<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-population-vs-sample-sd" target="_blank">Population vs Sample SD Exercise</a></p><p>From the previous mountain entry, we learned the steps to calculate standard deviation in more complex cases. </p>
<p>The difference between <strong><span style="color:#8e44ad;">Population vs Sample Standard Deviation</span></strong> hinges on one of these steps.</p>
<p><strong><span style="color:#2980b9;">Steps For Calculating Standard Deviation (Population vs Sample)</span></strong></p>
<ul>
<li><strong><u>Step 1</u></strong>: Find the mean of the data set.
<ul>
<li><strong><span style="color:#27ae60;">SAME</span></strong>.</li>
</ul>
</li>
<li><strong><u>Step 2</u></strong>: Subtract the mean from each number in the dataset to find the difference values.
<ul>
<li><strong><span style="color:#27ae60;">SAME</span></strong>.</li>
</ul>
</li>
<li><strong><u>Step 3</u></strong>: Add the difference values together.
<ul>
<li><strong><span style="color:#27ae60;">SAME</span></strong>.</li>
</ul>
</li>
<li><strong><u>Step 4</u></strong>: Divide the sum of the difference values by the number of items in the dataset.
<ul>
<li><strong><span style="color:#e74c3c;">DIFFERENT</span></strong>.
<ul>
<li>For $n$ items in a dataset...
<ul>
<li><span style="color:#8e44ad;">Population SD</span>: Divide by $n$</li>
<li><span style="color:#8e44ad;">Sample SD</span>: Divide by $(n-1)$</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li><strong><u>Step 5</u></strong>: Take the square root.
<ul>
<li><strong><span style="color:#27ae60;">SAME</span></strong>.</li>
</ul>
</li>
</ul>
<p><strong><span style="color:#8e44ad;">The Formulas</span></strong></p>
<p style="text-align: center;">$$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}$$</p>
<p style="text-align: center;"><span style="color:#2980b9;"><em>Population Standard Deviation</em></span></p>
<p style="text-align: center;"><em>$$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$</em></p>
<p style="text-align: center;"><span style="color:#2980b9;"><em>Sample Standard Deviation</em></span></p>
<p><span style="color:#e74c3c;"><strong><u>NOTE</u></strong></span>: While the sample standard deviation is generally larger than the population standard deviation, this is not always the case. Also, in very large datasets with many hundreds, thousands, or millions of entries, the standard and population deviation are essentially equal to each other.</p>