:download:`download notebook <../ipynb_files/descriptive_stats.ipynb>`

記述統計
========

データの要約
------------

次のようなデータが与えられたとする．

.. math::


   X = [4, 0, 7, 6, 5, 6, 5, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10],

.. math::


   Y = [9, 6, 1, 6, 5, 10, 3, 2, 7, 2, 5, 8, 7, 6, 1, 8, 4, 5, 4, 0, 9, 3, 4]

データを眺めていても傾向は読めないので，\ ``np.sort``\ 関数を使ってデータを順に並べてみる


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import numpy as np
    X = np.array([4, 1, 7, 6, 5, 6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    Y = np.array([9, 6, 1, 6, 5, 10, 3, 2, 7, 2, 5, 8, 7, 6, 1, 8, 4, 5, 4, 0, 9, 3, 4])
    print("X :=", np.sort(X) )
    print("Y :=", np.sort(Y) )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    X := [ 1  3  4  4  4  4  4  4  5  5  5  5  5  5  5  5  6  6  6  6  6  7 10]
    Y := [ 0  1  1  2  2  3  3  4  4  4  5  5  5  6  6  6  7  7  8  8  9  9 10]


すると最小値や最大値が一目でわかり


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    print( np.min(X), np.max(X) ) # = X.max(), X.min()
    print( np.min(Y), np.max(Y) ) # = Y.max(), Y.min()

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    1 10
    0 10


もう少し詳しい情報を\ ``pandas``\ パッケージを用いて要約させると


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import pandas as pd
    df = pd.DataFrame({
    '試験X': X,
    '試験Y': Y,
    })
    df.describe()

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>試験X</th>
          <th>試験Y</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>count</th>
          <td>23.000000</td>
          <td>23.000000</td>
        </tr>
        <tr>
          <th>mean</th>
          <td>5.000000</td>
          <td>5.000000</td>
        </tr>
        <tr>
          <th>std</th>
          <td>1.651446</td>
          <td>2.796101</td>
        </tr>
        <tr>
          <th>min</th>
          <td>1.000000</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>25%</th>
          <td>4.000000</td>
          <td>3.000000</td>
        </tr>
        <tr>
          <th>50%</th>
          <td>5.000000</td>
          <td>5.000000</td>
        </tr>
        <tr>
          <th>75%</th>
          <td>6.000000</td>
          <td>7.000000</td>
        </tr>
        <tr>
          <th>max</th>
          <td>10.000000</td>
          <td>10.000000</td>
        </tr>
      </tbody>
    </table>
    </div>



とわかる．mean以下の詳細は次節で詳細に説明するが，ここでは\ ``df.discribe()``\ の出力結果を簡潔に説明する．

まず，countはデータセットの大きさ（要素数，総度数）であり試験X，試験Yともに23個である．
meanはデータセットの平均値であり試験X，試験Yともに5点である．
stdは標準偏差であり，試験Xは約1.65点，試験Yは約2.80点である．
標準偏差はデータの散らばりを表す指標（散布度）であり，値が大きいほどデータのばらつきが大きいことを意味し，試験Yの方がデータが散らばっていうることがわかる（逆に試験Xは平均値の周りにデータが集まっている）．
minはデータセットの最小値， 25%，50%，75%
はデータセットの25%点，50%点（中央値），75%点，
maxはデータセットの最大値を意味する．
このようにデータセットの特徴量を要約することで，データの傾向を把握することができる．

代表値
~~~~~~


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    n = len(X)
    print("平均値 :", np.sum(X) / n )
    print("平均値 :", np.mean(X) )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    平均値 : 5.0
    平均値 : 5.0



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X = np.array([50, 100, 150, 200, 4500])
    print("平均値：", np.mean(X) )
    print("中央値：", np.median(X) )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    平均値： 1000.0
    中央値： 150.0


散布度
~~~~~~

平均偏差
^^^^^^^^


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X = np.array([4, 1, 7, 6, 5, 6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    Dx = X - X.mean()
    print("Dx = ", Dx)
    print("np.mean(Dx) = ", np.mean(Dx) )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    Dx =  [-1. -4.  2.  1.  0.  1.  0. -1. -1. -2. -1. -1.  0.  0.  0.  0.  1.  1.
      1.  0. -1.  0.  5.]
    np.mean(Dx) =  0.0


**問** Yのデータセットに対して同様に偏差の平均値が0になることを示せ．


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    Y = np.array([9, 6, 1, 6, 5, 10, 3, 2, 7, 2, 5, 8, 7, 6, 1, 8, 4, 5, 4, 0, 9, 3, 4])

.. raw:: html

     </div>
    </div>

２乗偏差と分散・標準偏差
^^^^^^^^^^^^^^^^^^^^^^^^


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X = np.array([4, 1, 7, 6, 5, 6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    Dx = X - X.mean() # = [𝑑_1, ... , 𝑑_𝑛] where 𝑑𝑖 = (𝑥_𝑖 - 𝑥)
    Dx2 = Dx ** 2 # = [𝑑^2_1 , ... ,𝑑^2_𝑛] where 𝑑^2_𝑖 = (𝑥_𝑖 - 𝑥)^2
    print("Dx2 = ", Dx2)
    print("np.mean(Dx2) = ", np.mean(Dx2) )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    Dx2 =  [ 1. 16.  4.  1.  0.  1.  0.  1.  1.  4.  1.  1.  0.  0.  0.  0.  1.  1.
      1.  0.  1.  0. 25.]
    np.mean(Dx2) =  2.608695652173913



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    print( X.var() )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    2.608695652173913


**問**
Yのデータセットに対して同様に２乗偏差の平均値を求め，np.var(Y)の値と一致することを示せ．


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    print("std = ", np.sqrt( np.var(X) ))

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    std =  1.6151457061744965



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    print(r"std = ", np.std(X) ) # = X.std()

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    std =  1.6151457061744965



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X2 = X**2

.. raw:: html

     </div>
    </div>

分散の計算公式
^^^^^^^^^^^^^^

:math:`s^2 = \overline{(x^2)} - \overline{(x)}^2`\ から分散の値を求める．


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X = np.array([4, 1, 7, 6, 5, 6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    X2 = X ** 2 # $= [4^2, 1^2, ..., 5^2, 10^2]$
    s2 = np.mean(X2) - np.mean(X) ** 2 # $= ¥overline{(x^2)} - (¥overline{x})^2$
    print( s2 )
    print( X.var() )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    2.608695652173914
    2.608695652173913


**問**
データセット\ :math:`Y`\ に対して同様に，\ :math:`Y^2`\ のデータセットを作成し，分散の計算公式から分散\ :math:`s^2`\ を評価せよ．またその値がnp.var(Y)の値に一致することを確認せよ．

データの整理
------------

名義尺度の場合
~~~~~~~~~~~~~~


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import pandas as pd
    import numpy as np
    X = ["A", "B", "O", "A", "O", "AB", "B", "O", "A", "O",
         "A", np.nan, "B", "AB", "O", "O", "A", "O", "B", "AB", "O"]
    bloodtypes = pd.Series(X)
    print("unique dataset =", bloodtypes.unique() )
    print("総度数 =", bloodtypes.count() )
    print("Xの大きさ = ", len(X) )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    unique dataset = ['A' 'B' 'O' 'AB' nan]
    総度数 = 20
    Xの大きさ =  21



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    bloodtypes = pd.Series(X).dropna()
    print("unique dataset =", bloodtypes.unique() )

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    unique dataset = ['A' 'B' 'O' 'AB']



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    bloodtypes.value_counts().reindex(["A", "B", "O", "AB"])

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    A     5
    B     4
    O     8
    AB    3
    Name: count, dtype: int64




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import matplotlib.pyplot as plt
    plt.ylabel("count", fontsize=20)
    bloodtypes.hist(xlabelsize=20)

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: ylabel='count'>




.. image:: descriptive_stats_files/descriptive_stats_32_1.png



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    bloodtypes.value_counts(normalize=True).reindex(["A", "B", "O", "AB"])

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>proportion</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>A</th>
          <td>0.25</td>
        </tr>
        <tr>
          <th>B</th>
          <td>0.20</td>
        </tr>
        <tr>
          <th>O</th>
          <td>0.40</td>
        </tr>
        <tr>
          <th>AB</th>
          <td>0.15</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> float64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    n = bloodtypes.count()
    w = np.ones(n) / n
    print(w)

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    [0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
     0.05 0.05 0.05 0.05 0.05 0.05]



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    plt.ylabel("propotion",fontsize=20)
    bloodtypes.hist(weights=w, xlabelsize=20)

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: ylabel='propotion'>




.. image:: descriptive_stats_files/descriptive_stats_35_1.png


量的データかつ離散データの場合
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    data = np.array([4, 1, 7, 6, 5,  6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    X = pd.Series(data)
    X.value_counts()

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>count</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>5</th>
          <td>8</td>
        </tr>
        <tr>
          <th>4</th>
          <td>6</td>
        </tr>
        <tr>
          <th>6</th>
          <td>5</td>
        </tr>
        <tr>
          <th>7</th>
          <td>1</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1</td>
        </tr>
        <tr>
          <th>3</th>
          <td>1</td>
        </tr>
        <tr>
          <th>10</th>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> int64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    Xe = np.arange(0, 11) # = [0, 1, ..., 10]
    X.value_counts().reindex(Xe)

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>count</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>NaN</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1.0</td>
        </tr>
        <tr>
          <th>2</th>
          <td>NaN</td>
        </tr>
        <tr>
          <th>3</th>
          <td>1.0</td>
        </tr>
        <tr>
          <th>4</th>
          <td>6.0</td>
        </tr>
        <tr>
          <th>5</th>
          <td>8.0</td>
        </tr>
        <tr>
          <th>6</th>
          <td>5.0</td>
        </tr>
        <tr>
          <th>7</th>
          <td>1.0</td>
        </tr>
        <tr>
          <th>8</th>
          <td>NaN</td>
        </tr>
        <tr>
          <th>9</th>
          <td>NaN</td>
        </tr>
        <tr>
          <th>10</th>
          <td>1.0</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> float64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X.value_counts().reindex(Xe, fill_value=0)

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>count</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>0</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1</td>
        </tr>
        <tr>
          <th>2</th>
          <td>0</td>
        </tr>
        <tr>
          <th>3</th>
          <td>1</td>
        </tr>
        <tr>
          <th>4</th>
          <td>6</td>
        </tr>
        <tr>
          <th>5</th>
          <td>8</td>
        </tr>
        <tr>
          <th>6</th>
          <td>5</td>
        </tr>
        <tr>
          <th>7</th>
          <td>1</td>
        </tr>
        <tr>
          <th>8</th>
          <td>0</td>
        </tr>
        <tr>
          <th>9</th>
          <td>0</td>
        </tr>
        <tr>
          <th>10</th>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> int64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X.value_counts(normalize=True).reindex(Xe, fill_value=0)

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>proportion</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>1</th>
          <td>0.043478</td>
        </tr>
        <tr>
          <th>2</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>3</th>
          <td>0.043478</td>
        </tr>
        <tr>
          <th>4</th>
          <td>0.260870</td>
        </tr>
        <tr>
          <th>5</th>
          <td>0.347826</td>
        </tr>
        <tr>
          <th>6</th>
          <td>0.217391</td>
        </tr>
        <tr>
          <th>7</th>
          <td>0.043478</td>
        </tr>
        <tr>
          <th>8</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>9</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>10</th>
          <td>0.043478</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> float64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    Xe = np.arange(0, 11) # 0, 1, ..., 10
    X.value_counts(normalize=True).reindex(Xe, fill_value=0).cumsum()

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
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      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>proportion</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>1</th>
          <td>0.043478</td>
        </tr>
        <tr>
          <th>2</th>
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        <tr>
          <th>3</th>
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          <th>6</th>
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        </tr>
        <tr>
          <th>7</th>
          <td>0.956522</td>
        </tr>
        <tr>
          <th>8</th>
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        </tr>
        <tr>
          <th>9</th>
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        </tr>
        <tr>
          <th>10</th>
          <td>1.000000</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> float64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import numpy as np
    import pandas as pd
    data = np.array([4, 1, 7, 6, 5,  6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    X = pd.Series(data)
    Xe = np.arange(0, 11) # = [0, 1, ..., 10]
    freq_tab = pd.DataFrame({
        '度数': X.value_counts().reindex(Xe, fill_value=0),
        '相対度数': X.value_counts(normalize=True).reindex(Xe, fill_value=0),
        '累積相対度数': X.value_counts(normalize=True).reindex(Xe, fill_value=0).cumsum()
        }, index = Xe)
    freq_tab

.. raw:: html

     </div>
    </div>




.. raw:: html

    
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        </tr>
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        <tr>
          <th>0</th>
          <td>0</td>
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          <td>1</td>
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          <th>2</th>
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          <th>3</th>
          <td>1</td>
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        </tr>
        <tr>
          <th>4</th>
          <td>6</td>
          <td>0.260870</td>
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        </tr>
        <tr>
          <th>5</th>
          <td>8</td>
          <td>0.347826</td>
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          <th>6</th>
          <td>5</td>
          <td>0.217391</td>
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          <th>7</th>
          <td>1</td>
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        </div>
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.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    !pip install japanize-matplotlib
    import japanize_matplotlib

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    Requirement already satisfied: japanize-matplotlib in /usr/local/lib/python3.11/dist-packages (1.1.3)
    Requirement already satisfied: matplotlib in /usr/local/lib/python3.11/dist-packages (from japanize-matplotlib) (3.10.0)
    Requirement already satisfied: contourpy>=1.0.1 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (1.3.2)
    Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (0.12.1)
    Requirement already satisfied: fonttools>=4.22.0 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (4.58.4)
    Requirement already satisfied: kiwisolver>=1.3.1 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (1.4.8)
    Requirement already satisfied: numpy>=1.23 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (2.0.2)
    Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (24.2)
    Requirement already satisfied: pillow>=8 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (11.2.1)
    Requirement already satisfied: pyparsing>=2.3.1 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (3.2.3)
    Requirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (2.9.0.post0)
    Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.11/dist-packages (from python-dateutil>=2.7->matplotlib->japanize-matplotlib) (1.17.0)



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    freq_tab.plot(kind='bar', subplots=True)

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    array([<Axes: title={'center': '度数'}>, <Axes: title={'center': '相対度数'}>,
           <Axes: title={'center': '累積相対度数'}>], dtype=object)




.. image:: descriptive_stats_files/descriptive_stats_44_1.png



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X.hist() # 度数

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: >




.. image:: descriptive_stats_files/descriptive_stats_45_1.png



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    w = np.ones(X.count()) / X.count()
    X.hist(weights=w) # 相対度数

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: >




.. image:: descriptive_stats_files/descriptive_stats_46_1.png


量的データかつ連続データの場合
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    data = [161.05, 176.75, 167.71, 180.49, 176.56, 165.79, 163.04,
            174.24, 171.99, 172.79, 163.93, 170.52, 173.68, 171.79, 164.07]
    X = pd.Series(data)
    X.value_counts()

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>count</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>161.05</th>
          <td>1</td>
        </tr>
        <tr>
          <th>176.75</th>
          <td>1</td>
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        <tr>
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          <td>1</td>
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          <td>1</td>
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          <td>1</td>
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          <th>163.04</th>
          <td>1</td>
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        <tr>
          <th>174.24</th>
          <td>1</td>
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          <th>171.99</th>
          <td>1</td>
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          <th>172.79</th>
          <td>1</td>
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          <th>163.93</th>
          <td>1</td>
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          <th>170.52</th>
          <td>1</td>
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          <th>173.68</th>
          <td>1</td>
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          <th>171.79</th>
          <td>1</td>
        </tr>
        <tr>
          <th>164.07</th>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> int64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    k = 5 # 階級の数
    X.value_counts(bins=k).sort_index() # sort_index()をつけない場合は度数の大きい順に並ぶ

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>count</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>(161.03, 164.938]</th>
          <td>4</td>
        </tr>
        <tr>
          <th>(164.938, 168.826]</th>
          <td>2</td>
        </tr>
        <tr>
          <th>(168.826, 172.714]</th>
          <td>3</td>
        </tr>
        <tr>
          <th>(172.714, 176.602]</th>
          <td>4</td>
        </tr>
        <tr>
          <th>(176.602, 180.49]</th>
          <td>2</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> int64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X.value_counts(bins=[160, 165, 170, 175, 180, 185]).sort_index()

.. raw:: html

     </div>
    </div>




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
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            text-align: right;
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    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>count</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>(159.999, 165.0]</th>
          <td>4</td>
        </tr>
        <tr>
          <th>(165.0, 170.0]</th>
          <td>2</td>
        </tr>
        <tr>
          <th>(170.0, 175.0]</th>
          <td>6</td>
        </tr>
        <tr>
          <th>(175.0, 180.0]</th>
          <td>2</td>
        </tr>
        <tr>
          <th>(180.0, 185.0]</th>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div><br><label><b>dtype:</b> int64</label>




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import pandas as pd
    data = [161.05, 176.75, 167.71, 180.49, 176.56, 165.79, 163.04,
            174.24, 171.99, 172.79, 163.93, 170.52, 173.68, 171.79, 164.07]
    X = pd.Series(data)
    cls = [160, 165, 170, 175, 180, 185]
    freq = X.value_counts(bins=cls).sort_index() # 度数
    prop = X.value_counts(bins=cls, normalize=True).sort_index() # 相対度数
    cprop = X.value_counts(bins=cls, normalize=True).cumsum().sort_index() # 累積相対度数
    freq_tab = pd.DataFrame({
    "階級値": freq.index.mid, # 階級区間の中央
    "度数": freq,
    "相対度数": prop,
    "累積相対度数": cprop
    }, index=freq.index)
    freq_tab

.. raw:: html

     </div>
    </div>




.. raw:: html

    
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        }
    
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            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>階級値</th>
          <th>度数</th>
          <th>相対度数</th>
          <th>累積相対度数</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>(159.999, 165.0]</th>
          <td>162.4995</td>
          <td>4</td>
          <td>0.266667</td>
          <td>0.666667</td>
        </tr>
        <tr>
          <th>(165.0, 170.0]</th>
          <td>167.5000</td>
          <td>2</td>
          <td>0.133333</td>
          <td>0.800000</td>
        </tr>
        <tr>
          <th>(170.0, 175.0]</th>
          <td>172.5000</td>
          <td>6</td>
          <td>0.400000</td>
          <td>0.400000</td>
        </tr>
        <tr>
          <th>(175.0, 180.0]</th>
          <td>177.5000</td>
          <td>2</td>
          <td>0.133333</td>
          <td>0.933333</td>
        </tr>
        <tr>
          <th>(180.0, 185.0]</th>
          <td>182.5000</td>
          <td>1</td>
          <td>0.066667</td>
          <td>1.000000</td>
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        </div>
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.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    X.hist(bins = [160, 165, 170, 175, 180, 185])

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: >




.. image:: descriptive_stats_files/descriptive_stats_52_1.png


度数分布表から平均・分散を求める．
----------------------------------


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import numpy as np
    import pandas as pd
    data = np.array([4, 1, 7, 6, 5,  6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    ds = pd.Series(data)
    Xe = np.arange(0, 11) # = [0, 1, ..., 10] # 期待しているユニークなデータセット
    f = ds.value_counts().reindex(Xe, fill_value=0).to_numpy() #　.to\_numpy()でnumpyの配列に変換
    p = ds.value_counts(normalize=True).reindex(Xe, fill_value=0).to_numpy()
    print("Xe = ", Xe)
    print("f = ", f)
    print("p = ", p)

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    Xe =  [ 0  1  2  3  4  5  6  7  8  9 10]
    f =  [0 1 0 1 6 8 5 1 0 0 1]
    p =  [0.         0.04347826 0.         0.04347826 0.26086957 0.34782609
     0.2173913  0.04347826 0.         0.         0.04347826]



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    x = np.array([1, 2, 3])
    y = np.array([2, 0, 10])
    x * y

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    array([ 2,  0, 30])




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    Xe * f

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    array([ 0,  1,  0,  3, 24, 40, 30,  7,  0,  0, 10])




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    n = X.count()
    mean1 = np.sum( Xe * f ) / n
    mean2 = np.sum( Xe * p )
    print(mean1, mean2)

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    5.0 5.0



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    D2 = ( Xe - X.mean() ) ** 2
    D2

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    array([25., 16.,  9.,  4.,  1.,  0.,  1.,  4.,  9., 16., 25.])




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    var1 = np.sum( D2 * f) / n
    var2 = np.sum( D2 * p )
    print(var1, var2)

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    2.608695652173913 2.608695652173913



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    Xe2 = Xe * Xe
    Xe2

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    array([  0,   1,   4,   9,  16,  25,  36,  49,  64,  81, 100])




.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    Xe2_mean = np.sum( Xe2 * p )
    Xe_mean  = np.sum( Xe * p)
    var = Xe2_mean - Xe_mean ** 2
    print(var)

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    2.608695652173914



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    np.abs(var1 - var)

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    np.float64(8.881784197001252e-16)



箱ひげ図
--------


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    !pip install japanize-matplotlib
    import japanize_matplotlib

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    Requirement already satisfied: japanize-matplotlib in /usr/local/lib/python3.11/dist-packages (1.1.3)
    Requirement already satisfied: matplotlib in /usr/local/lib/python3.11/dist-packages (from japanize-matplotlib) (3.10.0)
    Requirement already satisfied: contourpy>=1.0.1 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (1.3.2)
    Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (0.12.1)
    Requirement already satisfied: fonttools>=4.22.0 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (4.58.4)
    Requirement already satisfied: kiwisolver>=1.3.1 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (1.4.8)
    Requirement already satisfied: numpy>=1.23 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (2.0.2)
    Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (24.2)
    Requirement already satisfied: pillow>=8 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (11.2.1)
    Requirement already satisfied: pyparsing>=2.3.1 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (3.2.3)
    Requirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.11/dist-packages (from matplotlib->japanize-matplotlib) (2.9.0.post0)
    Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.11/dist-packages (from python-dateutil>=2.7->matplotlib->japanize-matplotlib) (1.17.0)



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import numpy as np
    import pandas as pd
    X = np.array([4, 1, 7, 6, 5,  6, 5, 4, 4, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 5, 4, 5, 10])
    Y = np.array([9, 6, 1, 6, 5, 10, 3, 2, 7, 2, 5, 8, 7, 6, 1, 8, 4, 5, 4, 0, 9, 3, 4])
    df = pd.DataFrame({
        '試験X': X,
        '試験Y': Y,
    })
    df.boxplot(showmeans=True) # showmeans=Trueで平均値$\bar{x}$が表示される．

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: >




.. image:: descriptive_stats_files/descriptive_stats_65_1.png


相関
----


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import numpy as np
    
    np.random.seed(8787)
    
    # 平均, 分散, 標準偏差
    mean_X, mean_Y = 170, 68
    var_X, var_Y = 5, 10
    std_X, std_Y  = np.sqrt(var_X), np.sqrt(var_Y)
    rho = 0.7  # 相関係数
    
    # 共分散行列
    cov = [
        [var_X,               rho * std_X * std_Y],
        [rho * std_X * std_Y, var_Y              ]
    ]
    L = np.linalg.cholesky(cov)
    
    # ランダムな非相関標準正規データを生成
    num_samples = 1000
    uncorr = np.random.standard_normal( (2, num_samples) )
    
    # 相関のあるデータに変換
    mean = np.array([mean_X, mean_Y]).reshape(2, 1)
    data = np.dot(L, uncorr) + mean
    
    import pandas as pd
    
    df = pd.DataFrame({
      "身長[cm]": data[0, :],
      "体重[kg]": data[1, :]
     })

.. raw:: html

     </div>
    </div>


.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    print(df.describe())

.. raw:: html

     </div>
    </div>


.. parsed-literal::

                身長[cm]       体重[kg]
    count  1000.000000  1000.000000
    mean    169.841922    67.796619
    std       2.267517     3.246836
    min     163.060816    57.335005
    25%     168.303610    65.626473
    50%     169.838044    67.734873
    75%     171.444423    70.033005
    max     179.247738    80.651467



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    !pip install japanize-matplotlib
    import japanize_matplotlib

.. raw:: html

     </div>
    </div>


.. parsed-literal::

    Requirement already satisfied: japanize-matplotlib in /usr/local/lib/python3.11/dist-packages (1.1.3)
    Requirement already satisfied: matplotlib in /usr/local/lib/python3.11/dist-packages (from japanize-matplotlib) (3.10.0)
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.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    df.plot(kind='scatter', x='身長[cm]', y='体重[kg]')

.. raw:: html

     </div>
    </div>




.. parsed-literal::

    <Axes: xlabel='身長[cm]', ylabel='体重[kg]'>




.. image:: descriptive_stats_files/descriptive_stats_70_1.png



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    import numpy as np
    import matplotlib.pyplot as plt
    
    np.random.seed(87)
    ns = 1000 # number of samples
    
    def gen_corr_data(ns, rho):
      # 平均, 分散, 標準偏差
      mean_X, mean_Y = 170, 68
      var_X, var_Y = 5, 10
      std_X, std_Y  = np.sqrt(var_X), np.sqrt(var_Y)
      rho = rho  # 相関係数
      # 共分散行列
      cov = [
          [var_X,               rho * std_X * std_Y],
          [rho * std_X * std_Y, var_Y              ]
      ]
      L = np.linalg.cholesky(cov)
    
      # ランダムな非相関標準正規データを生成
      uncorr = np.random.standard_normal( (2, num_samples) )
    
    # 相関のあるデータに変換
      mean = np.array([mean_X, mean_Y]).reshape(2, 1)
      data = np.dot(L, uncorr) + mean
      return data
    
    fig, ax = plt.subplots(2, 3, figsize=(10,6))
    X1, Y1 = gen_corr_data(ns, 0.9)
    X2, Y2 = gen_corr_data(ns, -0.9)
    X3, Y3 = gen_corr_data(ns, 0)
    X4, Y4 = np.random.random(ns), np.random.random(ns)
    x = np.linspace(-10, 10, ns)
    eps = np.random.normal(0, 3, ns) # noise
    y5 = - x ** 2 + eps
    eps = np.random.normal(0, 0.05, ns) # noise
    z = (1 + eps)*np.exp( 1.j * 2*np.pi * x)
    ax[0,0].scatter(X1, Y1)
    ax[0,1].scatter(X2, Y2)
    ax[0,2].scatter(X3, Y3)
    ax[1,0].scatter(X4, Y4)
    ax[1,1].scatter(x, y5)
    ax[1,2].scatter(z.real, z.imag)
    #ax[3].scatter(z.real, z.imag)
    tit=["(a)", "(b)" ,"(c)", "(d)", "(e)", "(f)"]
    for i, a in enumerate(ax.flat):
      a.set_xlabel(r"$x$", fontsize=16)
      a.set_ylabel(r"$y$", fontsize=16)
      a.set_title(tit[i])
      a.set_xticks([])
      a.set_yticks([])
    plt.tight_layout()

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     </div>
    </div>



.. image:: descriptive_stats_files/descriptive_stats_71_0.png



.. raw:: html

    <div class="input-with-prompt">
      <div class="prompt">In:</div>
      <div class="code">

.. code-block:: ipython

    for i,j in

.. raw:: html

     </div>
    </div>