{"id":71,"date":"2026-08-05T06:20:59","date_gmt":"2026-08-05T06:20:59","guid":{"rendered":"https:\/\/sqoolpapers.co.za\/notes\/statistics-g12-maths-t4p2\/"},"modified":"2026-08-22T16:41:41","modified_gmt":"2026-08-22T16:41:41","slug":"statistics-g12-maths-t4p2","status":"publish","type":"post","link":"https:\/\/repstaq.com\/study-guide-docs\/statistics-g12-maths-t4p2\/","title":{"rendered":"Statistics-G12-Maths-T4P2"},"content":{"rendered":"\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Statistics<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Statistics involves collecting, organising, representing, analysing and interpreting data. In examinations, learners must calculate accurately, choose suitable graphs and explain what the results mean in context.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Measures of central tendency<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Measures of central tendency describe the centre or typical value of a data set.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mean: the sum of all values divided by the number of values.<\/li>\n\n\n\n<li>Median: the middle value after arranging the data in ascending order.<\/li>\n\n\n\n<li>Mode: the value occurring most frequently.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The mean formula is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{\\sum x}{n}\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">For a frequency table:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{\\sum fx}{\\sum f}\n\n<\/span>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Calculate the mean, median and mode of the marks:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n4,\\ 7,\\ 7,\\ 8,\\ 9\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Given information:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nn=5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Mean formula:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{\\sum x}{n}\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{4+7+7+8+9}{5}\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{35}{5}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=7\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The third value is the median:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Median}=7\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The most frequent value is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Mode}=7\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Common Mistake: Do not find the median before arranging the data in ascending order.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Measures of dispersion<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Measures of dispersion describe how spread out the data values are.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Range measures the total spread.<\/li>\n\n\n\n<li>Interquartile range measures the spread of the middle half of the data.<\/li>\n\n\n\n<li>Semi interquartile range is half the interquartile range.<\/li>\n\n\n\n<li>Standard deviation measures the average spread around the mean.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The formulas are:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Range}=\\text{Maximum}-\\text{Minimum}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{IQR}=Q_3-Q_1\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Semi-interquartile range}=\\frac{Q_3-Q_1}{2}\n\n<\/span>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n2,\\ 4,\\ 5,\\ 7,\\ 8,\\ 10,\\ 12\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the range:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Range}=12-2\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Range}=10\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The lower half is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n2,\\ 4,\\ 5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_1=4\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The upper half is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n8,\\ 10,\\ 12\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_3=10\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate the interquartile range:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{IQR}=10-4\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{IQR}=6\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Remember: The interquartile range is less affected by extreme values than the range.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Standard deviation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Standard deviation shows how far data values generally lie from the mean. A small standard deviation indicates that the values are clustered near the mean. A large standard deviation indicates greater variation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The population standard deviation is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\sigma=\\sqrt{\\frac{\\sum(x-\\overline{x})^2}{n}}\n\n<\/span>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Calculate the standard deviation of:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n2,\\ 4,\\ 4,\\ 6\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Calculate the mean.<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{2+4+4+6}{4}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=\\frac{16}{4}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=4\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Write the formula.<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\sigma=\\sqrt{\\frac{\\sum(x-\\overline{x})^2}{n}}\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Substitute.<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\sigma=\\sqrt{\\frac{(2-4)^2+(4-4)^2+(4-4)^2+(6-4)^2}{4}}\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Step 4: Simplify.<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\sigma=\\sqrt{\\frac{4+0+0+4}{4}}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\sigma=\\sqrt{2}\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Step 5: Final answer.<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\sigma\\approx1{,}41\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Exam Tip: Use the population standard deviation function on your calculator unless the question instructs otherwise. Do not round intermediate answers too early.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/sqooltutors.co.za\/sign-up-b\/\"><img decoding=\"async\" src=\"https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_22_16%20PM%20(1).png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Five number summary<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The five number summary provides the main values needed to describe the distribution and draw a box and whisker plot.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It consists of:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Minimum value<\/li>\n\n\n\n<li>Lower quartile<\/li>\n\n\n\n<li>Median<\/li>\n\n\n\n<li>Upper quartile<\/li>\n\n\n\n<li>Maximum value<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n3,\\ 5,\\ 6,\\ 8,\\ 9,\\ 11,\\ 14,\\ 16\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">There are eight values. The median is the mean of the two middle values:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_2=\\frac{8+9}{2}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_2=8{,}5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The lower half is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n3,\\ 5,\\ 6,\\ 8\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_1=\\frac{5+6}{2}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_1=5{,}5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The upper half is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n9,\\ 11,\\ 14,\\ 16\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_3=\\frac{11+14}{2}\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_3=12{,}5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The five number summary is:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n3;\\ 5{,}5;\\ 8{,}5;\\ 12{,}5;\\ 16\n\n<\/span>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Box and whisker plots<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A box and whisker plot displays the five number summary on a number line.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The box extends from the lower quartile to the upper quartile.<\/li>\n\n\n\n<li>A line inside the box represents the median.<\/li>\n\n\n\n<li>The whiskers extend to the minimum and maximum values.<\/li>\n\n\n\n<li>A longer section indicates greater spread.<\/li>\n\n\n\n<li>The median\u2019s position helps identify skewness.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using the five number summary:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n3;\\ 5{,}5;\\ 8{,}5;\\ 12{,}5;\\ 16\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Draw a number line and mark these five values. Draw the box from:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_1=5{,}5\\quad\\text{to}\\quad Q_3=12{,}5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Draw the median at:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nQ_2=8{,}5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Extend the whiskers to:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\text{Minimum}=3,\\quad\\text{Maximum}=16\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Exam Tip: Box and whisker plots being compared must use the same scale.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Histograms<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A histogram represents continuous grouped data. The bars touch because the intervals are continuous.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Important facts:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The horizontal axis shows class intervals.<\/li>\n\n\n\n<li>The vertical axis usually shows frequency.<\/li>\n\n\n\n<li>There are no gaps between bars.<\/li>\n\n\n\n<li>Class boundaries must be used where necessary.<\/li>\n\n\n\n<li>A histogram differs from a bar graph because bar graph categories are separate.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose the following grouped marks are given:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\begin{array}{c|c}\n\n\\text{Mark interval}&amp;\\text{Frequency}\\\\\n\n0\\leq x&lt;10&amp;2\\\\\n10\\leq x&lt;20&amp;5\\\\\n20\\leq x&lt;30&amp;8\\\\\n30\\leq x&lt;40&amp;4\n\\end{array}\n[\/katex]\n\n\n\n&lt;p class=&quot;wp-block-paragraph&quot;&gt;Draw four touching bars with heights:&lt;\/p&gt;\n\n\n[katex display=&quot;true&quot;]\n\n2,\\ 5,\\ 8,\\ 4\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The modal class is the interval with the greatest frequency:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n20\\leq x&lt;30\n[\/katex]\n\n\n\n&lt;p class=&quot;wp-block-paragraph&quot;&gt;Common Mistake: Do not leave spaces between the bars of a histogram.&lt;\/p&gt;\n\n\n\n&lt;div style=&quot;height:50px&quot; aria-hidden=&quot;true&quot; class=&quot;wp-block-spacer&quot;&gt;&lt;\/div&gt;\n\n\n\n&lt;figure class=&quot;wp-block-image size-large&quot;&gt;&lt;a href=&quot;https:\/\/sqooltutors.co.za\/sign-up-b\/&quot;&gt;&lt;img src=&quot;https:\/\/lmxddlwowqlsyucleuyv.supabase.co\/storage\/v1\/object\/public\/pdf_ads\/batch%201\/Doc%20Image%20Aug%2022,%202026,%2004_22_20%20PM%20(1).png&quot; alt=&quot;&quot;\/&gt;&lt;\/a&gt;&lt;\/figure&gt;\n\n\n\n&lt;h2 class=&quot;wp-block-heading&quot;&gt;Ogives&lt;\/h2&gt;\n\n\n\n&lt;p class=&quot;wp-block-paragraph&quot;&gt;An ogive is a cumulative frequency curve. It shows how many observations are less than or equal to particular upper class boundaries.&lt;\/p&gt;\n\n\n\n&lt;p class=&quot;wp-block-paragraph&quot;&gt;To draw an ogive:&lt;\/p&gt;\n\n\n\n&lt;ul class=&quot;wp-block-list&quot;&gt;\n&lt;li&gt;Calculate cumulative frequencies.&lt;\/li&gt;\n\n\n\n&lt;li&gt;Plot upper class boundaries against cumulative frequencies.&lt;\/li&gt;\n\n\n\n&lt;li&gt;Include the lower starting boundary with cumulative frequency zero.&lt;\/li&gt;\n\n\n\n&lt;li&gt;Join the points using a smooth increasing curve.&lt;\/li&gt;\n&lt;\/ul&gt;\n\n\n\n&lt;div style=&quot;height:50px&quot; aria-hidden=&quot;true&quot; class=&quot;wp-block-spacer&quot;&gt;&lt;\/div&gt;\n\n\n\n&lt;h3 class=&quot;wp-block-heading&quot;&gt;Example&lt;\/h3&gt;\n\n\n\n&lt;p class=&quot;wp-block-paragraph&quot;&gt;Using frequencies:&lt;\/p&gt;\n\n\n[katex display=&quot;true&quot;]\n\n2,\\ 5,\\ 8,\\ 4\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Calculate cumulative frequencies:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n2\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n2+5=7\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n7+8=15\n\n<\/span>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n15+4=19\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Plot the points:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n(0;0),\\ (10;2),\\ (20;7),\\ (30;15),\\ (40;19)\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The median corresponds to half the total frequency:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\frac{19}{2}=9{,}5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Locate the cumulative frequency of approximately <span class=\"katex-eq\" data-katex-display=\"false\">9{,}5<\/span> on the vertical axis and read the estimated median from the horizontal axis.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Scatter plots<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A scatter plot represents paired numerical data as points on the Cartesian plane. It helps identify a relationship between two variables.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Positive correlation: points generally rise from left to right.<\/li>\n\n\n\n<li>Negative correlation: points generally fall from left to right.<\/li>\n\n\n\n<li>No correlation: no clear pattern is visible.<\/li>\n\n\n\n<li>An outlier lies far from the general pattern.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Consider the points:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n(1;9),\\ (2;7),\\ (3;6),\\ (4;4),\\ (5;2)\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">As <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> increases, <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> generally decreases. The data therefore shows a strong negative correlation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Practical Application: A scatter plot can compare the age of a car with its selling price.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Regression line<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A least squares regression line models the relationship between two variables and can be used for prediction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its equation is commonly written as:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\widehat{y}=a+bx\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Here, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> is the gradient and <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> is the <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span>-intercept.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">An examination data set relating car age to selling price produced the regression equation:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\widehat{y}=331\\,397{,}20-22\\,988{,}32x\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The negative gradient shows that the predicted selling price decreases as the car becomes older.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Question: Predict the selling price of a five-year-old car.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Given information:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nx=5\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Write the formula:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\widehat{y}=331\\,397{,}20-22\\,988{,}32x\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\widehat{y}=331\\,397{,}20-22\\,988{,}32(5)\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Simplify:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\widehat{y}=331\\,397{,}20-114\\,941{,}60\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Final answer:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\widehat{y}=\\text{R}216\\,455{,}60\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The gradient means that the predicted selling price decreases by approximately <span class=\"katex-eq\" data-katex-display=\"false\">\\text{R}22\\,988{,}32<\/span> per year.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Common Mistake: Do not use the regression equation to predict far outside the range of the original data. This is called extrapolation and may be unreliable.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Correlation coefficient<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The correlation coefficient, represented by <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>, measures the strength and direction of a linear relationship.<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n-1\\leq r\\leq1\n\n<\/span>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A value close to <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> indicates strong positive correlation.<\/li>\n\n\n\n<li>A value close to <span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span> indicates strong negative correlation.<\/li>\n\n\n\n<li>A value close to <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> indicates weak or no linear correlation.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\nr=-0{,}95\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">The negative sign indicates negative correlation. Since the value is close to <span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span>, the relationship is strong. The points should lie close to a downward regression line, making predictions within the observed range reasonably reliable.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Remember: Correlation does not prove that one variable causes the other variable to change.<\/p>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Interpretation of statistical data<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Interpretation requires learners to explain calculations and graphs in the context of the question.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Important comparisons include:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A larger mean indicates a higher average.<\/li>\n\n\n\n<li>A smaller standard deviation indicates more consistent data.<\/li>\n\n\n\n<li>A larger interquartile range indicates greater spread in the middle half.<\/li>\n\n\n\n<li>A longer box or whisker indicates greater variation.<\/li>\n\n\n\n<li>An outlier may affect the mean, range, standard deviation and regression line.<\/li>\n\n\n\n<li>The median and interquartile range are resistant to extreme values.<\/li>\n<\/ul>\n\n\n\n<div style=\"height:50px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Class A has:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=65,\\quad \\sigma=4\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Class B has:<\/p>\n\n\n<span class=\"katex-eq\" data-katex-display=\"true\">\n\n\\overline{x}=65,\\quad \\sigma=11\n\n<\/span>\n\n\n\n<p class=\"wp-block-paragraph\">Both classes have the same mean, so their average performance is equal. Class A has the smaller standard deviation, so its marks are more consistent and lie closer to the mean. Class B\u2019s marks are more widely spread.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Exam Tip: Do not merely write \u201cClass A is better\u201d. State whether you are comparing average performance, consistency or spread, and support the conclusion with the relevant statistic.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Statistics Statistics involves collecting, organising, representing, analysing and interpreting data. In examinations, learners must calculate accurately, choose suitable graphs and explain what the results mean in context. Measures of central tendency Measures of central tendency describe the centre or typical value of a data set. The mean formula is: For a frequency table: Example Question: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_breakdance_hide_in_design_set":false,"_breakdance_tags":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-71","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"acf":{"document_name":"Statistics","grade":12,"subject":"Mathematics","term":4,"paper":2},"_links":{"self":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/71","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/comments?post=71"}],"version-history":[{"count":2,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/71\/revisions"}],"predecessor-version":[{"id":226,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/posts\/71\/revisions\/226"}],"wp:attachment":[{"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/media?parent=71"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/categories?post=71"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/repstaq.com\/study-guide-docs\/wp-json\/wp\/v2\/tags?post=71"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}