tokyo Entrance Exam (Residual Analysis Supplement)

Previously, I performed a regression analysis based on data regarding the difficulty evaluations of the University of Tokyo entrance exam by the three major prep schools and the year-over-year fluctuations in the average scores of successful applicants, and calculated the regression prediction values for the 2026 minimum passing scores.

In response to this article, I received comments on both note and X asking, 'Could you tell us the difference between the 2025 regression prediction values and the actual values?' I infer that those who took this year's University of Tokyo entrance exam commented because they wanted to know the reliability and margin of error of the 2026 regression prediction values. It is typical of University of Tokyo examinees to focus on the residuals of a regression analysis and try to verify them properly.

Therefore, I have analyzed the residuals of the previous regression analysis, and I will write an article about that as well.

0. Summary

• The correlation between the 3 major prep schools' difficulty evaluations for the University of Tokyo entrance exam and the fluctuations in the average scores of successful applicants is weak (coefficient of determination: approximately 0.3), and residuals exist for each subject with a standard deviation of 4.5 (points).

• Since difficulty evaluations between subjects are considered independent, the standard deviation for the total of 5 subjects is 10.1 (points).

• For the 2026 University of Tokyo entrance exam, many companies evaluated it as having a trend toward increased difficulty, and the regression prediction values for both humanities and sciences across all categories are approximately -10 points lower than the previous year in terms of the total score (out of 550 points). When the residual distribution is added to this, it is considered that the 2026 minimum passing score will generally fall within a range of the same as the previous year to -20 points (probability of occurrence: approximately 70%).

1. Distribution of Residuals for Each Subject

In this regression analysis, I analyzed the relationship between the difficulty evaluations of the 3 major prep schools (Sundai, Kawaijuku, Yozemi) and the year-over-year fluctuations in the average scores of successful applicants for each subject of the University of Tokyo entrance exam from 2020 to 2025. The target data consisted of 71 items. For the regression analysis, I performed two patterns: a multiple regression analysis that separated the evaluations by prep school, and a simple regression analysis that combined the evaluations of the three companies.

This graph aggregates the distribution of the residuals calculated for both the multiple regression and simple regression for these 71 items.

The distribution is generally symmetrical on the left and right (positive and negative residuals), which is not a bad result. When I performed a statistical analysis of the residual distribution for both multiple regression and simple regression, the results were as follows.

• Residual distribution of multiple regression
  Mean: 0, Standard deviation: 4.589
  Kurtosis: 1.366, Skewness: 0.105

• Residual distribution of simple regression
  Mean: 0, Standard deviation: 4.443
  Kurtosis: 1.949, Skewness: -0.216

Although it is not a normal distribution, looking at the kurtosis and skewness, it is considered to be within a range that does not deviate significantly. Also, the standard deviation is almost the same value for both multiple regression and simple regression. Therefore, using the average of the two standard deviations, 4.5, I will proceed with the analysis by treating the residual of the regression prediction of the average score fluctuation of successful applicants from the prep school difficulty evaluation for each subject as following a normal distribution N(0, 4.5^2).

2. Impact of Residuals on Minimum Passing Score Prediction

Looking at each subject, I found that the residuals of the regression prediction values for the prep school difficulty evaluations and the year-over-year fluctuations in the average scores of successful applicants are distributed with a standard deviation of 4.5 (points). However, this is the residual for each subject. Since the minimum passing score prediction for each category is based on 4 subjects and 5 exams, it is necessary to look at the fluctuation range of the total residual for the 5 subjects.

It is considered that the difficulty evaluations for each subject by the prep schools are evaluated independently. Therefore, it seems appropriate to consider the distribution of residuals for each subject as independent as well. If so, since we can simply combine the mean and variance, the residual distribution for the total of 5 subjects can be calculated as Mean: 0, Standard deviation (σ): 4.5 × √5 = 10.1.

Previously, the predicted values for the 2026 minimum passing scores for each category were predicted via regression as follows.

Since a residual of σ: 10.1 exists in this secondary fluctuation part (regression prediction value), it can be said that the minimum passing score prediction value also has an equivalent residual of σ: 10.1. Assuming a normal distribution, ±1σ = 68% (about 70%), so it is considered sufficient to look at the buffer of the predicted value within this range. Calculating this ±1σ and showing it on a graph along with the regression prediction value results in this.

While the regression prediction value for the year-over-year fluctuation of the average passing score is around -10 points, and that is slid as-is to serve as the regression prediction value for the minimum passing score, the result is that the upper bound and the previous year's value are close because σ: 10.1 is almost the same as the predicted year-over-year decrease. However, it is best to view this as a coincidence.

Looking at the range of residual ±1σ (68% probability of occurrence), it seems that the minimum passing score for the 2026 entrance exam for any University of Tokyo department will fall within a range of the same as the previous year to -20 points. There are many posts on social media expecting the minimum passing score to fall below 300 points, but while that might be possible for Humanities III, Sciences I, and Sciences II, the impression is that Humanities I and Humanities II are unlikely to fall below 300 points.

Under the assumption that the residuals are normally distributed, calculating the probability of passing by score for each department results in this.

3. Summary

What I noticed while conducting the residual analysis is that when prep schools view the difficulty of the University of Tokyo entrance exam as 'on par with the previous year,' it does not mean an average score of ±0 points, but that some plus or minus fluctuation (±2 points = ±0.5σ) naturally occurs. Therefore, even if the evaluation for multiple subjects is 'on par with the previous year,' the sum of the average values does not necessarily end up near ±0 from the previous year.

For example, regarding the four subjects of English, Japanese, and two science subjects, even if the prep school's difficulty evaluation is 'on par with the previous year' for all of them, if the average score for each is +2 points from the previous year, the average score for the total of the four subjects will be an easier exam by +8 points from the previous year. At this time, even if the remaining subject, Mathematics, becomes more difficult and the average score is -6 points from the previous year, the average score for the total of the five subjects is an increase of +2 points from the previous year. Even if the prep school evaluation is superficially 'only Mathematics is more difficult, others are on par with the previous year,' the average score for the total of the five subjects can be higher than the previous year, and the minimum passing score can also rise.

It is possible to calculate the probability of such misreadings occurring, but since the accuracy of the original regression prediction is not high (coefficient of determination 0.3), I think there is no point in being too rigorous about it. I hope you will accept this as one prediction for the University of Tokyo's minimum passing score.

I hope that all the University of Tokyo examinees who have read this article will receive their acceptance notifications on 3/10 (the day after tomorrow).