問77
頻出度★★☆
受験者10,000人の4教科の試験結果は表のとおりであり，いずれの教科の得点分布も正規分布に従っていたとする。ある受験者の4教科の得点が全て71点であったとき，この受験者が最も高い偏差値を得た教科はどれか。 単位 点
| 平均点 | 標準偏差 | |
|---|---|---|
| 国語 | 62 | 5 |
| 社会 | 55 | 9 |
| 数学 | 58 | 6 |
| 理科 | 60 | 7 |
令和5年度 ITパスポート試験 問77
テクノロジ系基礎理論 › 応用数学
This question's explanation and your answers so far.
解説
問 77
Incorrectア
For Japanese, the mean is 62 and the standard deviation is 5. The difference from the mean is 71−62=9 points, the smallest among the 4 subjects, but because the standard deviation of 5 is also the smallest, 50+10×(71−62)÷5=68.0 comes out fairly high. Even so, it does not reach Mathematics’ roughly 71.7 and is not the highest, so it is wrong. This is a typical case where, even with a small difference, a small standard deviation makes the deviation value high.
Incorrectイ
For Social Studies, the mean of 55 is the lowest among the 4 subjects and the difference from the score (71−55=16 points) is the largest, but because the standard deviation of 9 is also the largest, the deviation value stays at 50+10×(71−55)÷9≈67.8 and is not the highest. Note that you must not only take the difference from the mean but also divide by the standard deviation.
- ウ
正解 The deviation value is found by “50+10×(score−mean)÷standard deviation”. Since the score is 71 in every subject, the deviation value of each subject is: Japanese=50+10×(71−62)÷5=68.0, Social Studies=50+10×(71−55)÷9≈67.8, Mathematics=50+10×(71−58)÷6≈71.7, Science=50+10×(71−60)÷7≈65.7. The highest is Mathematics at about 71.7, so the answer is ウ“Mathematics”. For Mathematics, the standardized score obtained by dividing the difference from the mean (71−58=13) by the standard deviation (6) is the largest among the 4 subjects, and because the gap measured by the spread from the mean is the greatest, its deviation value becomes the highest.
Incorrectエ
For Science, the mean is 60 and the standard deviation is 7. 50+10×(71−60)÷7≈65.7, which is the lowest among the 4 subjects. Relative to the difference from the mean (11 points), the standard deviation (7) is somewhat large, so the deviation value does not rise, and it is not the highest, so it is wrong.
ここがポイント
- Definition and formula of the deviation value: deviation value=50+10×(score−mean)÷standard deviation. If the score is exactly the mean, the deviation value is 50.
- The height of the deviation value is determined not only by “the difference from the mean” but also by “how small the standard deviation is” (standardization). Even if the difference is small, a small standard deviation can make the deviation value high, so compare by looking at both.
用語