Normal Distribution Calculations
<p><span style="font-size:22px"><a href="https://www.prepswift.com/quizzes/quiz/prepswift-normal-distribution-calculation" target="_blank">Normal Distribution Calculations Exercise</a></span></p><p>If you have an intuitive understanding of how the normal distribution works, how the middle is equal to the mean and the median, how it sits atop the number line, how if you subtract one standard deviation from the mean, your values are in an interval that comprises $34\%$ of the data, how the interval from the mean to two standard deviations above consists of roughly $48\%$ (34 + 14) of the data, etc., then you're pretty good. </p>
<p>With all this crap in mind, we can do <strong><span style="color:#8e44ad;">Normal Distribution Calculations</span></strong>.</p>
<p><strong><span style="color:#27ae60;">Calculation Type 1</span></strong></p>
<p style="margin-left: 40px;"><em>A certain dataset is normally distributed. The value one standard deviation below the mean is $38$ and the value two standard deviations above the mean is $50$. What is the mean of the data set?</em></p>
<p style="margin-left: 40px;"><strong><u>Step 1</u></strong>: Set up algebraic equations, where $x$ is the mean and $y$ is the standard deviation.</p>
<p>$$x - y = 38$$</p>
<p>$$x+2y = 50$$</p>
<p style="margin-left: 40px;"><strong><u>Step 2</u></strong>: Solve with systems of equation rules.</p>
<p><strong><span style="color:#27ae60;">Calculation Type 2</span></strong></p>
<p style="margin-left: 40px;"><i>If a normally distributed dataset consists of $95$,$400$ values and has a mean of $62$ and a standard deviation of $4$, approximately how many of the values are less than $54$?</i></p>
<p style="margin-left: 40px;"><strong><u>Step 1</u></strong>: Note that $54$ is two standard deviations below the mean $62$. If we draw our chart, we can see that values less than $54$ comprise $2\%$ of the data.</p>
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<p style="margin-left: 40px;"><strong><u>Step 2</u></strong>: Calculate $2\%$ of $95$,$400$</p>
<p><strong><span style="color:#27ae60;">Calculation Type 3</span></strong></p>
<p style="margin-left: 40px;"><em>A certain dataset is normally distributed. Its mean is $50$ and its standard deviation is $20$. Consider two subgroups, $A$ and $B$. Subgroup $A$ consists of all values from $50$ to $60$ and Subgroup $B$ consists of all values from $60$ to $70$. None of the values in the dataset equal $50$, $60$ or $70$. </em></p>
<p style="margin-left: 40px;"><em>So here's the question. Are there more values in $A$ or $B$? Or are they equal? Or is it sometimes $A$, sometimes $B$? What do you think?</em></p>
<p style="text-align: center;"><span style="color:#8e44ad;">Perhaps a diagram can help.</span></p>
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<p><span style="color:#27ae60;">The correct answer is actually Subgroup $A$</span>. Recall that if you have a segment of the bell-curve that is closer to the mean, where most of the values are "bunched up," there's going to be more values there. Another way to think about is with area. Which region has the larger area? Still $A$. Notice that the bases of regions $A$ and $B$ are equal at $10$. But the HEIGHT of region $A$ is larger, resulting in a larger area.</p>
<p>If you wanted to divide regions $A$ and $B$ into equal halves, you have to move the dividing line to the left a little bit. Yeah, region $B$ would now have a longer base, but region $A$ is taller, so it equals out.</p>
<p>That means that the true "dividing line" into equal halves is less than $60$ in this case.</p>