余談等
プロローグ
どうも、安田です。
私は現役受験生ですので、模試の結果を見て一喜一憂しています。マーク試験において、点数が高いと気にしない事ではありますが、特に暗記科目において偶然による正解というのは必ず存在します。
私は文系科目が苦手ですので、地理の点数は非常に悪いです。しかしながら、ほとんど何も分かっていない状況下でも30%を切った事はありません。マーク式なので一定値を下回りずらいのは当たり前の事です。
そういうわけで、今記事は実際に分かっている部分の割合はどの程度なのか、という軽い内容について考えます。
この記事の有用性
この記事を書くに至った経緯ですが、ほとんどが焦りです。その焦りも何によるものかというと、最近記事を更新していないという点に関してです。
よって、記事の内容を厳選していません。たまたま気になって調べていた事が個人的に面白かったから載せよう(載せれる)と思ったまでです。
それぞれのセクションの結論を軽く読み飛ばす程度の気持ちで読んで、面白そうな研究していると感じた方は、ページ最下部の「ランダム記事」から目についた記事に飛んでいただけると幸いです。
実質正答率
実質正答率とは私の造語です。実際は何パーセント程理解しているのかという指標を指す言葉と定義しておきます。
この値を$Z[\%]$とし、理解しているしていないに関わらず正解している割合を$P[\%]$, 選択肢の個数の平均値を $n$ とします。
実質正答率の一般式
考察は一般式を求めてからの事柄ですので、まずは一般式を求めます。
あくまでこれは理想値ですので、多少はブレが生じます。
一般的な観点
一般的に選択肢は4択以上かと思います。これは何故かを比較するため、Desmosを使って検証してみます。
※赤線は実質正答率を表します。
$P$の違いによる実質正答率のブレ
まずは、$P = 80$ と $P = 40$ について比較してみます。縦軸が$Z$で横軸が$n$です。
$P = 80$
$P = 40$
$P = 80$ の時は $n \geqq 3$ においてほとんど変化しないのに対し、$P = 40$ の時は $n$ が6から7あたりの時に安定し始めます。しかしながら、選択肢がその辺りに収束する事は非常に少ないです。
大学入試センターの資料によると、この調査された年の平均選択肢数は5.6択程度という事です。$n$ がそのあたりをうろつくと考えますと、実質正答率は非常に不安定に変化します。すなわち、模試がただの運によるものと化しかけているという事です。自分は地理がこのあたりの点数なので、自分で調査しておきながら悲しいですね。
$n = 5.6$ での実質正答率
先ほど述べた通り、大学入試センターの調査によりますと、ある年の平均選択肢数は5.6程度だそうです。
この時の$Z$と$P$の関係をDesmosを使って検証します。今回の横軸は$P$です。
誤差があることは分かりましたが特徴が分かりずらいと私は感じましたので、テストの結果の信頼度$R[\%]$を以下のように定義します。
\begin{equation}
R = \frac{Z}{P} \times 100
\end{equation}
この$R$のグラフは顕著に特徴が現れます。縦軸を$R$で横軸を$P$とすると、
模試の正答率が40点付近を行き来している場合、現状把握という点において模試を受ける意味は皆無に近いものという事が分かります。
これは、信頼度の変化率が大きいからです。
結論
グラフを用いて可視化する事により、いくつかの。
そして今回ひたすらに見えた結論は、「正答率が40%付近またはそれ以下の場合の模試の結果に特に意味はない」という事でした。
それでは、お疲れさまでした。

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