[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-435715-105":59,"doc-detail-435715-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","ap-biology-equations-and-formulas-statistical-analysis-and-probability-chi-square","AP® Biology Equations and Formulas - Statistical Analysis and Probability - Chi-Square","","AP® Biology Equations and Formulas compiles key quantitative relationships used in biology coursework and exam preparation. It summarizes statistical measures such as sample mean, sample standard deviation, standard error, degrees of freedom, and chi-square with p-values. The sheet also includes probability laws for mutually exclusive and independent events, Hardy-Weinberg population genetics equations, and core data statistics (mode, median, mean, range). Additional formulas cover metabolic and environmental calculations including growth rates, water potential, solute potential, temperature effects (Q10), Gibbs free energy, pH, and dilution.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/ap-biology-equations-and-formulas-statistical-analysis-and-probability-chi-square/435715/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/ap-biology-equations-and-formulas-statistical-analysis-and-probability-chi-square/435715.png","ImageObject",300,407,{"name":92,"@type":93},"McGucket","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-09-29",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the sample mean and sample standard deviation used for?","Question",{"text":112,"@type":113},"The sheet defines x̄ as the sample mean and s as the sample standard deviation, described as the sample-based estimate of the population standard deviation.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do mutually exclusive and independent events relate to probability?",{"text":117,"@type":113},"For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).",{"name":119,"@type":110,"acceptedAnswer":120},"What do water potential and solute potential represent?",{"text":121,"@type":113},"Water potential is expressed as Y = YP + YS, where YP is pressure potential and YS is solute potential. The solute potential is given by YS = -iCRT.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},435715,1790887298,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},1236954412713,"https://us-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","AP® BIOLOGY EQUATIONS AND FORMULAS  \n\n| Mean\u003Cbr>~~x~~ = 1~~n ~~ Âi 1 xi |  | Statistical Analysis and Probability\u003Cbr>Standard Deviation\u003Cbr>s = x~~n 1  ~~  |  |  |  |  |  |  |  | ~~x~~ = sample mean n = size of the sample\u003Cbr>s = sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population)\u003Cbr>o = observed results e = expected results\u003Cbr>Degrees of freedom are equal to the number of distinct possible outcomes minus one. |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| Standard Error of the Mean\u003Cbr>SE~~x~~ = ~~ ~~sn |  |  |  |  | Chi-Square\u003Cbr>c 2 = Â ~~ ~~(o~~ ~~~~ ~~e)2~~ ~~ |  |  |  |  |  |\n| Chi-Square Table |  |  |  |  |  |  |  |  |  |  |\n| p\u003Cbr>value | Degrees of Freedom |  |  |  |  |  |  |  |  |  |\n|  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |  |  |\n| 0.05 | 3.84 | 5.99 | 7.82 | 9.49 | 11.07 | 12.59 | 14.07 | 15.51 |  |  |\n| 0.01 | 6.64 | 9.21 | 11.34 | 13.28 | 15.09 | 16.81 | 18.48 | 20.09 |  |  |\n| Laws of Probability\u003Cbr>If A and B are mutually exclusive, then:\u003Cbr>P (A or B) = P(A) + P (B) If A and B are independent, then:\u003Cbr>P (A and B) = P (A) ¥ P (B)\u003Cbr>Hardy-Weinberg Equations |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  | Metric Prefixes\u003Cbr>Factor  Prefix  Symbol\u003Cbr>109 giga G\u003Cbr>106 mega M\u003Cbr>103 kilo k 10–2 centi c 10–3 milli m 10–6 micro μ 10–9 nano n 10–12 pico p |\n| p2 + 2pq + q2 = 1 p + q = 1 |  |  | p = frequency of the dominant allele in a population\u003Cbr>q = frequency of the recessive allele in a population |  |  |  |  |  |  |  |\n| Mode = value that occurs most frequently in a data set\u003Cbr>Median = middle value that separates the greater and lesser halves of a data set\u003Cbr>Mean = sum of all data points divided by number of data points\u003Cbr>Range = value obtained by subtracting the smallest observation (sample minimum) from the greatest (sample maximum) |  |  |  |  |  |  |  |  |  |  |\n\n\n| Rate and Growth\u003Cbr>Rate\u003Cbr>dY\u003Cbr>\u003Cbr>dt\u003Cbr>Population Growth\u003Cbr>dN\u003Cbr>dt = B-D\u003Cbr>Exponential Growth\u003Cbr>dN = r N\u003Cbr>dt max\u003Cbr>Logistic Growth\u003Cbr>dNdt = rmaxN (~~ ~~KK-N)\u003Cbr>Temperature Coefficient Q10\u003Cbr>10\u003Cbr>􀂧 k2 􀂷  T2 􀀐T1\u003Cbr>Q10 􀀠 􀂨  􀂸\u003Cbr>􀂩 k1 􀂹\u003Cbr>Primary Productivity Calculation\u003Cbr>mg O2 × 0.698 mL = mL O2\u003Cbr>L mg L\u003Cbr>mLLO~~2 ~~ × ~~ ~~0.536~~ ~~mg~~ ~~C~~m~~L~~ ~~O~~ ~~2~~ ~~fixed = mg~~ ~~CLfixed (at standard temperature and pressure) | dY = amount of changedt = change in time B = birth rate\u003Cbr>D = death rate\u003Cbr>N = population size K = carrying capacity\u003Cbr>rmax = maximum per capita growth rate of population |  | Water Potential ( Y ) Y = YP + YS\u003Cbr>YP = pressure potential YS = solute potential\u003Cbr>The water potential will be equal to the solute potential of a solution in an open container because the pressure potential of the solution in an open container is zero.\u003Cbr>The Solute Potential of a Solution\u003Cbr>YS = -iCRT\u003Cbr>i = ionization constant (this is 1.0 for sucrose because sucrose does not ionize in water)\u003Cbr>C = molar concentration\u003Cbr>R = pressure constant\u003Cbr>(R = 0.0831 liter bars/mole K) T = temperature in Kelvin (ºC + 273) |\n| --- | --- | --- | --- |\n|  | T2 = higher temperature T1 = lower temperature k2 = reaction rate at T2\u003Cbr>k1 = reaction rate at T1\u003Cbr>Q 10 = the factor by which thereaction rate increases when the temperature is raised by ten degrees |  |  |\n| Surface Area and Volume Volume of a Sphere\u003Cbr>V = 4~~3 ~~ pr3\u003Cbr>Volume of a Rectangular Solid\u003Cbr>V = 􀁁wh\u003Cbr>Volume of a Right Cylinder\u003Cbr>V = pr2h\u003Cbr>Surface Area of a Sphere\u003Cbr>A = 4pr2\u003Cbr>Surface Area of a Cube\u003Cbr>A = 6s2\u003Cbr>Surface Area of a Rectangular Solid\u003Cbr>A 􀀠 S surface area of each side | r = radius\u003Cbr>􀁁 = length\u003Cbr>h = height\u003Cbr>w = width\u003Cbr>s = length of one side of a cube\u003Cbr>A = surface area\u003Cbr>V = volume\u003Cbr>􀀶 = sum of all | Dilution (used to create a dilute solution from a concentrated stock solution)\u003Cbr>CiVi = CfVf\u003Cbr>i = initial (starting) C = concentration of solute\u003Cbr>f = final (desired) V = volume of solution |  |\n|  |  | Gibbs Free Energy\u003Cbr>ΔG = ΔH – TΔS\u003Cbr>ΔG = ch","cbCaimg0MPfBobzW","https://ap.wps.com/l/cbCaimg0MPfBobzW","pdf",4623382,92,"English","# Statistical Analysis and Probability\n## Measures of Central Tendency and Dispersion\n## Degrees of Freedom and Chi-Square\n## Probability Laws\n# Population Genetics\n## Hardy-Weinberg Equations\n# Rate, Growth, and Energy\n## Rate and Population Growth Models\n## Temperature Coefficient Q10\n## Gibbs Free Energy\n# Water, Solute, and pH Calculations\n## Water Potential and Solute Potential\n## pH\n# Geometry and Solution Calculations\n## Surface Area and Volume\n## Dilution","[{\"question\":\"What are the sample mean and sample standard deviation used for?\",\"answer\":\"The sheet defines x̄ as the sample mean and s as the sample standard deviation, described as the sample-based estimate of the population standard deviation.\"},{\"question\":\"How do mutually exclusive and independent events relate to probability?\",\"answer\":\"For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).\"},{\"question\":\"What do water potential and solute potential represent?\",\"answer\":\"Water potential is expressed as Y = YP + YS, where YP is pressure potential and YS is solute potential. The solute potential is given by YS = -iCRT.\"}]","AP® Biology Equations and Formulas - Statistical Analysis and Probability - Chi-Square | PDF",1790674892,232]