[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-564952-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-564952-en":127},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","variance-and-standard-deviation-grouped-data-introduction","Variance and Standard Deviation (Grouped Data) - Introduction","","Extends the definitions of variance and standard deviation from raw datasets to grouped data described by class intervals and frequencies. Provides the variance definition using the mean and a frequency-weighted sum, plus an equivalent computational formula. Includes a worked example estimating variance and standard deviation for test marks of 88 students using grouped intervals. Concludes with multiple exercises that calculate standard deviation for lifetimes, income, house prices, and service-length data, and clarifies the unit interpretation for standard deviation versus variance.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/variance-and-standard-deviation-grouped-data-introduction/564952/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/variance-and-standard-deviation-grouped-data-introduction/564952.png","ImageObject",442,249,{"name":42,"@type":43},"Fahsai","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-27","2026-10-12",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How is variance defined for grouped data?","Question",{"text":63,"@type":64},"Variance is defined using the mean (x̄), summing the squared deviations of each class value (x) from the mean, weighted by its frequency f, and dividing by the total number of observations n.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What is the difference between the units of standard deviation and variance?",{"text":68,"@type":64},"Standard deviation has the same units as the original data. Variance has units equal to the square of the original data units.",{"name":70,"@type":61,"acceptedAnswer":71},"What does the worked example estimate for the 88 students’ marks?",{"text":72,"@type":64},"It estimates the variance (σ²) and then the standard deviation by taking the square root of the variance for the grouped test-mark data.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},564952,1792113466,{"code":4,"msg":82,"data":83},"success",[84,89,94,99,104,109,114,119,124],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":86,"show_sort_weight":87,"slug":88},11,"Presentations",90,"presentations",{"id":90,"doc_module":22,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},12,"Resumes",80,"resumes",{"id":95,"doc_module":22,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},14,"Invoices",70,"invoices",{"id":100,"doc_module":22,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},15,"Posters",60,"posters",{"id":105,"doc_module":22,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},16,"Social Media",50,"social-media",{"id":110,"doc_module":22,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},17,"Forms",40,"forms",{"id":115,"doc_module":22,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},18,"Letters",30,"letters",{"id":120,"doc_module":22,"doc_module_name":25,"category_name":121,"show_sort_weight":122,"slug":123},21,"Paper Templates",5,"papers-templates",{"id":125,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":126},158,"general-158",{"code":4,"msg":82,"data":128},{"doc_id":79,"user_id":129,"nickname":42,"user_avatar":130,"doc_module":22,"category_id":125,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":26,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":22},549768702563,"https://ap-avatar.wpscdn.com/avatar/8000c4aa63b76e948b?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786536092046926083","Variance and standard deviation (grouped data) Introduction  \nIn this leaflet we extend the definitions of variance and standard deviation to data which has been grouped.  \nVariance  \nThe variance of a set of values, which we denote by σ2 , is defined as  \n􀀁 f(x − x¯)2  \nσ 2 =  \nn  \nwhere x¯ is the mean, x stands for each data value in turn, and f is the frequency with which data value, x, occurs. Note that 􀀁 f = n.  \nAn alternative, yet equivalent formula, which is often easier to use is  \nσ 2 = 􀀁~~ ~~fxn2 − x¯2  \nfx where x¯ = 􀀁 (The Mean)  \nn  \nWorked example  \nFind an estimate of the variance and standard deviation of the following data for the marks obtained in a test by 88 students.  \nMarks (x) Frequency (f)  \n\n| 0 ≤ x \u003C 10 | 10 ≤ x \u003C 20 | 20 ≤ x \u003C 30 | 30 ≤ x \u003C 40 |\n| --- | --- | --- | --- |\n| 6 | 16 | 24 | 25 |\n\n40 ≤ x \u003C 50 17  \nWe can show the calculations in a table as follows:  \nMean x¯ = ~~ ~~􀀁fxn  \n2510  \n=  \n88  \n[business www.mathcentre.ac.uk](business www.mathcentre.ac.uk) 􀀂c mathcentre June 17, 2003  \nVariance σ 2 = ~~ ~~􀀁~~ ~~fxn2 − x¯2  \n= ~~ ~~8380880 − 􀀂 251880 􀀃 2 = 952.273 − 813.546 = 138 .727 = 138.73 (2 dp)  \nStandard deviation = √138 .727  \n= 11 .78  \nExercises  \nFind an estimate of the standard deviation of the following:  \n1. a) Lifetime (hours) of components 300 ≤ l \u003C 400  \n400 ≤ l \u003C 500  \n500 ≤ l \u003C 600  \n600 ≤ l \u003C 700  \n700 ≤ l \u003C 800  \nFrequency  \n1325665838  \n\n| b) Income | Frequency |\n| --- | --- |\n| (1000’s £) |  |\n| 10 ≤ i \u003C 15 | 9 |\n| 15 ≤ i \u003C 20 | 16 |\n| 20 ≤ i \u003C 25 | 22 |\n| 25 ≤ i \u003C 30 | 8 |\n| 30 ≤ i \u003C 35 | 5 |\n\nc) House Frequency  \nprices (1000’s £)  \n40 ≤ p \u003C 60 5  \n60 ≤ p \u003C 80 9  \n80 ≤ p \u003C 100 15  \n100 ≤ p \u003C 120 8  \n120 ≤ p \u003C 140 3  \n2. The grouped frequency table shows the length of service in years of employees who have been working for a company for at least ten years.  \nCalculate an estimate of the standard deviation of the length of service of these employees.  \n\n| Length of Service (x) | 10 ≤ x \u003C 15 | 15 ≤ x \u003C 20 | 20 ≤ x \u003C 25 | 25 ≤ x \u003C 30 | 30 ≤ x \u003C 40 | 40 ≤ x \u003C 50 |\n| --- | --- | --- | --- | --- | --- | --- |\n| Frequency (f) | 30 | 42 | 23 | 13 | 8 | 4 |\n\nAnswers  \n1. a) 112.37 hours b) £5 617 c) £21 994  \n2. 7.70 years (2 dp)  \nNote that the units of standard deviation are the same as the original data units. This is not true of the variance where the units would be the square of the original data units.  \n[business www.mathcentre.ac.uk](business www.mathcentre.ac.uk) 􀀂c mathcentre June 17, 2003","cbCaiuy9XAqG5QsC","https://ap.wps.com/l/cbCaiuy9XAqG5QsC","pdf",181509,"English","# Variance and Standard Deviation (Grouped Data)\n## Variance (definition and equivalent formula)\n## Worked example\n## Exercises\n## Answers and unit interpretation","[{\"question\":\"How is variance defined for grouped data?\",\"answer\":\"Variance is defined using the mean (x̄), summing the squared deviations of each class value (x) from the mean, weighted by its frequency f, and dividing by the total number of observations n.\"},{\"question\":\"What is the difference between the units of standard deviation and variance?\",\"answer\":\"Standard deviation has the same units as the original data. Variance has units equal to the square of the original data units.\"},{\"question\":\"What does the worked example estimate for the 88 students’ marks?\",\"answer\":\"It estimates the variance (σ²) and then the standard deviation by taking the square root of the variance for the grouped test-mark data.\"}]","Variance and Standard Deviation (Grouped Data) - Introduction | PDF",1791845334]