[{"data":1,"prerenderedAt":179},["ShallowReactive",2],{"topic-page:\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis\u002Fvariance-basics":3},{"topic":4,"category":146,"seo":150,"breadcrumbs":155,"prerequisites":162,"nextTopics":172,"relatedTopics":178},{"id":5,"slug":6,"title":7,"summary":8,"canonicalPath":9,"categoryPath":10,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"generatedAt":14,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":16,"expectedKnowledge":21,"learningGoal":25,"blocks":26},"high-school-mathematics-i-data-analysis-variance-basics","variance-basics","偏差と分散","各値が平均からどれだけ離れているかを偏差で表し、正負の相殺を二乗で防いで、その平均を分散として求めます。","\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis\u002Fvariance-basics","\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis","beginner","student_high",6,"2026-07-27","ai_generated",[17,18,19,20],"数学I","データの分析","偏差","分散",[22,23,24],"平均値","正負の数","二乗の計算","偏差を求めて二乗し、その平均として分散を計算し、散らばりの大小として説明できるようになる。",[27,32,41,63,69,79,85,95,105,111,127,136],{"id":28,"type":29,"title":7,"subtitle":30,"shortDefinition":31},"hero","HeroBlock","平均からの離れ方を一つの数にする","散らばりを測るには、各値が平均からどれだけ離れたかを集めます。正負のずれが打ち消し合わないよう二乗し、その平均を分散にします。",{"id":33,"type":34,"term":7,"definition":35,"plainExplanation":36,"keywords":37},"definition","DefinitionBlock","偏差は各値と平均の差、分散は偏差を二乗した値の平均です。","平均より小さい値の偏差は負、大きい値は正になります。偏差をそのまま合計すると必ず0になり、散らばりを測れません。そこで離れた向きではなく大きさを残すために二乗し、全データについて平均します。",[38,39,40],"偏差=値-平均","偏差和は0","二乗して平均",{"id":42,"type":43,"title":44,"formulas":45,"strategy":62},"formula","FormulaBlock","分散の定義",[46,59],{"label":19,"expression":47,"meaning":48,"symbols":49},"d_i = x_i - x_bar","各値xᵢが平均x̄からどちらへどれだけ離れたか。",[50,53,56],{"symbol":51,"meaning":52},"xᵢ","i番目のデータ",{"symbol":54,"meaning":55},"x̄","データの平均",{"symbol":57,"meaning":58},"dᵢ","i番目の偏差",{"label":20,"expression":60,"meaning":61},"variance = sum(d_i^2) \u002F n","偏差の二乗をすべて足し、データの個数nで割る。","平均、偏差、二乗、平均の順を表にして計算します。",{"id":64,"type":65,"src":66,"alt":67,"caption":68},"diagram","DiagramBlock","\u002Fassets\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis\u002Fvariance-basics\u002Fdeviation-squares.svg","平均4から値2、4、6への偏差と二乗を示す数直線の図","平均4から左へ2、同じ位置、右へ2という偏差を示します。二乗すると4、0、4となり、離れた向きに関係なく大きさが残ります。",{"id":70,"type":71,"steps":72,"title":78},"steps","StepBlock",[73,74,75,76,77],"データの平均を求める","各値から平均を引く","偏差を二乗する","二乗した値を合計する","データの個数で割る","分散を求める5段階",{"id":80,"type":81,"scenario":82,"explanation":83,"result":84},"example","ExampleBlock","データ2、4、6の分散を求める。","平均は(2+4+6)\u002F3=4です。偏差は-2、0、2で、合計は0になります。偏差の二乗は4、0、4、合計8。個数3で割り、分散は8\u002F3です。偏差の符号は消えますが、平均から2離れた二つの値が散らばりとして反映されます。","分散8\u002F3は0より大きく、三つの値が平均4だけに一致していないことを表します。",{"id":86,"type":87,"title":88,"points":89,"body":94},"key-points","KeyPointBlock","計算後の意味を読む",[90,91,92,93],"分散は0以上","全値が同じなら分散0","大きいほど平均から広く離れる","単位は元データの二乗","同じ種類・単位のデータなら、分散が大きい方が平均から広く散らばっています。",{"id":96,"type":87,"title":97,"points":98,"body":104},"variance-verification","分散の計算を三つの性質で検算する",[99,100,101,102,103],"平均との差である偏差を各値について求める","偏差の合計が0になることを途中確認に使う","二乗後の値を合計し、このデータの個数で割る","答えが0以上か、散らばりの見た目と矛盾しないかを見る","この単元では与えられたデータ全体の分散として個数nで割る","偏差の二乗はすべて0以上なので、分散が負になることはありません。全値が平均と一致するときだけ分散は0です。また元データをすべて同じだけ平行移動しても、平均からの距離は変わらないため分散も変わりません。",{"id":106,"type":107,"warningTitle":108,"message":109,"severity":106,"action":110},"warning","WarningBlock","偏差をそのまま平均しない","偏差の合計は平均の性質により0です。0を散らばりなしと読むのではなく、偏差を二乗してから平均します。","計算表に「偏差」と「偏差²」の列を分けて書きましょう。",{"id":112,"type":113,"title":114,"questions":115},"quiz","QuizBlock","確認テスト",[116],{"question":117,"choices":118,"answerIndex":122,"choiceExplanations":123},"偏差を二乗してから平均する主な理由はどれですか。",[119,120,121],"正負の偏差が打ち消し合うのを防ぐため","平均値を大きくするため","データの個数を減らすため",0,[124,125,126],"正解です。二乗で離れた方向をそろえ、離れ方の大きさを残します。","平均値そのものを変える操作ではありません。","すべてのデータを使い、最後に個数で割ります。",{"id":128,"type":129,"title":130,"points":131},"summary","SummaryBlock","まとめ",[132,133,134,135],"偏差は値から平均を引く","偏差和は0になる","二乗で離れ方の大きさを残す","偏差²の平均が分散",{"id":137,"type":138,"title":139,"nextTopics":140,"relatedKeywords":142},"next-action","NextActionBlock","元の単位で散らばりを読む",[141],"high-school-mathematics-i-data-analysis-standard-deviation-basics",[143,144,145],"**標準偏差**","平方根","単位",{"path":10,"title":18,"description":147,"parentPath":148,"accentToken":149},"分散、標準偏差、散布図、相関係数、仮説検定の考え方、外れ値を整理していくカテゴリです。","\u002Fhigh-school\u002Fmathematics-i","signal-blue",{"title":151,"description":152,"canonicalUrl":153,"ogImage":154},"偏差と分散 | 数学I データの分析","平均からの偏差、偏差の二乗、分散の計算と意味を小さなデータで学びます。","https:\u002F\u002Fzeqnilo.com\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis\u002Fvariance-basics","https:\u002F\u002Fzeqnilo.com\u002Fassets\u002Fsite\u002Fdefault-og.svg",[156,159,160,161],{"title":157,"path":158},"高校","\u002Fhigh-school",{"title":17,"path":148},{"title":18,"path":10},{"title":7,"path":9},[163],{"id":164,"title":165,"summary":166,"canonicalPath":167,"categoryPath":10,"categoryTitle":18,"difficulty":11,"targetAudience":12,"estimatedMinutes":168,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":169},"high-school-mathematics-i-data-analysis-center-and-spread-review","代表値と散らばりを分けて読む","平均・中央値などの代表値と散らばりを別の観点として読み、同じ平均でも分布の形が異なることを具体例で確認します。","\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis\u002Fcenter-and-spread-review",5,[17,18,170,171],"代表値","散らばり",[173],{"id":141,"title":174,"summary":175,"canonicalPath":176,"categoryPath":10,"categoryTitle":18,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":177},"標準偏差","分散の平方根を標準偏差として求め、元データと同じ単位に戻して、平均からの散らばりを比較します。","\u002Fhigh-school\u002Fmathematics-i\u002Fdata-analysis\u002Fstandard-deviation-basics",[17,18,174,20],[],1785565322186]