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| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.stat_tut.weg.cs_eg.chi_sq_intervals"></a><a class="link" href="chi_sq_intervals.html" title="Confidence Intervals on the Standard Deviation"> |
| Confidence Intervals on the Standard Deviation</a> |
| </h6></div></div></div> |
| <p> |
| Once you have calculated the standard deviation for your data, a legitimate |
| question to ask is "How reliable is the calculated standard deviation?". |
| For this situation the Chi Squared distribution can be used to calculate |
| confidence intervals for the standard deviation. |
| </p> |
| <p> |
| The full example code & sample output is in <a href="../../../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_deviation_test.cpp</a>. |
| </p> |
| <p> |
| We'll begin by defining the procedure that will calculate and print |
| out the confidence intervals: |
| </p> |
| <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">confidence_limits_on_std_deviation</span><span class="special">(</span> |
| <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span> <span class="comment">// Sample Standard Deviation |
| </span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">)</span> <span class="comment">// Sample size |
| </span><span class="special">{</span> |
| </pre> |
| <p> |
| We'll begin by printing out some general information: |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> |
| <span class="string">"________________________________________________\n"</span> |
| <span class="string">"2-Sided Confidence Limits For Standard Deviation\n"</span> |
| <span class="string">"________________________________________________\n\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Number of Observations"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">N</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">Sd</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| and then define a table of significance levels for which we'll calculate |
| intervals: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span> |
| </pre> |
| <p> |
| The distribution we'll need to calculate the confidence intervals is |
| a Chi Squared distribution, with N-1 degrees of freedom: |
| </p> |
| <pre class="programlisting"><span class="identifier">chi_squared</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span> |
| </pre> |
| <p> |
| For each value of alpha, the formula for the confidence interval is |
| given by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../../equations/chi_squ_tut1.png"></span> |
| </p> |
| <p> |
| Where <span class="inlinemediaobject"><img src="../../../../../../equations/chi_squ_tut2.png"></span> is the upper critical value, and <span class="inlinemediaobject"><img src="../../../../../../equations/chi_squ_tut3.png"></span> is |
| the lower critical value of the Chi Squared distribution. |
| </p> |
| <p> |
| In code we begin by printing out a table header: |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n\n"</span> |
| <span class="string">"_____________________________________________\n"</span> |
| <span class="string">"Confidence Lower Upper\n"</span> |
| <span class="string">" Value (%) Limit Limit\n"</span> |
| <span class="string">"_____________________________________________\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| and then loop over the values of alpha and calculate the intervals |
| for each: remember that the lower critical value is the same as the |
| quantile, and the upper critical value is the same as the quantile |
| from the complement of the probability: |
| </p> |
| <pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="comment">// Confidence value: |
| </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span> |
| <span class="comment">// Calculate limits: |
| </span> <span class="keyword">double</span> <span class="identifier">lower_limit</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">)));</span> |
| <span class="keyword">double</span> <span class="identifier">upper_limit</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span> |
| <span class="comment">// Print Limits: |
| </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">lower_limit</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">upper_limit</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| To see some example output we'll use the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm" target="_top">gear |
| data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH |
| e-Handbook of Statistical Methods.</a>. The data represents measurements |
| of gear diameter from a manufacturing process. |
| </p> |
| <pre class="programlisting">________________________________________________ |
| 2-Sided Confidence Limits For Standard Deviation |
| ________________________________________________ |
| |
| Number of Observations = 100 |
| Standard Deviation = 0.006278908 |
| |
| |
| _____________________________________________ |
| Confidence Lower Upper |
| Value (%) Limit Limit |
| _____________________________________________ |
| 50.000 0.00601 0.00662 |
| 75.000 0.00582 0.00685 |
| 90.000 0.00563 0.00712 |
| 95.000 0.00551 0.00729 |
| 99.000 0.00530 0.00766 |
| 99.900 0.00507 0.00812 |
| 99.990 0.00489 0.00855 |
| 99.999 0.00474 0.00895 |
| </pre> |
| <p> |
| So at the 95% confidence level we conclude that the standard deviation |
| is between 0.00551 and 0.00729. |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.cs_eg.chi_sq_intervals.confidence_intervals_as_a_function_of_the_number_of_observations"></a><h5> |
| <a name="id938955"></a> |
| <a class="link" href="chi_sq_intervals.html#math_toolkit.dist.stat_tut.weg.cs_eg.chi_sq_intervals.confidence_intervals_as_a_function_of_the_number_of_observations">Confidence |
| intervals as a function of the number of observations</a> |
| </h5> |
| <p> |
| Similarly, we can also list the confidence intervals for the standard |
| deviation for the common confidence levels 95%, for increasing numbers |
| of observations. |
| </p> |
| <p> |
| The standard deviation used to compute these values is unity, so the |
| limits listed are <span class="bold"><strong>multipliers</strong></span> for |
| any particular standard deviation. For example, given a standard deviation |
| of 0.0062789 as in the example above; for 100 observations the multiplier |
| is 0.8780 giving the lower confidence limit of 0.8780 * 0.006728 = |
| 0.00551. |
| </p> |
| <pre class="programlisting">____________________________________________________ |
| Confidence level (two-sided) = 0.0500000 |
| Standard Deviation = 1.0000000 |
| ________________________________________ |
| Observations Lower Upper |
| Limit Limit |
| ________________________________________ |
| 2 0.4461 31.9102 |
| 3 0.5207 6.2847 |
| 4 0.5665 3.7285 |
| 5 0.5991 2.8736 |
| 6 0.6242 2.4526 |
| 7 0.6444 2.2021 |
| 8 0.6612 2.0353 |
| 9 0.6755 1.9158 |
| 10 0.6878 1.8256 |
| 15 0.7321 1.5771 |
| 20 0.7605 1.4606 |
| 30 0.7964 1.3443 |
| 40 0.8192 1.2840 |
| 50 0.8353 1.2461 |
| 60 0.8476 1.2197 |
| 100 0.8780 1.1617 |
| 120 0.8875 1.1454 |
| 1000 0.9580 1.0459 |
| 10000 0.9863 1.0141 |
| 50000 0.9938 1.0062 |
| 100000 0.9956 1.0044 |
| 1000000 0.9986 1.0014 |
| </pre> |
| <p> |
| With just 2 observations the limits are from <span class="bold"><strong>0.445</strong></span> |
| up to to <span class="bold"><strong>31.9</strong></span>, so the standard deviation |
| might be about <span class="bold"><strong>half</strong></span> the observed value |
| up to <span class="bold"><strong>30 times</strong></span> the observed value! |
| </p> |
| <p> |
| Estimating a standard deviation with just a handful of values leaves |
| a very great uncertainty, especially the upper limit. Note especially |
| how far the upper limit is skewed from the most likely standard deviation. |
| </p> |
| <p> |
| Even for 10 observations, normally considered a reasonable number, |
| the range is still from 0.69 to 1.8, about a range of 0.7 to 2, and |
| is still highly skewed with an upper limit <span class="bold"><strong>twice</strong></span> |
| the median. |
| </p> |
| <p> |
| When we have 1000 observations, the estimate of the standard deviation |
| is starting to look convincing, with a range from 0.95 to 1.05 - now |
| near symmetrical, but still about + or - 5%. |
| </p> |
| <p> |
| Only when we have 10000 or more repeated observations can we start |
| to be reasonably confident (provided we are sure that other factors |
| like drift are not creeping in). |
| </p> |
| <p> |
| For 10000 observations, the interval is 0.99 to 1.1 - finally a really |
| convincing + or -1% confidence. |
| </p> |
| </div> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
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