<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-standardization" target="_blank">Standardization Exercise</a></p><p><strong><span style="color:#8e44ad;">Standardization</span></strong> is a process that tells you "how many standard deviations" from the mean a number in a dataset is. For example, imagine we have a list of numbers in which the average is $30$ and the standard deviation is $5$. The number $25$ is exactly one standard deviation <u><strong>below</strong></u> the mean, so we would give it the "standard" value of $-1$. The number $35$ is exactly one standard deviation <u><strong>above</strong></u> the mean, so we would give it the standard value of $+1$, or simply $1$. What standardized value would we give to $20$? $-2$. How about $45$? $+3$.</p>
<p><strong><span style="color:#e74c3c;">Example</span></strong></p>
<p><em>Standardize the values below. The average is $3$ and the standard deviation, $\sigma$, is $\sqrt{2}$, or approximately $1.4$.</em></p>
<p>$$1, 2, 3, 4, 5$$</p>
<p style="margin-left: 40px;"><strong><u>Step 1</u></strong>: Subtract the mean from each number in the dataset:</p>
<p>$$-2, -1, 0, 1, 2$$</p>
<p style="margin-left: 40px;"><strong><u>Step 2</u></strong>: Divide each of the difference values from step 2 by the standard deviation, $1.4$.</p>
<p>$$\frac{-2}{1.4}, \frac{-1}{1.4}, \frac{0}{1.4}, \frac{1}{1.4}, \frac{2}{1.4}$$</p>
<p>$$-1.43, -0.714, 0, 0.714, 1.43$$ </p>
<p><strong>NOTE: </strong>The average of a standardized data set is always $0$ and the sum of the standardized values always sum to $0$ as well.</p>