示例：峰态和偏斜度

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示例：峰态和偏斜度

使用 kurt 和 skew 函数对比韦伯分布和正态分布的形状。

1. 定义下列分布并绘制其图像：

<region id="ID0ESI3AB" top="192" left="24" width="600" height="219" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="false">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="10">
          <traceStyle color="#FF00008B" symbol="none" line-weight="1" line-style="Solid">columns</traceStyle>
        </trace>
        <trace resultRef="11">
          <traceStyle color="#FFF79646" symbol="none" line-weight="1" line-style="Solid">columns</traceStyle>
        </trace>
      </traces>
      <axes>
        <xAxis legend-position="PlotBoundaryBottom" rank="1">
          <axisLine position="bottom" positionticmark="-1" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <matcol />
                  <id>WH</id>
                  <real>0</real>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <matcol />
                  <id>NH</id>
                  <real>0</real>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis legend-position="PlotBoundaryLeft" rank="1">
          <axisLine position="ticknumberlock" positionticmark="2" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <matcol />
                  <id>WH</id>
                  <real>1</real>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <matcol />
                  <id>NH</id>
                  <real>1</real>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

2. 通过计算分布的峰态来对比分布的峰值。

柱状图的柱说明正态分布的峰态稍微偏离零。韦伯分布的峰态表明其峰值小于正态分布的峰值。

可通过下列样本峰态系数估算总体峰态。

<region id="ID0EDM3AB" top="672" left="48" width="679.516666666667" height="80" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math>
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id label="None">kurtosis</id>
        <boundVars>
          <id label="None">v</id>
        </boundVars>
      </function>
      <apply>
        <minus />
        <apply>
          <div />
          <apply>
            <mult />
            <apply>
              <mult />
              <apply>
                <id>rows</id>
                <id>v</id>
              </apply>
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <id>rows</id>
                    <id>v</id>
                  </apply>
                  <real>1</real>
                </apply>
              </parens>
            </apply>
            <apply>
              <summation />
              <lambda>
                <boundVars>
                  <placeholder />
                </boundVars>
                <apply>
                  <pow />
                  <parens>
                    <apply>
                      <minus />
                      <id>v</id>
                      <apply>
                        <id>mean</id>
                        <id>v</id>
                      </apply>
                    </apply>
                  </parens>
                  <real>4</real>
                </apply>
              </lambda>
              <upperBound>
                <placeholder />
              </upperBound>
            </apply>
          </apply>
          <apply>
            <mult />
            <apply>
              <mult />
              <apply>
                <mult />
                <parens>
                  <apply>
                    <minus />
                    <apply>
                      <id>rows</id>
                      <id>v</id>
                    </apply>
                    <real>1</real>
                  </apply>
                </parens>
                <parens>
                  <apply>
                    <minus />
                    <apply>
                      <id>rows</id>
                      <id>v</id>
                    </apply>
                    <real>2</real>
                  </apply>
                </parens>
              </apply>
              <parens>
                <apply>
                  <minus />
                  <apply>
                    <id>rows</id>
                    <id>v</id>
                  </apply>
                  <real>3</real>
                </apply>
              </parens>
            </apply>
            <apply>
              <pow />
              <apply>
                <id>Stdev</id>
                <id>v</id>
              </apply>
              <real>4</real>
            </apply>
          </apply>
        </apply>
        <apply>
          <div />
          <apply>
            <mult />
            <real>3</real>
            <apply>
              <pow />
              <parens>
                <apply>
                  <minus />
                  <apply>
                    <id>rows</id>
                    <id>v</id>
                  </apply>
                  <real>1</real>
                </apply>
              </parens>
              <real>2</real>
            </apply>
          </apply>
          <apply>
            <mult />
            <parens>
              <apply>
                <minus />
                <apply>
                  <id>rows</id>
                  <id>v</id>
                </apply>
                <real>2</real>
              </apply>
            </parens>
            <parens>
              <apply>
                <minus />
                <apply>
                  <id>rows</id>
                  <id>v</id>
                </apply>
                <real>3</real>
              </apply>
            </parens>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" />
    </resultFormat>
  </math>
</region>

3. 比较分布的偏斜度。

韦伯分布向右拖尾。

可通过下列样本偏斜系数估算总体偏斜度。

<region id="ID0ENR3AB" top="1152" left="24" width="379.883333333333" height="80" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math>
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id label="None">skewness</id>
        <boundVars>
          <id label="None">v</id>
        </boundVars>
      </function>
      <apply>
        <div />
        <apply>
          <mult />
          <apply>
            <id>rows</id>
            <id>v</id>
          </apply>
          <apply>
            <summation />
            <lambda>
              <boundVars>
                <placeholder />
              </boundVars>
              <apply>
                <pow />
                <parens>
                  <apply>
                    <minus />
                    <id>v</id>
                    <apply>
                      <id>mean</id>
                      <id>v</id>
                    </apply>
                  </apply>
                </parens>
                <real>3</real>
              </apply>
            </lambda>
            <upperBound>
              <placeholder />
            </upperBound>
          </apply>
        </apply>
        <apply>
          <mult />
          <apply>
            <mult />
            <parens>
              <apply>
                <minus />
                <apply>
                  <id>rows</id>
                  <id>v</id>
                </apply>
                <real>1</real>
              </apply>
            </parens>
            <parens>
              <apply>
                <minus />
                <apply>
                  <id>rows</id>
                  <id>v</id>
                </apply>
                <real>2</real>
              </apply>
            </parens>
          </apply>
          <apply>
            <pow />
            <apply>
              <id>Stdev</id>
              <id>v</id>
            </apply>
            <real>3</real>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" />
    </resultFormat>
  </math>
</region>