<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&bull; provides a numerical measure of the overall amount of variation in a data set and</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&bull; can be used to determine whether a particular data value is close to or far from the mean.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>The standard deviation provides a measure of the overall variation in a data set.</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The standard deviation is always positive or zero. The standard deviation is small when all the data are concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Suppose that we are studying the amount of time customers wait in line at the checkout at Supermarket A and Supermarket&nbsp;B. The average wait time at both supermarkets is five minutes. At Supermarket A, the standard deviation for the wait time is two minutes; at Supermarket B, the standard deviation for the wait time is four minutes.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at Supermarket A are more concentrated near the average.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>The standard deviation can be used to determine whether a data value is close to or far from the mean.</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Suppose that both Roja and Binu shop at Supermarket A. Roja waits at the checkout counter for seven minutes, and Binu waits for one minute. At Supermarket A, the mean waiting time is five minutes, and the standard deviation is two minutes.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The standard deviation can be used to determine whether a data value is close to or far from the mean. A </span><em><span style="color:black">z</span></em><span style="color:black">-score is a standardized score that lets us compare data sets. It tells us how many standard deviations a data value is from the mean and is calculated as the ratio of the difference in a particular score and the population mean to the population standard deviation.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">We can use the given information to create the table below.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238383151623838315.png" style="height:152px; width:800px" /></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px">Since Roja and Binu only shop at Supermarket A, we can ignore the row for Supermarket B.</span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">We need the values from the first row to determine the number of standard deviations above or below the mean each individual wait time is; we can do so by calculating two different <em>z</em>-scores.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px">Rosa waited for seven minutes, so the <em>z</em>-score representing this deviation from the population mean may be calculated as&nbsp;<span class="math-tex">\(Z=\frac{x-\mu}{\sigma}=\frac{7-5}{2}=1\)</span>&nbsp;.</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px">The<em> z</em>-score of one tells us that Rosa&rsquo;s wait time is one standard deviation above the mean wait time of five minutes.</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px">Binh waited for one minute, so the <em>z</em>-score representing this deviation from the population mean may be calculated as&nbsp;<span class="math-tex">\(Z=\frac{x-\mu}{\sigma}=\frac{1-5}{2}=-2\)</span>&nbsp;.</span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The <em>z</em>-score of &minus;2 tells us that Binh&rsquo;s wait time is two standard deviations below the mean wait time of five minutes.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if they are more than two standard deviations away is more of an approximate <em>rule of thumb </em>than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is farther away than two standard deviations. You will learn more about this in later chapters.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is <strong>one </strong>standard deviation to the <strong>right </strong>of five because 5 + (1)(2) = 7.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px">If one were also part of the data set, then one is two standard deviations to the left of five because 5 + (&ndash;2)(2) = 1.</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238386711623838671.png" style="height:77px; width:800px" /></span></p>

<p>&nbsp;</p>

<h3>&nbsp;</h3>

<p><span style="font-size:14px"><strong><span style="color:#26397f">Figure 2.26</span></strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; In general, a </span><strong><span style="color:black">value = mean + (#ofSTDEV)(standard deviation)</span></strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; where #ofSTDEVs = the number of standard deviations</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; #ofSTDEV does not need to be an integer</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; One is </span><strong><span style="color:black">two standard deviations less than the mean </span></strong><span style="color:black">of five because 1 = 5 + (&ndash;2)(2).</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">The equation </span><strong><span style="color:black">value = mean + (#ofSTDEVs)(standard deviation) </span></strong><span style="color:black">can be expressed for a sample and for a population as&nbsp;follows:</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; </span><strong><span style="color:black">Sample</span></strong><em> </em><span class="math-tex">\(x =\bar{x}+\)</span>( # <em>o f STDEV</em>)(<em>s</em>)</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">The lowercase letter </span><em><span style="color:black">s </span></em><span style="color:black">represents the sample standard deviation and the Greek letter </span><em><span style="color:black">&sigma; </span></em><span style="color:black">(lower case) represents the population standard deviation. The symbol </span><span class="math-tex">\(\bar{x}\)</span>&nbsp;<span style="color:black">is the sample mean, and the Greek symbol </span><em><span style="color:black">&mu; </span></em><span style="color:black">is the population mean.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>Calculating the Standard Deviation</strong></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">If </span><em><span style="color:black">x </span></em><span style="color:black">is a number, then the difference </span><em><span style="color:black">x</span></em><em><span style="color:black">&nbsp;mean </span></em><span style="color:black">is called its </span><strong><span style="color:black">deviation</span></strong><span style="color:black">. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols, a deviation is </span><em><span style="color:black">x </span></em><span style="color:black">&ndash; </span><em><span style="color:black">&mu;</span></em><span style="color:black">. For sample data, in symbols, a deviation is </span><em><span style="color:black">x </span></em><span style="color:black">&ndash;&nbsp;</span><span class="math-tex">\(\bar{x}\)</span>.</span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar but not identical. Therefore, the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lowercase letter </span><em><span style="color:black">s </span></em><span style="color:black">represents the sample standard deviation and the Greek letter </span><em><span style="color:black">&sigma; </span></em><span style="color:black">(lowercase sigma) represents the population standard deviation. If the sample has the same characteristics as the population, then </span><em><span style="color:black">s </span></em><span style="color:black">should be a good estimate of </span><em><span style="color:black">&sigma;</span></em><span style="color:black">.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">To calculate the standard deviation, we need to calculate the variance first. The </span><strong><span style="color:black">variance </span></strong><span style="color:black">is the </span><strong><span style="color:black">average of the squares of the deviations </span></strong><span style="color:black">(the </span><em><span style="color:black">x&nbsp;</span></em><span class="math-tex">\(-\bar{x}\)</span>&nbsp;</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">values for a sample or the </span><em><span style="color:black">x </span></em><span style="color:black">&ndash; </span><em><span style="color:black">&mu; </span></em><span style="color:black">values for a population). The symbol </span><em><span style="color:black">&sigma;</span></em><sup><span style="color:black">2</span></sup><span style="color:black"> represents the population variance; the population standard deviation </span><em><span style="color:black">&sigma; </span></em><span style="color:black">is the square root of the population variance. The symbol </span><em><span style="color:black">s</span></em><span style="color:black">2 represents the&nbsp;sample variance; the sample standard deviation </span><em><span style="color:black">s </span></em><span style="color:black">is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">If the numbers come from a census of the entire </span><strong><span style="color:black">population </span></strong><span style="color:black">and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by </span><em><span style="color:black">N</span></em><span style="color:black">, the number of items in the population. If the data are from a </span><strong><span style="color:black">sample </span></strong><span style="color:black">rather than a population, when we calculate the average of the squared deviations, we divide by <strong><em>n </em></strong></span><strong><span style="color:black">&ndash; 1</span></strong><span style="color:black">, one less than the number of&nbsp; items in the sample.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000">Formulas for the Sample Standard Deviation</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#26397f"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238390581623839058.png" style="height:81px; width:400px" /></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px">&bull; For the sample standard deviation, the denominator is <em>n</em>&minus;; that is, the sample size minus 1.</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000">Formulas for the Population Standard Deviation</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#26397f"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238390941623839094.png" style="height:54px; width:400px" /></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; For the population standard deviation, the denominator is </span><em><span style="color:black">N</span></em><span style="color:black">, the number of items in the population.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">In these formulas, </span><em><span style="color:black">f </span></em><span style="color:black">represents the frequency with which a value appears. For example, if a value appears once, </span><em><span style="color:black">f </span></em><span style="color:black">is one. If a value appears three times in the data set or population, </span><em><span style="color:black">f </span></em><span style="color:black">is three.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>Types of Variability in Samples</strong></span></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. </span><em><span style="color:black">Variability </span></em><span style="color:black">is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following:</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">&bull; Observational or measurement variability</span></span></p>

<p><span style="font-size:14px"><span style="color:black">&bull; Natural variability</span></span></p>

<p><span style="font-size:14px"><span style="color:black">&bull; Induced variability</span></span></p>

<p><span style="font-size:14px"><span style="color:black">&bull; Sample variability</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">Here are some examples to describe each type of variability:</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:black">Example 1: Measurement variability</span></strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers. For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:black">Example 2: Natural variability</span></strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">Natural variability arises from the differences that naturally occur because members of a population differ from each other.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:black">Example 3: Induced variability</span></strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Induced variability is the counterpart to natural variability. This occurs because we have &nbsp;artificially induced an element of variation that, by definition, was not present naturally. For example, we assign people to two different groups to study memory, and we induce a variable in one group by limiting the amount of sleep they get. The difference in outcomes may&nbsp;be affected by induced variability.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:black">Example 4: Sample variability</span></strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Sample variability occurs when multiple random samples are taken from the same population. For example, if I conduct four surveys of 50 people randomly selected from a given population, the differences in outcomes may be affected by sample variability.</span></span></p>