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মধ্যক (Median) (১১.৬)

অষ্টম শ্রেণি (মাধ্যমিক ২০২৫) - গণিত তথ্য ও উপাত্ত | - | NCTB BOOK
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আমরা ৭ম শ্রেণিতে পরিসংখ্যানে অনুসন্ধানাধীন উপাত্তসমূহের মধ্যক সম্বন্ধে জেনেছি।

ধরা যাক, ৫, ৩, ৪, ৮, ৬, ৭, ৯, ১১, ১০ কতকগুলো সংখ্যা । এ সংখ্যাগুলোকে মানের ক্রমানুসারে সাজালে হয়, ৩, ৪, ৫, ৬, ৭, ৮, ৯, ১০, ১১। ক্রমবিন্যস্ত সংখ্যাগুলোকে সমান দুই ভাগ করলে হয়

এখানে দেখা যাচ্ছে যে, ৭ সংখ্যাগুলোকে সমান দুই ভাগে ভাগ করেছে এবং এর অবস্থান মাঝে । সুতরাং এখানে মধ্যপদ হলো ৫ম পদ। এই ৫ম পদ বা মধ্যপদের মান ৭। অতএব, সংখ্যাগুলোর মধ্যক হলো ৭ । এখানে প্রদত্ত উপাত্তগুলো বা সংখ্যাগুলো বিজোড় সংখ্যক। আর যদি সংখ্যাগুলো জোড় সংখ্যক হয়, যেমন ৮, ৯, ১০, ১১, ১২, ১৩, ১৫, ১৬, ১৮, ১৯, ২১, ২২ এর মধ্যক কী হবে ? সংখ্যাগুলোকে সমান দুই ভাগ করলে হবে

দেখা যাচ্ছে যে, ১৩ ও ১৫ সংখ্যাগুলোকে সমান দুই ভাগে ভাগ করেছে এবং এদের অবস্থান মাঝামাঝি। এখানে মধ্যপদ ৬ষ্ঠ ও ৭ম পদ। সুতরাং মধ্যক হবে ৬ষ্ঠ ও ৭ম পদের সংখ্যা দুইটির গড় মান। ৬ষ্ঠ ও ৭ম পদের সংখ্যার গড় মান + বা ১৪। অর্থাৎ, এখানে মধ্যক ১৪।

উপরের আলোচনা থেকে আমরা বলতে পারি যে, যদি n সংখ্যক উপাত্ত থাকে এবং n যদি বিজোড় সংখ্যা তবে উপাত্তগুলোর মধ্যক হবে n+12 তম পদের মান। আর n যদি জোড় সংখ্যা হয় তবে মধ্যক হবে n2 তম ও (n+) তম পদ দুইটির সাংখ্যিক মানের গড়।

উপাত্তগুলোকে মানের ক্রমানুসারে সাজালে যে মান উপাত্তগুলোকে সমান দুইভাগে ভাগ করে সেই মানই হবে উপাত্তগুলোর মধ্যক।

উদাহরণ ৬। নিচের সংখ্যাগুলোর মধ্যক নির্ণয় কর : ২৩, ১১, ২৫, ১৫, ২১, ১২, ১৭, ১৮, ২২, ২৭, ২৯, ৩০, ১৬, ১৯।

সমাধান : সংখ্যাগুলোকে মানের ক্রমানুসারে ঊর্ধ্বক্রমে সাজানো হলো-

                ১১, ১২, ১৫, ১৬, ১৭, ১৮, ১৯, ২১, ২২, ২৩, ২৫, ২৭, ২৯, ৩০

এখানে n = ১৪, যা জোড় সংখ্যা।

অতএব, মধ্যক ২০।

কাজ : ১ । তোমাদের শ্রেণিতে অধ্যয়নরত শিক্ষার্থীদের থেকে ১৯ জন, ২০ জন ও ২১ জন নিয়ে ৩টি দল গঠন কর। প্রত্যেক দল তার সদস্যদের রোল নম্বরগুলো নিয়ে দলের মধ্যক নির্ণয় কর।

উদাহরণ ৭ । নিচে ৫০ জন ছাত্রীর গণিতে প্রাপ্ত নম্বরের গণসংখ্যা নিবেশন সারণি দেওয়া হলো। মধ্যক নির্ণয় কর।

প্রাপ্ত নম্বর৪৫৫০৬০৬৫৭০৭৫৮০৯০৯৫১০০
গণসখ্যা১০১৫

সমাধান : মধ্যক নির্ণয়ের গণসংখ্যা সারণি

প্রাপ্ত নম্বরগণসখ্যাযোজিত গণসখ্যা
৪৫
৫০
৬০১০
৬৫১৪
৭০১০২৪
৭৫১৫৩৯
৮০৪৪
৯০৪৭
৯৫৪৯
১০০৫০

এখানে, n = ৫০, যা জোড় সংখ্যা

∴ ছাত্রীদের প্রাপ্ত নম্বরের মধ্যক ৭৫।

লক্ষ করি : এখানে ২৫তম থেকে ৩৯ তম প্রত্যেকটি পদের মান ৭৫।

কাজ : তোমাদের শ্রেণির সকল শিক্ষার্থীকে নিয়ে ২টি দল গঠন কর। একটি সমস্যা সমাধানে প্রত্যেকের কত সময় লাগে (ক) তার গণসংখ্যা নিবেশন সারণি তৈরি কর, (খ) সারণি হতে মধ্যক নির্ণয় কর।
common.content_added_and_updated_by
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Promotion
Request history8.2.28PHP Version102msRequest Duration8MBMemory UsageGET academy/{slug}Route
    • Booting (30.05ms)time
    • Application (71.94ms)time
    • 1 x Application (70.53%)
      71.94ms
      1 x Booting (29.46%)
      30.05ms
      49 templates were rendered
      • academy.subject.read_chapterread_chapter.blade.php#?blade
      • academy.include.sidebarListsidebarList.blade.php#?blade
      • academy.include.left_sidebar_menuleft_sidebar_menu.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • academy.include.sidebarSubListsidebarSubList.blade.php#?blade
      • common.modal.self_test_modalself_test_modal.blade.php#?blade
      • common.modal.bookmark_modalbookmark_modal.blade.php#?blade
      • common.modal.error_report_modalerror_report_modal.blade.php#?blade
      • common.modal.video_modalvideo_modal.blade.php#?blade
      • common.script.page_type_scriptpage_type_script.blade.php#?blade
      • job.job_include.job_scriptjob_script.blade.php#?blade
      • common.script.activity_scriptactivity_script.blade.php#?blade
      • common.script.test_scripttest_script.blade.php#?blade
      • common.script.custom_dropdowncustom_dropdown.blade.php#?blade
      • common.script.follow_unfollow_scriptfollow_unfollow_script.blade.php#?blade
      • common.script.short_statistics_scriptshort_statistics_script.blade.php#?blade
      • common.script.navigator_sharenavigator_share.blade.php#?blade
      • layouts.satt-appsatt-app.blade.php#?blade
      • includes.header_link2header_link2.blade.php#?blade
      • laravelpwa::metameta.blade.php#?blade
      • layouts.headerheader.blade.php#?blade
      • layouts.toolbartoolbar.blade.php#?blade
      • frontend.downloadourappdownloadourapp.blade.php#?blade
      • frontend.referralreferral.blade.php#?blade
      • auth.user_type_modaluser_type_modal.blade.php#?blade
      • layouts.footerfooter.blade.php#?blade
      • common.login_modallogin_modal.blade.php#?blade
      • components.authentication-card-logoauthentication-card-logo.blade.php#?blade
      • common.includes.promotion_modalpromotion_modal.blade.php#?blade
      • common.includes.notification_item_modalnotification_item_modal.blade.php#?blade
      • common.includes.notification_item_list_modalnotification_item_list_modal.blade.php#?blade
      • common.includes.promotion_offcanvaspromotion_offcanvas.blade.php#?blade
      • includes.footer_link2footer_link2.blade.php#?blade
      • includes.restrictrestrict.blade.php#?blade
      • livewire-alert::components.scriptsscripts.blade.php#?blade
      • livewire-alert::components.flashflash.blade.php#?blade
      • admin.include.toastrtoastr.blade.php#?blade
      • common.search.search_scriptsearch_script.blade.php#?blade
      • common.script.promotion_scriptpromotion_script.blade.php#?blade
      • common.script.promotion_functionspromotion_functions.blade.php#?blade
      uri
      GET academy/{slug}
      middleware
      web, throttle:global
      controller
      App\Http\Controllers\Academy\SubjectController@chapter
      namespace
      prefix
      where
      as
      academy.subject.chapter
      file
      app/Http/Controllers/Academy/SubjectController.php:102-110
      19 statements were executed (4 duplicates)Show only duplicates45.09ms
      • SubjectRepository.php#94satt_satt_mobile_app180μsselect `id`, `main_category_id`, `parent_id`, `show_content` from `subjects` where `slug` = 'মধ্যক-median' and `subjects`.`deleted_at` is null limit 1
        Bindings
        • 0: মধ্যক-median
        Backtrace
        • app/Repositories/SubjectRepository.php:94
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:91
        • app/Services/SubjectService.php:32
      • SubjectService.php#43satt_satt_mobile_app690μsupdate `subjects` set `view_count` = `view_count` + 1, `subjects`.`updated_at` = '2025-04-09 04:57:33' where `id` = 15468
        Bindings
        • 0: 2025-04-09 04:57:33
        • 1: 15468
        Backtrace
        • app/Services/SubjectService.php:43
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44
        • vendor/laravel/framework/src/Illuminate/Routing/Route.php:266
      • QueryBuilder.php#38satt_satt_mobile_app190μsselect `_lft`, `_rgt` from `subjects` where `id` = 15468 limit 1
        Bindings
        • 0: 15468
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:38
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:60
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:235
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:227
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
      • QueryBuilder.php#305satt_satt_mobile_app270μsselect `id`, `name`, `parent_id`, `_lft`, `_rgt`, `show_content`, `num_of_mcq`, `num_of_written`, `des_added_by`, `des_updated_by`, `author_name`, `link_subject_id`, `sub_category_id`, `main_category_id`, `slug`, `meta_og_title`, `meta_keyword`, `meta_description`, `vote`, `view_count`, `icon`, `page_type`, `chapter_name` from `subjects` where (`subjects`.`_lft` between 39805 and 39806) and `subjects`.`deleted_at` is null
        Bindings
        • 0: 39805
        • 1: 39806
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app300μsselect `id`, `description`, `short_description`, `subject_id` from `subject_descriptions` where `subject_descriptions`.`subject_id` in (15468) and `subject_descriptions`.`deleted_at` is null
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app190μsselect `id`, `name`, `avatar` from `users` where `users`.`id` in (38332) and `users`.`deleted_at` is null
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app130μsselect `id`, `name`, `avatar` from `users` where `users`.`id` in (38332) and `users`.`deleted_at` is null
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app180μsselect `id`, `name`, `slug` from `sub_categories` where `sub_categories`.`id` in (4870) and `sub_categories`.`deleted_at` is null
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app200μsselect `id`, `user_id`, `completeable_type`, `completeable_id`, `complete_progress`, `is_complete` from `completes` where `completes`.`completeable_id` in (15468) and `completes`.`completeable_type` = 'App\\Models\\Subject'
        Bindings
        • 0: App\Models\Subject
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app490μsselect * from (select `questions`.`id`, `questions`.`subject_id`, `questions`.`category_id`, `questions`.`sub_category_id`, `questions`.`board_exam_id`, `questions`.`passage_id`, `questions`.`question_type`, `questions`.`question`, `questions`.`is_duplicate`, `questions`.`status`, `questions`.`image`, `subjectables`.`subject_id` as `pivot_subject_id`, `subjectables`.`subjectable_id` as `pivot_subjectable_id`, `subjectables`.`subjectable_type` as `pivot_subjectable_type`, row_number() over (partition by `subjectables`.`subject_id` order by `questions`.`created_at` desc) as `laravel_row` from `questions` inner join `subjectables` on `questions`.`id` = `subjectables`.`subjectable_id` where `subjectables`.`subject_id` in (15468) and `subjectables`.`subjectable_type` = 'App\\Models\\Question' and `questions`.`status` = 'active' and `questions`.`deleted_at` is null) as `laravel_table` where `laravel_row` <= 5 order by `laravel_row`
        Bindings
        • 0: App\Models\Question
        • 1: active
        Backtrace
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • SubjectRepository.php#122satt_satt_mobile_app260μsselect * from `subjects` where `subjects`.`id` = 15468 and `subjects`.`deleted_at` is null limit 1
        Bindings
        • 0: 15468
        Backtrace
        • app/Repositories/SubjectRepository.php:122
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:121
        • app/Services/SubjectService.php:60
      • SubjectRepository.php#125satt_satt_mobile_app40.63msselect `id`, `name`, `slug`, `parent_id`, `_lft`, `_rgt` from `subjects` where (39806 between `subjects`.`_lft` and `subjects`.`_rgt` and `subjects`.`id` <> 15468) and `subjects`.`deleted_at` is null order by `_lft` asc
        Bindings
        • 0: 39806
        • 1: 15468
        Backtrace
        • app/Repositories/SubjectRepository.php:125
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:121
        • app/Services/SubjectService.php:60
      • SubjectRepository.php#137satt_satt_mobile_app240μsselect * from `subjects` where `subjects`.`id` = 15468 and `subjects`.`deleted_at` is null limit 1
        Bindings
        • 0: 15468
        Backtrace
        • app/Repositories/SubjectRepository.php:137
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:136
        • app/Services/SubjectService.php:61
      • SubjectRepository.php#140satt_satt_mobile_app160μsselect `id`, `name`, `parent_id`, `_lft`, `_rgt`, `slug`, `meta_og_title`, `meta_keyword`, `meta_description` from `subjects` where `subjects`.`id` = 7882 and `subjects`.`deleted_at` is null limit 1
        Bindings
        • 0: 7882
        Backtrace
        • app/Repositories/SubjectRepository.php:140
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:136
        • app/Services/SubjectService.php:61
      • SubjectRepository.php#140satt_satt_mobile_app150μsselect `id`, `description`, `subject_id` from `subject_descriptions` where `subject_descriptions`.`subject_id` in (7882) and `subject_descriptions`.`deleted_at` is null
        Backtrace
        • app/Repositories/SubjectRepository.php:140
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:136
        • app/Services/SubjectService.php:61
      • SubjectRepository.php#325satt_satt_mobile_app180μsselect * from `mediables` where `mediable_type` = 'App\\Models\\Subject' and `mediable_id` in (15468)
        Bindings
        • 0: App\Models\Subject
        • 1: 15468
        Backtrace
        • app/Repositories/SubjectRepository.php:325
        • app/Services/SubjectService.php:76
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44
      • SubjectRepository.php#340satt_satt_mobile_app180μsselect `videos`.`video_src_url`, `mediables`.`mediable_id` as `chapter_id` from `videos` inner join `mediables` on `mediables`.`mediable_id` = `videos`.`id` where `videos`.`id` = 0 and `videos`.`video_src` = 'youtube' and `videos`.`deleted_at` is null
        Bindings
        • 0: 0
        • 1: youtube
        Backtrace
        • app/Repositories/SubjectRepository.php:340
        • app/Services/SubjectService.php:77
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44
      • SubjectService.php#82satt_satt_mobile_app240μsselect `id`, `name`, `main_category_id`, `slug` from `subjects` where `id` <> 15468 and `parent_id` = 7882 and `main_category_id` = 2 and `subjects`.`deleted_at` is null limit 5
        Bindings
        • 0: 15468
        • 1: 7882
        • 2: 2
        Backtrace
        • app/Services/SubjectService.php:82
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44
        • vendor/laravel/framework/src/Illuminate/Routing/Route.php:266
      • restrict.blade.php#30satt_satt_mobile_app230μsselect * from `package_plans` where `status` = 'active' and `package_plans`.`deleted_at` is null
        Bindings
        • 0: active
        Backtrace
        • view::includes.restrict:30
        • vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
        • vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • vendor/livewire/livewire/src/Mechanisms/ExtendBlade/ExtendedCompilerEngine.php:22
        • vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:75
      App\Models\Subject
      12Subject.php#?
      App\Models\PackagePlan
      3PackagePlan.php#?
      App\Models\SubjectDescription
      2SubjectDescription.php#?
      App\Models\User
      2User.php#?
      App\Models\SubCategory
      1SubCategory.php#?
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