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323 lines
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<h1 class="center">Confidence Intervals</h1>
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<p class="center"><img src="images/ci4pm2.svg" alt="confidence interval 4 plus or minus 2" height="109" width="292"><br>
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An interval of <b>4 plus or minus 2</b></p>
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<p>A Confidence Interval is a <b>range of values</b> we are fairly sure our <b>true value</b> lies in.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/men.jpg" alt="men running" height="155" width="240"></p>
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<h3>Example: Average Height</h3>
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<p>We measure the heights of <b>40</b> randomly chosen men, and get a <a href="../mean.html">mean</a> height of <b>175cm</b>,</p>
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<p>We also know the <a href="standard-deviation.html">standard deviation</a> of men's heights is <b>20cm</b>.</p>
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<p>The <b>95% Confidence Interval</b> (we show how to calculate it later) is:</p>
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<p class="center"><img src="images/ci175pm.svg" alt="confidence interval 175 plus minus 6.2" height="93" width="206"></p>
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<p>The "<b>±</b>" means "plus or minus", so <b>175cm ± 6.2cm</b> means</p>
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<ul>
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<li>175cm − 6.2cm = 168.8cm <b>to </b></li>
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<li>175cm + 6.2cm = 181.2cm</li></ul>
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<p>And our result says the <b>true mean</b> of ALL men (if we could measure all their heights) is likely to be between 168.8cm and 181.2cm</p>
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<p><i>But it might not be!</i></p>
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<p>The "95%" says that 95% of experiments like we just did will include the true mean, but <b>5% won't</b>.</p>
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<p>So there is a 1-in-20 chance (5%) that our Confidence Interval does NOT include the true mean.</p>
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</div>
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<h2>Calculating the Confidence Interval</h2>
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<p><b>Step 1</b>: start with</p>
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<ul>
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<li>the number of observations <b>n</b></li>
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<li>the mean <b><span style="border-top: 1px solid black;">X</span></b></li>
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<li>and the <a href="standard-deviation.html">standard deviation</a> <b>s</b></li>
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</ul>
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<div class="center80">
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<p>Note: we should use the standard deviation of the entire <b>population</b>, but in many cases we won't know it.</p>
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<p>We can use the standard deviation for the <b>sample</b> if we have enough observations (at least n=30, hopefully more).</p>
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</div>
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<p>Using our example:</p>
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<ul>
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<li>number of observations <b>n = 40</b></li>
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<li>mean <b><span style="border-top: 1px solid black;">X</span> = 175</b></li>
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<li>standard deviation <b>s = 20</b></li>
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</ul>
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<p><b>Step 2</b>: decide what Confidence Interval we want: 95% or 99% are common choices. Then find the "Z" value for that Confidence Interval here:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"><b>Confidence<br>
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Interval</b></td>
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<td style="text-align:center;"><b>Z</b></td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">80%</td>
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<td style="text-align:center; width:100px;">1.282</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">85%</td>
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<td style="text-align:center;">1.440</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">90%</td>
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<td style="text-align:center;">1.645</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">95%</td>
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<td style="text-align:center;">1.960</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">99%</td>
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<td style="text-align:center;">2.576</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">99.5%</td>
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<td style="text-align:center;">2.807</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">99.9%</td>
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<td style="text-align:center;">3.291</td>
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</tr>
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</tbody></table>
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</div>
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<p>For 95% the Z value is <b>1.960</b></p>
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<p><b>Step 3</b>: use that Z value in this formula for the Confidence Interval</p>
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<p class="center larger"><span style="border-top: 1px solid black;">X</span> ± Z<span class="intbl"><em>s</em><strong>√n</strong></span></p>
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<p>Where:</p>
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<ul>
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<li><span style="border-top: 1px solid black;"><b>X</b></span> is the mean</li>
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<li><b>Z</b> is the chosen Z-value from the table above</li>
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<li><b>s</b> is the standard deviation</li>
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<li><b>n</b> is the number of observations</li>
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</ul>
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<p>And we have:</p>
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<p class="center larger">175 ± 1.960 × <span class="intbl"><em>20</em><strong>√40</strong></span></p>
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<p>Which is:</p>
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<p class="center larger"><b>175cm ± 6.20cm</b></p>
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<p>In other words: from 168.8cm to 181.2cm</p>
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<div class="words">
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<p>The value after the ± is called the <b>margin of error</b></p>
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<p>The margin of error in our example is 6.20cm</p>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><a href="confidence-interval-calculator.html"><img src="images/conf-interval-calc.gif" alt="confidence interval calculator" height="144" width="150"></a></p>
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<h2>Calculator</h2>
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<p>We have a <a href="confidence-interval-calculator.html">Confidence Interval Calculator</a> to make life easier for you.</p>
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<div style="clear:both"></div>
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<h2>Simulator</h2>
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<p>We also have a very interesting <a href="normal-distribution-simulator.html">Normal Distribution Simulator</a>. where we can start with some theoretical "true" mean and standard deviation, and then take random samples.</p>
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<p>It helps us to understand how random samples can sometimes be very good or bad at representing the underlying true values.</p>
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<h2>Another Example</h2>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/apple-tree.jpg" alt="apple tree" height="134" width="186"></p>
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<h3>Example: Apple Orchard</h3>
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<p>Are the apples big enough?</p>
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<p>There are hundreds of apples on the trees, so you randomly choose just <b>46</b> apples and get:</p>
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<ul>
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<li>a Mean of <b>86</b></li>
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<li>a Standard Deviation of <b>6.2</b></li>
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</ul>
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<p>So let's calculate:</p>
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<p class="center larger"><span style="border-top: 1px solid black;">X</span> ± Z<span class="intbl"><em>s</em><strong>√n</strong></span></p>
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<p>We know:</p>
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<ul>
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<li><span style="border-top: 1px solid black;"><b>X</b></span> is the mean = 86</li>
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<li><b>Z</b> is the Z-value = 1.960 (from the table above for 95%)</li>
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<li><b>s</b> is the standard deviation = 6.2</li>
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<li><b>n</b> is the number of observations = 46</li>
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</ul>
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<p class="center larger">86 ± 1.960 × <span class="intbl"><em>6.2</em><strong>√46</strong></span> = <b>86 ± 1.79</b></p>
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<p>So the true mean (of all the hundreds of apples) is <b>likely</b> to be between 84.21 and 87.79</p>
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<h3>True Mean</h3>
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<p>Now imagine we get to pick ALL the apples straight away, and get them ALL measured by the packing machine (this is a luxury not normally found in statistics!)</p>
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<p>And the <b>true mean</b> turns out to be<b> 84.9</b></p>
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<p>Let's lay all the apples on the ground from smallest to largest:</p>
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<p class="center"><img src="images/ci1.svg" alt="confidence interval 86 plus minus 1.79" height="205" width="256"><br>
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Each apple is a green dot,<br>
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our observations are marked blue</p>
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<p>Our result was not exact ... it is random after all ... but the true mean is inside our confidence interval of 86 ± 1.79 (in other words 84.21 to 87.79)</p>
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<p>Now the true mean<b> might not</b> be inside the confidence interval, but in <b>95% of the cases it will be!</b></p>
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<p class="center large">95% of all "95% Confidence Intervals" will include the true mean.</p>
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<p>Maybe we had this sample, with a mean of 83.5:</p>
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<p class="center"><img src="images/ci3.svg" alt="confidence interval 83.5 plus minus 1.25" height="212" width="256"><br>
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Each apple is a green dot,<br>
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our observations are marked purple</p>
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<p>That <b>does not include the true mean.</b> That can happen about 5% of the time for a 95% confidence interval.</p>
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<p>So how do we know if our sample is one of the "lucky" 95% or the unlucky 5%? Unless we get to measure the whole population like above we simply <b>don't know</b>.</p>
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<p>This is the risk in <a href="sampling.html">sampling</a>, we might have a "bad" sample.</p>
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</div>
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<h2>Example in Research</h2>
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<div class="example">
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<p>Here is Confidence Interval used in actual research on <b>extra exercise for older people</b>:</p>
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<p class="center"><img src="images/ci-extract2.gif" alt="confidence interval extract" style="border: 1px solid blue;" height="106" width="549"></p>
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<p>What is it saying? Looking at the "Male" line we see:</p>
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<ul>
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<li>1,226 Men (47.6% of all people)</li>
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<li>had a "HR" (see below) with a <b>mean of 0.92</b>,</li>
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<li>and a <b>95% Confidence Interval (95% CI) of 0.88 to 0.97</b> (which is also 0.92±0.05)</li>
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</ul>
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<p>"HR" is a measure of health benefit (lower is better), so it says that the <b>true benefit of exercise</b> for the wider population of men has a <b>95% chance</b> of being between 0.88 and 0.97</p>
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<p><i>* Note for the curious: "HR" is used a lot in health research and means "Hazard Ratio" where lower is better. So an HR of 0.92 means the subjects were better off, and a 1.03 means slightly worse off.</i></p>
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</div>
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<h2>Standard Normal Distribution</h2>
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<p>It is all based on the idea of the <a href="standard-normal-distribution-table.html">Standard Normal Distribution</a>, where the <span class="large">Z</span> value is the "Z-score"</p>
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<p>For example the <span class="large">Z</span> for 95% is 1.960, and here we see the range from -1.96 to +1.96 includes 95% of all values:</p>
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<p class="center"><img src="images/ci95.svg" alt="confidence interval 95%" height="142" width="295"><br>
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From -1.96 to +1.96 standard deviations is 95%</p>
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<p>Applying that to our sample looks like this:</p>
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<p class="center"><img src="images/ci4.svg" alt="confidence interval 86 plus minus 1.79 bell" height="212" width="256"><br>
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Also from -1.96 to +1.96 standard deviations, so includes 95%</p>
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<h2>Conclusion</h2>
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<p>The Confidence Interval is based on Mean and Standard Deviation. Its formula is:</p>
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<p class="center large"><span style="border-top: 1px solid black;">X</span> ± Z<span class="intbl"><em>s</em><strong>√n</strong></span></p>
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<p>Where:</p>
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<ul>
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<li><span style="border-top: 1px solid black;"><b>X</b></span> is the mean</li>
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<li><b>Z</b> is the Z-value from the table below</li>
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<li><b>s</b> is the standard deviation</li>
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<li><b>n</b> is the number of observations</li>
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</ul>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"><b>Confidence<br>
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Interval</b></td>
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<td style="text-align:center;"><b>Z</b></td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">80%</td>
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<td style="text-align:center; width:100px;">1.282</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">85%</td>
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<td style="text-align:center;">1.440</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">90%</td>
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<td style="text-align:center;">1.645</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">95%</td>
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<td style="text-align:center;">1.960</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">99%</td>
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<td style="text-align:center;">2.576</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">99.5%</td>
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<td style="text-align:center;">2.807</td>
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</tr>
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<tr>
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<td style="text-align:center; width:100px;">99.9%</td>
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<td style="text-align:center;">3.291</td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<div class="questions">11285, 11286, 11287, 11288, 11289, 11290, 11291, 11292</div>
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<div class="related">
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<a href="confidence-interval-calculator.html">Confidence Interval Calculator</a>
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<a href="standard-normal-distribution-table.html">Standard Normal Distribution</a>
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<a href="../mean.html">Mean</a>
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<a href="standard-deviation.html">Standard Deviation</a>
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