{"id":299,"date":"2017-08-24T07:08:50","date_gmt":"2017-08-24T07:08:50","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=299"},"modified":"2017-09-22T18:00:07","modified_gmt":"2017-09-22T18:00:07","slug":"data-displays-normal-distributions-and-lines-of-best-fit","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/data-displays-normal-distributions-and-lines-of-best-fit\/","title":{"rendered":"Data Displays, Normal Distributions and Lines of Best Fit"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/beginning-statistics\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/calculus\">Next\u00a0Workshop \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Data Displays, Normal Distributions and Lines of Best Fit<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study how to organize data sets using methods such as frequency tables, histograms, standard\u00a0line graphs, bar graphs, stem-and-leaf displays, and scatter plots. In addition, you will discuss normal\u00a0distributions, as well as how to find a line of best fit using least squares regression.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A <strong><em>data set<\/em><\/strong>,\u00a0or <em><strong>data distribution<\/strong><\/em>, is a collection\u00a0of values representing a population. It is usually represented as\u00a0a range of figures or terms enclosed in braces, e.g.,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p1_clip_image003.gif\" width=\"96\" height=\"16\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>mean <\/em><\/strong>of\u00a0a data set is the average value. It is defined by the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p1_clip_image006.gif\" width=\"65\" height=\"38\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <em><strong>mode <\/strong><\/em>of\u00a0a data set is the value that occurs most often. Remember that the\u00a0mode can be represented by more than one number, because more\u00a0than one element in a set might recur an equal number of times.<\/li>\n<li>The <strong><em>range <\/em><\/strong>is\u00a0the difference between the smallest number in a data set (the\u00a0minimum) and the largest number in that data set (the maximum).<\/li>\n<li>The <strong><em>median\u00a0<\/em><\/strong>of a data set is the number that falls in the middle of the data\u00a0set.<\/li>\n<li>The <strong><em>variance\u00a0<\/em><\/strong>measures the way that a data set&#8217;s elements are dispersed. It is\u00a0defined by the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p1_clip_image009.gif\" width=\"112\" height=\"38\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li>The <strong><em>standard\u00a0deviation <\/em><\/strong>is the square root of the variance. It is\u00a0defined by the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p1_clip_image012.gif\" width=\"119\" height=\"46\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>maximum\u00a0<\/em><\/strong>of a data set is the largest element in that data set.<\/li>\n<li>The <strong><em>minimum\u00a0<\/em><\/strong>of a data set is the smallest element in that data set.<\/li>\n<li>The <strong><em>lower quartile\u00a0<\/em><\/strong>of a data set is the median of the subset of elements between the\u00a0data set&#8217;s median and its minimum. These elements are greater\u00a0than the minimum and less than the median.<\/li>\n<li>The <em><strong>upper quartile\u00a0<\/strong><\/em>of a data set is the median of the subset of elements between the\u00a0data set&#8217;s median and its maximum. These elements are greater\u00a0than the median and less than the maximum.<\/li>\n<li>A <strong><em>box-and-whisker plot<\/em><\/strong> organizes\u00a0the information in a data set graphically.<\/li>\n<\/ul>\n<section>\n<h3>Data Displays, Normal Distributions and Lines of Best Fit<\/h3>\n<p>The box-and-whisker plot is a useful way to display information\u00a0relating to the distribution of elements in a data set. There are\u00a0many other ways to graphically display the information in a data\u00a0set. Different display methods can highlight different important\u00a0properties of the data distribution.<\/p>\n<p>What are histograms and frequency tables, and how are they related?<\/p>\n<p><span style=\"text-decoration: none;\">Just as a box-and-whisker\u00a0plot arranges information in relation to the <\/span><em><span style=\"text-decoration: none;\">median\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of a data set, a\u00a0<\/span><abbr title=\" provides information relating to the mode of a data set\"><span style=\"text-decoration: none;\">frequency\u00a0table<\/span><\/abbr><span style=\"text-decoration: none;\"> provides\u00a0information relating to the <\/span><em><span style=\"text-decoration: none;\">mode\u00a0<\/span><\/em><span style=\"text-decoration: none;\">of a data set. A frequency table is a data display that lists the\u00a0times that each element in a data set occurs. Often, the <\/span><em><span style=\"text-decoration: none;\">relative\u00a0frequency<\/span><\/em><span style=\"text-decoration: none;\"> is also\u00a0displayed in a frequency table. The relative frequency is a value,\u00a0given as a percent, that represents the number of times an element\u00a0occurs in a data set. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">For example,<\/span> the\u00a0number of points a local soccer team scored in their last 35 games\u00a0is listed below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p2_clip_image003.gif\" width=\"277\" height=\"35\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Arrange this information into a frequency table.<\/p>\n<p>The elements of this data set range from 0 to 5. The score 0\u00a0occurs 5 times. The score 1 occurs 6 times, 2 occurs 8 times, and\u00a0so on.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"3\">Frequency Table<\/th>\n<\/tr>\n<tr>\n<th>Score<\/th>\n<th>Frequency<\/th>\n<th>Relative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>5<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>6<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>8<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>5<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>4<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To compute the relative frequency of the score 0, we must\u00a0divide its frequency (5) by the total number of elements in the\u00a0data set (35). Therefore, we have\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p2_clip_image006.gif\" width=\"103\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So, 0 occurs approximately 14.3% of the time. To compute the\u00a0relative frequency of the score 1, we must divide its frequency\u00a0(6) by the total elements,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p2_clip_image009.gif\" width=\"76\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So, 1 occurs approximately 17.1% of the time. Following the same\u00a0method, we find that 2 occurs approximately 22.9% of the time, and\u00a03 occurs approximately 20% of the time.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"3\">Frequency Table<\/th>\n<\/tr>\n<tr>\n<th>Score<\/th>\n<th>Frequency<\/th>\n<th>Relative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>5<\/td>\n<td>14.3%<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>6<\/td>\n<td>17.1%<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>8<\/td>\n<td>22.9%<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7<\/td>\n<td>20%<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>5<\/td>\n<td>14.3%<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>4<\/td>\n<td>11.4%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Data Displays, Normal Distributions and Lines of Best Fit<\/h3>\n<p>A <abbr title=\" a type of data display that can be used to graph information relating to frequency. In a histogram, bars are used to represent the number of times an element of the data set occurs. \">histogram<\/abbr> is another type of display that can be used to graph information\u00a0relating to frequency. In a histogram, bars are used to represent\u00a0the number of times an element of a data set occurs.<\/p>\n<p><span style=\"text-decoration: none;\">Histograms sort elements.\u00a0The <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0denotes the categories, or classes, the elements will be sorted\u00a0into, and the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0tells how many elements fall into each specific class. The <\/span><abbr title=\" term for the bars of a histogram that denote the way we define each of the bins, or classes, that the elements are sorted into \"><span style=\"text-decoration: none;\">class\u00a0interval<\/span><\/abbr><span style=\"text-decoration: none;\"> is a rule by which we define each of the classes that the elements\u00a0of a data set will be sorted into when we organize a histogram.\u00a0Many times the class interval of a histogram is a range of values,\u00a0though this is not always the case. <\/span><\/p>\n<p>The histogram below displays the information from the\u00a0frequency table above. In this histogram, the class interval is\u00a0defined as one goal. We could display the same information in a\u00a0histogram with a different class interval.<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/histogram1v1_110305.jpg\" width=\"325\" height=\"325\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>Histograms are often displayed with a class interval larger\u00a0than one unit. Thus, each class denotes a range of values rather\u00a0than just one value. The histogram below displays the same\u00a0information above with a class interval of 2 goals.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/histogram2v1_110305.jpg\" width=\"200\" height=\"325\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>As a rule, histograms are drawn with no gap between the bars.. So, be\u00a0sure to place the bars of a histogram right next to each other.<\/p>\n<p class=\"notebox_text\" align=\"center\">Occasionally, histograms are drawn with extra space between bars,<br \/>\nmaking them look more like bar graphs. We discuss the difference between histograms and bar graphs\u00a0below.<\/p>\n<\/div>\n<h4>How are bar graphs and histograms related?<\/h4>\n<p>An important thing to remember is that a histogram is a certain\u00a0type of <abbr title=\"a data display that uses bars to relate different measurements for many different items. \">bar graph<\/abbr>. In a histogram, we are using bars to display specific information\u00a0about a data set. A histogram is like a frequency table that uses\u00a0bars to represent the frequency. In histograms, there is only one\u00a0variable to consider, and we categorize the elements of the data\u00a0set using this one variable.<\/p>\n<p>A bar graph, on the other hand, is more general. A bar graph\u00a0uses bars to display information relating many measurements to\u00a0many different items. For example, we would use a bar graph like\u00a0the one below, not a histogram, to display how many points each\u00a0soccer team scored at the last tournament.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20003.JPG\" alt=\"Bar Graph\" width=\"400\" height=\"289\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The big difference between histograms and bar graphs is that a\u00a0histogram uses bars to display the frequency of just one variable,\u00a0while a bar graph uses bars to relate many different measurements\u00a0to many different items.<\/p>\n<h3>What are line graphs?<\/h3>\n<p><span style=\"text-decoration: none;\">A <\/span><abbr title=\"a type of data display that relates two variables, an independent variable and a dependent variable, within a data distribution\"><span style=\"text-decoration: none;\">line\u00a0graph<\/span><\/abbr><span style=\"text-decoration: none;\"> provides another method to display the information contained in a\u00a0data set. More specifically, a line graph relates two variables,\u00a0an independent variable and a dependent variable, within a data\u00a0distribution. The <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0denotes the independent variable, and the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0denotes the dependent variable. <\/span><\/p>\n<p style=\"text-decoration: none;\">Line graphs are very useful\u00a0because not only do they provide a clear and concise way to\u00a0represent data, they are also very useful in extrapolating and\u00a0interpolating more refined information from a given data set. In\u00a0addition, line graphs are often used to recognize a relationship\u00a0between two variables and to make informed predictions for the\u00a0future based upon the relationship displayed in the graph.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Dr. Connolly teaches Honors Calculus at a university. Over\u00a0several years, Dr. Connolly has monitored the number of women that\u00a0enroll in his class. He has compiled this information into the\u00a0line graph below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20004.JPG\" alt=\" Class enrollment line graph\" width=\"400\" height=\"294\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>According to the line graph Dr. Connolly\u00a0created, which statement below is NOT correct?<\/p>\n<ol>\n<li>The number of women enrolled in Dr. Connolly\u2019s Honors\u00a0Calculus class stayed the same from 1999 to 2000.<\/li>\n<li>One year, there were no women enrolled in Dr. Connolly\u2019s\u00a0Honor Calculus class.<\/li>\n<li>The number of women in Dr. Connolly&#8217;s Honors Calculus class has\u00a0continuously increased since 1997.<\/li>\n<li>7 women were enrolled in Dr. Connolly\u2019s Honors Calculus\u00a0class in 2003.<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is C. The number of women in the Honors\u00a0Caclulus class has not be<span style=\"text-decoration: none;\">en\u00a0<\/span><em><span style=\"text-decoration: none;\">continuously\u00a0<\/span><\/em>increasing. In fact, there was a decline from 2003 to 2004.<\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>We can extrapolate information that is not explicitly contained in a\u00a0line graph\u00a0by recognizing patterns in the data relationships. Though the number of enrolled women is not strictly\u00a0increasing,\u00a0there is an obvious trend toward more women in the Honors Calculus class. Based on this line graph, we\u00a0can\u00a0conjecture that in the coming years, there will be an even greater number of women enrolled in Dr. Connolly\u2019s Honors\u00a0Calculus class.<\/p>\n<\/div>\n<h3>What are some other methods used to display data?<\/h3>\n<p>As previously stated, line graphs are very useful in noticing\u00a0trends in a data set and making predictions for future\u00a0developments based upon those trends. Another data display that is\u00a0useful when determining the existence of a relationship between\u00a0two variables is a scatter diagram, also called a <abbr title=\"a coordinate plane with points plotting one set of data values against another \">scatter\u00a0plot<\/abbr>.<\/p>\n<p><span style=\"text-decoration: none;\">A scatter plot is a\u00a0coordinate plane with points plotting one set of data values\u00a0against another. Whereas the other methods for displaying data\u00a0sets we have examined are based upon displaying a single variable\u00a0in a data set., a scatter plot is used to compare two data sets,\u00a0with one set being represented on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0and the other on the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis.<br \/>\n<\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">Suppose we measured the\u00a0height and shoe size of a large sample group of people. By\u00a0plotting the height on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0and the shoe size on the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0we create the scatter plot below. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20005.JPG\" alt=\"Scatter plot of shoe size vs. height\" width=\"400\" height=\"344\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><strong><br \/>\n<\/strong><\/p>\n<p>Which statement below is supported by the\u00a0information in the scatter plot?<\/p>\n<ol>\n<li>Anyone with a shoe size over 12 is taller than 5\u201910\u2019\u2019.<\/li>\n<li>As a person&#8217;s height increases, their shoe size decreases.<\/li>\n<li>There is no clear relationship between a person\u2019s height\u00a0and their shoe size.<\/li>\n<li>In general, the taller a person is, the larger their shoe size\u00a0is.<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. The scatter plot indicates that\u00a0height and shoe size are directly proportional.<\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Be Aware!<\/h4>\n<p>When interpreting a scatter plot, be wary of absolute statements, such as the statement made in choice<br \/>\nA. The lone\u00a0point in the upper left corner shows that there was one person measured who is about 5 ft 2 in tall,\u00a0with a shoe\u00a0size of about <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p5_clip_image003.gif\" width=\"26\" height=\"33\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p class=\"notebox_text\" align=\"center\">Always be in the lookout for aberrational elements on a scatter plot. They can\u00a0provide counterexamples to absolute statements. When making general statements, it is alright to ignore\u00a0these\u00a0aberrational elements, and describe the general trend of the data.<\/p>\n<\/div>\n<h3>Why are there so many ways to display a data set?<\/h3>\n<p>A data set can be very large and, by simply listing the\u00a0elements in a data set, it can be hard to deduce any useful\u00a0information. All the methods we have discussed provide ways to\u00a0simplify the information in a data set. However, as we have seen,\u00a0different display methods better at highlighting different\u00a0properties of a data set. It is important to be familiar with each\u00a0of the display methods so that we know how to best highlight a\u00a0certain aspect of a data set.<\/p>\n<p>Another method that is often used to arrange data into a more\u00a0accessible format is the <abbr title=\" a technique used to organize data that provides a quick reference for comparison of the elements in larger categories \">stem-and-leaf\u00a0display<\/abbr>. A stem-and-leaf display is similar to a histogram\u00a0because it allows us to quickly count the number of elements in a\u00a0data set that fall within a specific range.<\/p>\n<p>Consider the data distribution {5, 16, 18, 4, 23, 25, 29, 31,\u00a024, 35, 44, 42, 39, 51, 40, 50, 39, 22, 48, 57, 12, 65, 44, 33,\u00a028, 29, 10, 9, 27, 8}.<\/p>\n<p>To organize this data as a stem-and-leaf display, we let the\u00a0stem be the ten\u2019s place, and the leaf be the one\u2019s\u00a0place. Then we create the following table.<\/p>\n<table width=\"50%\">\n<thead>\n<tr>\n<th colspan=\"3\">Stem and Leaf Display<\/th>\n<\/tr>\n<tr>\n<th>Stem<\/th>\n<th>Leaf<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>4, 5, 8, 9<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>0, 2, 6, 8,<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>2, 3, 4, 5, 7, 8, 9, 9<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>1, 3, 5, 9, 9<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>0, 2, 4, 4, 8<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>0, 1, 7<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"text-decoration: none;\">Each element of the data\u00a0set is represented, and we can quickly see that the greatest\u00a0number of elements is between 20 and 30, because the <\/span><em><span style=\"text-decoration: none;\">leaf\u00a0<\/span><\/em><span style=\"text-decoration: none;\">adjacent to the \u201c2\u201d <\/span><em><span style=\"text-decoration: none;\">stem\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the longest. <\/span><\/p>\n<h3>What are normal distributions?<\/h3>\n<p>If we define the <abbr title=\" term for the bars of a histogram that denote the way we define each of the bins, or classes, that the elements are sorted into \">class\u00a0interval<\/abbr> of a histogram as a very small interval, and create\u00a0correspondingly small bars, we could easily imagine a curve\u00a0created by the bars of the histogram. In fact, the smaller we make\u00a0our class interval, the more accurate our curve will be.<\/p>\n<p>Suppose we want to make a histogram that displays the heights\u00a0of all the students in a particular high school. If we define the\u00a0class interval as 6 inches, we get the following histogram.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20006.JPG\" alt=\"Histogram with class interval 6 inches\" width=\"400\" height=\"217\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>If we define the class interval as 0.1 inches, however, we get\u00a0the histogram below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20007.jpg\" alt=\" Histogram with class interval 0.1\" width=\"400\" height=\"247\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The resultant curve is called the normal curve, or the bell\u00a0curve. The histogram tells us that the majority of the students in\u00a0the high school are of average height, yet there are a couple of\u00a0very tall and very short students. Data sets which follow this\u00a0pattern are called <abbr title=\"a theoretical frequency distribution for a set of variable data, usually represented by a bell-shaped curve that is symmetrical about the mean \">normal\u00a0distributions<\/abbr>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Many properties in nature and in behavioral and\u00a0social sciences follow a normal distribution and produce a normal\u00a0curve when mapped in this way.<\/p>\n<ul>\n<li>Height<\/li>\n<li>IQ<\/li>\n<li>SAT score<\/li>\n<\/ul>\n<\/div>\n<h3>How does all this relate to probability?<\/h3>\n<p>The normal curve makes very natural statements about\u00a0probability. Consider, for example, the histogram relating the IQ\u00a0of a population, in which the class interval is one point. As with\u00a0many phenomena in nature, a bell curve results.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20008.JPG\" alt=\" Bell curve\" width=\"400\" height=\"265\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Probabilistically speaking, this curve states that the citizens\u00a0in this population have an average IQ of about 100. As the IQ\u00a0increases, the number of citizens with this IQ decreases. The bell\u00a0curve tells us that there are very few citizens who qualify as\u00a0genius(have an IQ above 140). However, there are equally few\u00a0citizens with an IQ below 55. Thus, a citizen chosen at random\u00a0will most likely have an average IQ, and it would be extremely\u00a0unlikely to randomly pick a genius out of the population.<\/p>\n<p>Using the language of standard deviations, we can make these\u00a0statements much more precise. Examine the graph below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20009.JPG\" alt=\"Bell curve with probabilities\" width=\"400\" height=\"236\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Each interval of standard deviation displays the probability\u00a0that one of its elements will be randomly chosen. Note that the\u00a0probabilities add up to 100% with 50% on either side of the mean\u00a0value.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Recall: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p8_clip_image003.gif\" width=\"7\" height=\"17\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/> is the mean\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p8_clip_image006.gif\" width=\"11\" height=\"8\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/> is the standard deviation<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>The lengths of time for telephone calls in the Jones household\u00a0approximate a normal distribution. If the mean length is 4.5\u00a0minutes, with a standard deviation of 1.5 minutes, about 84% of\u00a0the calls are\u2026<\/p>\n<ol>\n<li>more than 9 minutes long<\/li>\n<li>between 4.5 and 9 minutes long<\/li>\n<li>less than 4.5 minutes long<\/li>\n<li>between 3 and 9 minutes long<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. Use the normal curve, and fill in the\u00a0appropriate values for the mean and standard deviations.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20010.JPG\" alt=\" Normal Curve with telephone calls\" width=\"400\" height=\"236\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><strong><br \/>\n<\/strong><\/p>\n<p>According to the chart, only 0.15% of the calls made are more\u00a0than 9 minutes long, eliminating choice A. The percentage of\u00a0calls between 4.5 and 9 minutes is 34 + 13.5 + 2.35 = 49.85%,\u00a0which eliminates choice B. For choice C, according to the chart,\u00a0the percentage of calls less than 4.5 minutes is 34 + 13.5 + 2.35\u00a0+ 0.15 = 50%. Finally, according to the chart, the percentage of\u00a0calls between 3 and 9 minutes is 34 + 34 + 13.5 + 2.35 = 83.85%,\u00a0which is about 84%, confirming D as the correct choice.<\/p>\n<\/div>\n<\/section>\n<p>The normal distributions and the normal curve it generates are\u00a0very useful patterns of data distribution. These patterns of\u00a0distribution can be found in naturally occurring phenomena, such\u00a0as height, and in data sets used in the behavioral and social\u00a0sciences, such as sets of IQ scores.<\/p>\n<h3>What if the data set does not have a normal distribution? Can\u00a0we find another curve that approximates the data set?<\/h3>\n<p>Much research in mathematics is devoted to finding a curve that\u00a0accurately describes discrete data sets. A <abbr title=\" for a collection of data points is a linear equation that closely approximates the behavior of that collection of data points. \">linear\u00a0regression<\/abbr> for a collection of data points is a <abbr title=\"an algebraic expression of the form y = ax + b, where a and b are constants. When graphed on the x- and y- axes, this function produces a straight line with slope a and y-intercept b. \">linear\u00a0equation <\/abbr>that closely approximates the behavior of a\u00a0collection of data points. The most popular method of finding a\u00a0linear regression is the least squares method.<\/p>\n<p><span style=\"text-decoration: none;\">The term <\/span><em><span style=\"text-decoration: none;\">regression\u00a0<\/span><\/em><span style=\"text-decoration: none;\">indicates that a linear equation is a less-than-perfect\u00a0approximation of a data set. Indeed, it is very rare that a data\u00a0distribution precisely mimics linear behavior. We can only create\u00a0a good guess. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Recall that a linear\u00a0equation is defined by the formula <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= <\/span><em><span style=\"text-decoration: none;\">ax <\/span><\/em><span style=\"text-decoration: none;\">+\u00a0<\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0where <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">are constants.<\/span><\/p>\n<h3>How exactly does a least squares regression work?<\/h3>\n<p>The method of <abbr title=\"creates a linear equation that minimizes the square of the vertical distance between points in the data set and the corresponding values on the line \">least\u00a0squares regression<\/abbr> creates a linear equation that\u00a0minimizes the square of the vertical distance between points in\u00a0the data set and the corresponding values on the line.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20011.JPG\" alt=\"Least squares regression\" width=\"251\" height=\"156\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>In the figure above, the points in the data set are black and\u00a0the corresponding values on the line are blue. The resulting least\u00a0squares regression line minimizes the value of the square of the\u00a0distance, (denoted in red) between these two points.<\/p>\n<p><span style=\"text-decoration: none;\">Since a linear equation has\u00a0the general form <\/span><em><span style=\"text-decoration: none;\">y\u00a0= ax + b<\/span><\/em><span style=\"text-decoration: none;\">, we need\u00a0to determine how to calculate the values of <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b <\/span><\/em><span style=\"text-decoration: none;\">in\u00a0order to find a linear regression. Luckily, there are formulas we\u00a0can use to calculate the values of <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">which will result in the least squares regression. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Sup<\/span>pose we are given\u00a0the data set\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image003.gif\" width=\"100\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and corresponding to each element of the data set, we have the\u00a0ordered pairs\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image006.gif\" width=\"228\" height=\"26\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The following equations are used to compute the v<span style=\"text-decoration: none;\">alues\u00a0of <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p><span style=\"text-decoration: none;\">Use the equations below to solve for <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\"> and <\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\"> in the linear regression. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image009.gif\" width=\"160\" height=\"89\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image012.gif\" width=\"136\" height=\"38\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the least squares regression line for the\u00a0data set below?<\/p>\n<p align=\"CENTER\">{(1, 1), (2, 4), (3, 2), (4, 4), (5, 3), (7, 9),<br \/>\n(8, 5), (9, 10)}<\/p>\n<ol>\n<li><em><span style=\"text-decoration: none;\">y = x<\/span><\/em><\/li>\n<li><em><span style=\"text-decoration: none;\">y = x <\/span><\/em>+ 0.299<\/li>\n<li><em><span style=\"text-decoration: none;\">y = <\/span><\/em><span style=\"text-decoration: none;\">2<\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ 1<\/span><\/li>\n<li><em><span style=\"text-decoration: none;\">y =\u00a0x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">\u2013 0.225 <\/span><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p align=\"LEFT\">The correct choice is B. Before we begin plugging\u00a0values into the equations above, let\u2019s organize what we\u00a0need to know by creating a table.<\/p>\n<table border=\"1\" width=\"75%\" cellspacing=\"0\" cellpadding=\"5\" align=\"center\">\n<tbody>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\"><em><span style=\"text-decoration: none;\">i <\/span><\/em><\/div>\n<\/td>\n<td>\n<div align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image018.gif\" width=\"9\" height=\"12\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/div>\n<\/td>\n<td>\n<div align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image021.gif\" width=\"11\" height=\"12\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/div>\n<\/td>\n<td>\n<div align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image024.gif\" width=\"14\" height=\"19\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/div>\n<\/td>\n<td>\n<div align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image027.gif\" width=\"21\" height=\"12\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<td>\n<div align=\"center\">1<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">8<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">9<\/div>\n<\/td>\n<td>\n<div align=\"center\">6<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">4<\/div>\n<\/td>\n<td>\n<div align=\"center\">16<\/div>\n<\/td>\n<td>\n<div align=\"center\">16<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">5<\/div>\n<\/td>\n<td>\n<div align=\"center\">5<\/div>\n<\/td>\n<td>\n<div align=\"center\">3<\/div>\n<\/td>\n<td>\n<div align=\"center\">25<\/div>\n<\/td>\n<td>\n<div align=\"center\">15<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">6<\/div>\n<\/td>\n<td>\n<div align=\"center\">7<\/div>\n<\/td>\n<td>\n<div align=\"center\">9<\/div>\n<\/td>\n<td>\n<div align=\"center\">49<\/div>\n<\/td>\n<td>\n<div align=\"center\">63<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">7<\/div>\n<\/td>\n<td>\n<div align=\"center\">8<\/div>\n<\/td>\n<td>\n<div align=\"center\">5<\/div>\n<\/td>\n<td>\n<div align=\"center\">64<\/div>\n<\/td>\n<td>\n<div align=\"center\">40<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\">8<\/div>\n<\/td>\n<td>\n<div align=\"center\">9<\/div>\n<\/td>\n<td>\n<div align=\"center\">10<\/div>\n<\/td>\n<td>\n<div align=\"center\">81<\/div>\n<\/td>\n<td>\n<div align=\"center\">90<\/div>\n<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>\n<div align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image030.gif\" width=\"16\" height=\"18\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/div>\n<\/td>\n<td>\n<div align=\"center\">39<\/div>\n<\/td>\n<td>\n<div align=\"center\">38<\/div>\n<\/td>\n<td>\n<div align=\"center\">249<\/div>\n<\/td>\n<td>\n<div align=\"center\">239<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can more easily compute the\u00a0values for <em>a\u00a0<\/em>and <em>b<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/s6_p10_html_m37012829.gif\" width=\"352\" height=\"95\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s6_p10_clip_image037.gif\" width=\"325\" height=\"45\" \/><\/p>\n<p>Thus the equation of the least squares regression line is y = x + 0.299.<\/p>\n<\/div>\n<\/section>\n<p>In the figure below, the points of the data set are mapped with\u00a0the corresponding least square regression line in red.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/Math%20Mod%207.4%20Art%20012.JPG\" alt=\" Least squares example\" width=\"300\" height=\"284\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We now have a linear equation that closely resembles the data\u00a0we have collected. Using this data, we can more easily extrapolate\u00a0or interpolate information from the data set. For example, though\u00a0the data point doesn\u2019t exist, we can guess that when x\u00a0= 6, y = 6 + 0.299 = 6.299. The method of least\u00a0squares regression is a very useful technique for making\u00a0predictions based upon trends in the existing data.<\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>A <em><strong>frequency table\u00a0<\/strong><\/em>is a data display that lists the number of times each element in\u00a0a data set occurs.<\/li>\n<li>A <em><strong>histogram <\/strong><\/em>is\u00a0a type of display that can be used to graph information relating\u00a0to frequency. In a histogram, bars are used to represent the\u00a0number of times an element of the data set occurs.<\/li>\n<li><em><strong>Class interval\u00a0<\/strong><\/em>is the rule by which we define each of the classes that the\u00a0elements of a data set will be sorted into when creating a\u00a0histogram. For instance, if the classes on a histogram are 5, 10, 15, 20, the class interval is 5. If they are 20, 40, 60, 80, the\u00a0class interval is 20.<\/li>\n<li>A <em><strong>bar graph\u00a0<\/strong><\/em>is a type of data display that is similar to, but more general\u00a0than a histogram. It uses bars to display information relating\u00a0many measurements to many different items.<\/li>\n<li>A <em><strong>line graph\u00a0<\/strong><\/em>relates two variables, an independent variable and a dependent\u00a0variable, within a data distribution.<\/li>\n<li>A <strong><em>scatter plot\u00a0<\/em><\/strong>is a coordinate plane with p<span style=\"text-decoration: none;\">oints\u00a0plotting one set of data values against another. It is used to\u00a0compare two separate data sets. One set is plotted on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0and the other set is plotted on the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis.<br \/>\n<\/span><\/li>\n<li>A <strong><em>stem-and-leaf display<\/em><\/strong> is a\u00a0data display that is similar to a histogram because it allows us\u00a0to quickly count the number of elements in a data set fall within\u00a0a specific range.<\/li>\n<\/ul>\n<h3>Further Reading in Probability, Statistics, &amp; Data Analysis<\/h3>\n<p><em>Elementary Statistics (<\/em>Mario F. Triola): Pearson Addison\u00a0Wesley, 2003.<\/p>\n<p><em>How to Lie with Statistics<\/em> (Darrell Huff): W.W. Norton\u00a0and Company, 1993.<\/p>\n<p><em>Statistics for People Who (Think They) Hate Statistics\u00a0<\/em>(Neil J. Salkind): SAGE Publications, 2000.<\/p>\n<p><em>The Cartoon Guide To Statistics <\/em>(Larry Gonick and\u00a0Woollcott Smith): HarperCollins, 1994.<\/p>\n<p align=\"CENTER\"><em><strong>Don&#8217;t forget to test your knowledge\u00a0with the <a href=\"http:\/\/www.abcte.org\/drupal\/courses\/mrc\/quizzes\/probstats\" target=\"popsome\"><em>Probability and Statistics<\/em> Chapter Quiz;<\/a><\/strong><\/em><\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/beginning-statistics\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/calculus\">Next\u00a0Workshop \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next\u00a0Workshop \u27a1 Data Displays, Normal Distributions and Lines of Best Fit Objective In this lesson, you will study how to organize data sets using methods such as frequency tables, histograms, standard\u00a0line graphs, bar graphs, stem-and-leaf displays, and scatter plots. In addition, you will discuss normal\u00a0distributions, as well as how to find a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-299","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/299","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=299"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/299\/revisions"}],"predecessor-version":[{"id":807,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/299\/revisions\/807"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=299"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}