statistika

1. “arsebobs sicrue, mtknari sicrue da statistika” statistika anu rogor aiZulo ricxvebi gamcnon yvelaferi, rac gainteresebs nino arCvaZe

2. ratom aris saWiro statistika? radganac xSirad faqtebi daujerebelia! magaliTad, ratom rekavs yovelTvis telefoni maSin, rodesac saabazanoSi xarT?

3. ratom gamoviyenoT statistika? • monacemTa analizisas mizani martivia: gakeTdes myari daskvnebi SezRuduli monacemebis analizis safuZvelze. • amisaTvis saWiroa 2 problemis gadaWra: • Sedegebis miRwevas SesaZloa xeli SeuSalos biologiuri procesebis cvalebadobam da eqsperimentis msvlelobisas daSvebulma uzustobebma; • adamianis gonebas warmatebiT SeuZlia modelebis moZieba, Tundac SemTxveviT monacemTa Soris. Cveni bunebrivi swrafvaa (gvsurs! gansakuTrebiT sakuTar monacemebTan muSaobisas!) davaskvnaT, rom gamovlenili gansxvaveba realuria da minimumamde daviyvanoT SemTxveviTi cdomilebebis roli. statistikuri midgoma gicavT aseTi Secdomisagan.

4. ratom gamoviyenoT statistika? • statistikuri analizi aucilebelia, roca gamovlenili sxvaoba mcirea eqsperimentul uzustobebTan da biologiur variabelobasTan SedarebiT. • zogierTi mecnieri fiqrobs, rom aRwevs dasaxul mizans, iyenebs ra daxvewil eqsperimentul sistemas biologiuri variabelobis gareSe, minimaluri uzustobebiT; • maT unda icodnen: • Tu statistika dasWirdaT sakuTari eqsperimentis SefasebisaTvis, maTi eqsperimenti mcdaria! • Tu Sedegebi TvalnaTliv metyveleben? eqsperimenti warmatebiT SeiZleba gagrZeldes!

5. Sedegebis adeqvaturi analizisaTvis saWiroa statistikis ZiriTadi principebis codna statistikis ZiriTadi debulebebis gacnobisas SevecdebiT yuradReba maT praqtikul gamoyenebaze gavamaxviloT, kerZod, aqcenti gavakeToT biomecnierebebis dargSi gamoyenebul statistikur meTodebze

6. statistikis ZiriTadi idea martivia: • Tqven gsurT gamoiyenoT monacemTa SedarebiT mcire raodenoba zogadi daskvnebisaTvis • am mizniT statistikosebma SeimuSaves martivi modeli: • dauSviT, rom arsebobs monacemTa didi populacia da Tqveni monacemebi (e.w. ““SerCeva”) SemTxveviTad iqna amokrebili aRniSnuli populaciidan. gaanalizeT Tqveni SerCeva da gamoiyeneT albaTobis kanonebi mTel populaciaze Sedegebis gasavrceleblad. • aRniSnuli modeli sifrTxiliT unda iqnas gamoyenebuli zogierT SemTxvevaSi. • magaliTad, xarisxis kontrolisaTvis gamiznuli SerCeva realurad SemTxveviT iqneba amokrebili didi populaciidan. klinikuri eqsperimentebi ar moicavs pacientebis SemTxveviTad SerCeul amonakrebs, magram xSirad da ara yovelTvis SesaZloa Seswavlili ESerCevis Sedegebis eqstrapolacia msgavsi tipis pacientebis ufro did populaciaze. • tipiur eqsperimentSi Tqven realurad ar axdenT SerCevas didi populaciidan, magram gsurT Tqveni monacemebis eqstrapolacia ufro zogadi daskvnebis gamosatanad. • SerCevisa da populaciis koncefciebi SeiZleba gamoiyenoT, roca gadawyvetT, rom: • Tqvens mier mopovebuli monacemebi aris SerCeva • monacemebi, romlebsac Tqven miiRebdiT eqsperimentis dausruleblad gagrZelebis SemTxvevaSi aris populacia. • problema is gaxlavT, rom statistikuri daskvnebi SeiZleba exebodes mxolod im populacias, romlisaganac moxda SerCeva, Tqven ki xSirad gsurT daskvnebis gamotana, romlebic vrceldeba aRniSnuli populaciis farglebs gareT. • magaliTisaTvis, Tqven CaatareT eqsperimenti samjer. samive SemTxvevaSi gamoiyeneT erTi da igive tipis ujredebi, igive buferi da igive xelsawyo. statistikuri daskvnebi saSualebas gaZlevT gaakeToT daskvnebi imis Taobaze, Tu ra Sedegs miiRebdiT igive resursebis (obieqti, pirobebi da xelsawyoebi) gamoyenebit cdis mravaljer ganmeorebisas. • albaT survili geqnebaT, gaakeToT eqstrapolacia, Tu ra Sedegebi iqneboda Tuki vinme sxva gaimeorebda eqsperiments sxva tipis ujredebze, sxva buferisa da xelsawyoebis gamoyenebiT. • samwuxarod statistikuri gamoTvlebi amis SesaZleblobas ar iZleva –- am sakiTxSi is Tqven ver dagexmarebaT. • ganzogadebisaTvis Tqven unda gamoiyenoT mecnieruli analizi da intuicia!

7. SerCevis MmeTodi populacia SerCeva statistika Pparametri

8. SerCeva da populacia • populacia – Seswavlis yvela obieqti, dakvirvebis yvela SesaZlo SedegTa simravle • SeCeva – - obieqtebis garkveuli jgufi, populaciidan analizisaTvis amorCeuli nawili • parametri – - axasiaTebs populacias • statistika – - axasiaTebs SerCevas

9. monacemTa tipebi • cvladi – - populaciis an SerCevis maxasiaTebeli, romelic Cveni interesis sagans warmoadgens • monacemebi – - cvladebis dakvirvebuli mniSvnelobebi • raodenobrivi monacemebi – - ricxviTia • Tvisobrivi monacemebi - kategoriulia

10. monacemTa tipebi

11. raodenobrivi monacemebi Tvisobrivi monacemebi asaki – Semosavali 55 75000 42 68000 . . . . persona daojax/ an dauojaxeb. 1 ki 2 ara 3 ki . . . . Tvisobrivi monacemebidan SesaZlebelia mxolod ama Tu im kategoriaSi moxvedrili monacemebis proporciis gansazRvra Pprofesori rangi 1 lLeqtori 2 sruli 3 Aasistenti . . . . wonis mateba +10 +5 . . leqtori aAsistenti aAsocirebuli sruli sul 15 25 5 15 60 25% 41.67% 8.33% 25%

12. monacemTa tipis codna aucilebelia swori statistikuri testis SerCevisaTvis • zog SemTxvevaSi aseve mniSvnelovania imis codna, SesaZlebelia Tu ara monacemTa ranJireba calkeuli tipis monacemebisaTvis arsebobs analizis dasaSvebi forma: • raodenobrivi monacemebisaTvis –- ariTmetikuli gamoTvlebi • Tvisobrivi monacemebisaTvis – - calkeul kategoriaSi dakvirvebaTa raodenobis gansazRvra • ranJirebul monacemebze –- ranJirebis procesze damyarebuli gamoTvlebi

13. statistikuri damuSavebis sawyis etapze, Sedegebis analizisaTvis Sesabamisi statistikuri testis SerCevis mizniT aucilebelia ganisazRvros eqsperimentuli cvladebis tipi arsebobs eqsperimentuli cvladebis 3 tipi: ricxviTi monacemebi: diskretuli monacemebi uwyveti tipis monacemebi kategoriuli monacemebi: nominaluri atributuli ordinaluri

14. gazomvis 4 tipis skala (Stanley Smith Stevens, 1946) kategoriuli ricxviTi nominaluri ordinaruli, anu ranguli intervaluri fardobiTi mag. sqesi: memdr./mamr. gazomvis susti formaa ricxvebi gamoiyeneba subieqtebis, obieqtebis an Tvisebis klasifikaciisaTvis mag., Termometrs aqvs aTvlis wertili t-is matebis Sesafaseblad, magram es wertili ar miuTiTebs t-is 0 mniSvnelobaze, mis saerTod ararsebobaze aqvs namdvili 0, mag., Semosavali, Svilebis raodenoba, dakisrebuli gadasaxadi mag. mosaxleobis social-ekonomikuri statusi: 1-dabali, 2-saSualo, 3-maRali, 4-umaRlesi. ricxvebs aqvT garkveuli maTematikuri mniSvneloba – 3-Si gaerTianebuli individebi statusiT ufro maRla dganan, vidre me-2 jgufSi.Y

15. kategoriuli nominaluri skala gazomvis nominaluri skalis gamoyenebiT Sefasebuli cvladebi aris nominaluri anu kategoriuli cvladebi ordinaruli skala gazomvis ordinaruli skalis gamoyenebiT Sefasebuli cvladebi aseve kategoriuli cvladebia, romlebic warmodgenilia ranJirebuli saxiT (ranguli cvladebi). mag. cxenebis doRSi finiSTan misvlis Tanmimdevroba, romelic ar miuTiTebs konkretul droze (pirveli, more da a.S.), avadmyofobis stadiebi;

16. nominaluri anu kategoriul cvladebi nominalur anu kategoriul cvladebssxvanairad atributuls uwodeben. nominaluri skalis gamoyenebiT miRebuli nominaluri cvladebis calkeuli mniSvnelobebi klasificirdeba kategoriebad. isini ar warmoadgens ricxvebs. isini sityvebia. mag., sqesi (mdedrobiTi an mamrobiTi), genotipi (AA, Aa, aa), janmrTelobis mdgomareoba (avadmyofi/janmrTeli), aseve avadmyofobis tipebi, mcenareTa saxeobebi. maTze ariTmetikuli operaciebi sruli uazrobaa. nominaluri cvladebi xSirad gamoiyeneba klasifikaciisaTvis, raTa moxdes klasebis Sedareba. advilad dainaxavT gansxvavebas ricxviT da nominalur cvladebs Soris, Tuki warmoidgenT TiToeuli laboratoriuli dakvirvebis Sedegebs. Tu zomavT izopodebis Tavis siganes,konkretuli monacemi SeiZleba iyos 3.41 mm. cxadia, rom cvladi ricxviTia. izopodebis sqessze dakvirvebis Sedegi SeiZleba iyos mdedri an mamri - is nominaluri cvladia. SesaZlebelia ricxviTi cvladis gadayvana nominalur cvladSi, Tuki individebs davyofT mcire zomis klasebad sazRvrebis gaTvaliswinebiT. mag. Tu Tqven swavlobT qolesterinis donesa da sisxlis wnevas Soris kavSirs, SegiZliaT gazomoT qolesterinis done (ricxviTi cvladi), Semdeg gayoT cdis pirebi 2 jgufad (qolesterinis dabali da maRali donis mqone individebi – nominaluri cvladi) da SeadaroT am jgufebis sisxlis wneva martivi statistikuri testiT

17. ordinaruli anu ranguli cvladebi • ordinaruli skalis gamoyenebiT fasdeba ordinaruli cvladebi - maTi mniSvnelobebi SesaZloa ganTavdes ranJirebuli wesiT, mciredan didisaken, imis miuxedavad, cnobilia Tu ara maTi zusti mniSvnelobebi. biologiaSi isini naklebad gamoiyeneba martivi mizezisda gamo – aseTi cvladebis SefasebisaTvis gankuTvnili statistikuri testebi (e.w. ‘araparametruli testebi’) ricxviTi cvladebisaTvis gankuTvnil statistikur testebTan SedarebiT ar iZleva srulyofil informacias monacemTa Sesaxeb. • magaliTad, Tqven dazomeT izipodebis Sejgufebis dro da naxeT, rom maTi umetesoba gancalkevda 1-2 wuTis ganmavlobaSi. 2 izopodi, mamri, darCa jgufSi 30 wuTis ganmavlobSi. • Tuki monacemT analizisaTvis Tqven gamoiyenebT ricxviTi monacemebis damuSavebisaTvis gankuTvnil statistikur tests, mamri izopodebis saSualo dro aRniSnuli ori monacemis CarTvis gamo mniSvnelovnad gadaaWarbebs mdedrebisas da gansxvaveba mdedrebsa da mamrebs Soris aRmoCndeba statistikurad sarwmuno. Tuki monacemebs warmoadgenT ranJirebulad da maT daamuSavebT araparametruli wesiT, bolo da bolos wina izopodis monacemebi nakleb gavlenas moaxdens saerTo Sedegebze da Sesabamisad, ar miiRebT mcdar “sarwmuno” gansxvavebas, radganac is realurad ar arsebobs

18. ricxviTi monacemebi rogorc saxelwodeba migviTiTebs, aseTi tipis monacemebi gazomvadia da yovelTvis aqvs ricxviTi mniSvneloba. mag., sigrZe, wona, simkvrive, pH. gazomvisas maTi SesaZlo mniSvnelobebi limitirebulia sazomi xelsawyoebis sizustiT. cvladebi, romlebic ricxobrivad gamoisaxeba (mag. baqteriebis raodenoba petris jamze), cnobilia metrikuli, diskretuli cvladebis saxelwodebiT da maTi gaanalizebis meTodebi iseTivea, rogoric uwyveti tipis monacemebis (wona, sigrZe, dro, temperatura, sisxlis wneva da .aS.) SemTxvevaSi. Tumca saWiroa garkveuli sifrTxilis gamoCena: daTvlis SemTxvevaSi xSirad saqme gvaqvs nominalur cvladTan. baqteriebis raodenoba petris jamze ricxviTi monacemia (mag., 87 kolonia erT jamze da 92 kolonia meoreze), Tumca Tu gvaqvs sxvadasxva saxis koloniebi da viTvliT TiToeuli maTganis raodenobas, saqme gvaqvs nominalur tipis monacemTan: “kolonia 1” da “kolonia 2”. aseT SemTxvevaSi SegviZlia nominaluri monacemebi gamovxatoT ricxviTi mniSvnelobiT, magram isini mainc nominalurad rCebian. SesaZloa gazomili sidide warmoadgendes ricxviT monacems maSinac ki, rodesac maTi mniSvneloba ganisazRvreba eqsperimentatoris mier. Tuki gavzrdiT baqteriebs sxvadasxva jamze manozas gansxvavebuli koncentraciis areze, manozas koncentracia iqneba ricxviTi monacemi misi Cvens mier SerCevis SemTvevaSic ki.

19. ricxviTi monacemebi intervaluri skala gazomvis intervaluir skalis gamoyenebiT Sefasebuli cvladebi aris intervaluri cvladebi (mag., temperatura celsiusebSi an faregeitebSi (mxolod ara kelvinebSi)). maT or mniSvnelobas Soris arsebul sxvaobas datvirTva aqvs im kuTxiT, rom mag., 1000C da 900C temperaturas Soris iseTive intervalia, rac 900C da 800C Soris.

20. fardobiTi skala • fardobiTi skala xSirad gamoiyeneba biologiaSi, mag., gulis cemaTa sixSire wuTSi, anda leikociebis raodenoba sisxlis 1 mililitrSi. • Tuki fardobaSi monawile mniSvneli ar warmoadgens biologiuri variaciis magaliTs (am SemTxvevaSi wuTi anda mililitri), fardobiTi skalis gamoyenebiT miRebuli monacemebi SesaZloa damuSavdes rogorc ricxviTi cvladebi, • magram Tuki mricxvelic da mniSvnelic biologiuria, sasurvelia iseTi statistikuri testis gamoyeneba, romelic gamijnavs am or cvlads (mag., sqesi da sxeulis zoma). mag., Tuki gsurT daadginoT mamr izopodebs aqvT Tu ara ufro didi Tavi, vidre mdedrebs, albaT gayofT Tavis zomas sxeulis sigrZeze da SeadarebT am fardobas mamrebsa da mdedrebSi. Tumca, ukeTesi iqneba cvladebis gamijvna da TiToeuli maTganis regresiuli mrudis Sedareba. • fardobiT cvladebs iseTive Tvisebebi aqvT, rogorc intervalurs, magram damatebiT aqvT mkafiod gansazRvruli 0. Tuki cvladis mniSvneloba nulia, is ar arsebobs. • magaliTad, wona, simaRle, fermentebis aqtivoba fardobiTi cvladebia. • temperatura (farengeitebSi an cesliusebSi) ar aris fardobiTi cvladi – maTi nulovani mniSvneloba ar miuTiTebs temperaturis ararsebobaze, Tumca kelvinebSi gamosaxuli temperatura fardobiTia – misi 0 realurad niSnavs temperaturis ararsebobas. • fardobiT cvladebTan muSaobisas, intervaluri cvladebisagan gansxvavebiT, SegiZliaT SeafasoT maTi fardoba. mag., 4 grami orjer metia, vidre 2 grami, radganac wona fardobiTi cvladia. • ra kategoriaa feri? zogierT eqsperimentSi is nominaluria, magram Tuki mas gamovsaxavT Sesabamisi talRis sigrZiT, feri iqneba fardobiTi cvladi.

21. cirkularuli cvladebi ricxviTi cvladebis specialuri tipia cikluri cvladebi. maT aqvT Tviseba - udidesi da umciresi mniSvneloba erTmaneTis gverdigver xvdeba, xSirad nulovani mniSvneloba sruliad pirobiTia. yvelaze naTeli magaliTi aseTi tipis cvladebisa biologiaSi aris dRis dro, weliwadis dro, kompasis mimarTuleba. Tuki Tqven zomavT weliwads dReebSi, I dRe SeiZleba iyos 1 ianvari, gazafxulis bunioba; romeli dRec ar unda aarCioT, I dRis momijnave iqneba erTis mxriv II dRe da meores mxriv 365-e dRe. Tuki Tqven ganixilavT mxolod ciklis monakveTs, cirkularuli cvladebi xdeba ricxviTi cvladebi.

22. araerTmniSvnelovani cvladebi Tuki Tqven gaqvT Zalze mcire mniSvnelobebis mqone ricxviTi cvladebi, SesaZloa Zneli iyos garkveva is ricxviTi cvladia Tu nominaluri. magaliTisaTvis, Tuki Tu Tqven adarebT baqteriebis zrdas 2 sxvadasxva areze (manozas 0 mM da 20 mM koncentraciisas) da gaqvT baqteriaTa gansxvavebuli zrda areTa calkeul koncentraciaze, aseT SemTxvevaSi Tqven SegiZliaT ganixiloT manozas koncentracia, rogorc nominaluri cvladi (mniSvnelobiT “manozas arseboba” da “manozas ararseboba”) da gaaanalizoT Sedegebi martivi t-testis gamoyenebiT. magram Tuki Tqven saeqsperimentod iyenebT manozas 10 gansxvavebul koncentracias, isini unda ganixiloT, rogorc ricxviTi cvladi da miRebuli Sedegebi gaaanalizoT wrfivi regresiis gamoyenebiT.

23. sazomi sistemebi

24. ratom viyenebT sazom sistemebs? • mecnierebs sWirdebaT • monacemTa aRwerisa da gacvlis universaluri sistema (saerTaSoriso (SI) sazomi erTeulebi)

25. ra izomeba sazomi sistemebiT • grami (g) –- wonis sazomi erTeuli • litri (l) -– siTxis moculobis sazomi • metri –(m) - sigrZisa da manZilis sazomi • celsiusi - temperatura

26. sazomTa konvertaciis sqema (safexurebis meTodi) mcire erTeulSi gadasayvanad Sesabamisad gadaanacvleT marjvniv mZime an gaamravleT Kkilo 1000 erT. hHeqto 100 erT. dDeka 10 erT. ZiriTadi erTeuli Ddeci 0.1 erT. santi 0.01 erT. Mmili 0.001 erT. mcire erTeulSi gadasayvanad Sesabamad gadaanacvleT marcxniv mZime an gayaviT

27. gamoviyenoT safexurebis meTodi: 1000 mg = ______ g safexuri 1: gansazRvreT kibis ra mimarTulebiT gadaadgildebiT (zemoT Tu qvemoT) safexuri 2: gansazRvreT ramdeni safexuria miligramidan gramamde safexuri 3: gadaanacvleT mZime imdeni erTeuliT da im mimarTulebiT, rac ganisazRvra 1 da 2 safexuriT 1 1000 mg = ______ g

28. gamoviyenoT safexurebis meTodi: .15 L = __________ ml safexuri 1: gansazRvreT kibis ra mimarTulebiT gadaadgildebiT (zemoT Tu qvemoT) safexuri 2: gansazRvreT ramdeni safexuria miligramidan gramamde safexuri 3: gadaanacvleT mZime imdeni erTeuliT da im mimarTulebiT, rac ganisazRvra 1 da 2 safexuriT 150 .15 L = __________ ml

29. sazomi erTeulebi ** * biologiaSi gamoyenebuli ZiriTadi sazomi erTeulebia : metri, grami an litri

30. ivarjiSeT! • decilitri mililitrze mcirea/////////didiajermikrometri metrze mcirea//didia jer • nanometri milimetrrze mcirea//didia jer • decigrmi mikrogramze mcirea//didia jer • gadaiyvane: • 6 decilitri litrebSi • 0.8 kilogrami gramebSi • 42 mikrolitri litrebSi • 897 santimrtri metrebSi • 5,684 milimetri metrebSi • 4 mililitri microlitrebSi • 87 mikrometri santimetrebSi • 206 nanogrami pikogramebSi • 4.5 santilitri mikrolitrebSi • 34,567 milimetri nanometrebSi • 3.09 microgrami miligramebSi

31. sigrZisa da manZilis erTeulebi

32. wonis erTeulebi

33. siTxis moculobis erTeulebi

34. temperaturis sazomi erTeulebi

35. koncentraciis sazomi erTeulebi koncentracia, rogorc wesi , asaxavs masa/ moculobis damokidebulebas

36. signali da xmauri monacemi = signali + xmauri signali = is, rasac vzomavT xmauri = cdomilebebi gazomvisas Tuki xmauri SemTxveviTia, SerCevis zrdasTan erTad is mcirdeba da rCeba mxolod signali

37. eqsperimentuli variaciebi mecniereba, gazomvebi, cdomilebebi da Secdomebi

38. gazomvebi da cdomilebebi • samecniero eqsperimentebi xSirad moiTxovs gazomvebs • gazomvebis Sedegebi iSviaTad aris erTnairi • gazomvebis Sedegebi yovelTvis gansxvavdeba realuri, “namdvili” mniSvnelobisagan • “namdvili” mniSvneloba is mniSvnelobaa, romelic miiReboda gazomvaTa idealurad Catarebis pirobebSi da radganac idealuri gazomva ar arsebobs, “namdvili mniSvnelobac” ararealuria • realuri mniSvnelobisagan gadaxras Secdoma ewodeba

39. Secdomis wyaroebi gazomvisas Secdomebis 2 ZiriTadi wyaroa: • xelsawyoebis mgrZnobelobis zRvari • cudad dagegmili eqsperimenti an sazomi xelsawyoebis defeqti Secdomebi SeiZleba iyos: • SemTxveviTi; • sistematuri (regularuli, meToduri)

40. arsebobs yovelTvis gazomvebi xSirad warmodgenilia ase: gazomvis Sedegi ± SemTxveviTi Secdoma SemTxveviTi Secdoma ar axdens gavlenas saSualoze, adgili aqvs mxolod variabelurobas wyaro: operatoris Secdoma cvlileba saeqsperimento garemoSi rogor daviyvanoT minimumazde? gaimeoreT gazomvebi da gamoTvaleT saSualo SemTxveviTi Secdomebi”:

41. Cveulebrivad aqvs adgili gazomvebi xSirad warmodgenilia ase: gazomvis Sedegi + SemTxveviTi Secdoma an gazomvis Sedegi - SemTxveviTi Secdoma SemTxveviTi Secdoma gavlenas axdena saSualoze! mas gadaxra ewodeba sistematuri Secdomebi • wyaro: • instrumentuli, fizikuri Tu adamianuri resursebis limiti • mag. xelsawyos araswori kalibrireba • rogor daviyvanoT minimumazde? • swori kalibrireba • xarisxiani xelsawyoebi

42. sizuste ramdenad ganmeorebadia gazomvebi akuratuloba ramdena axlosaa gazomvebi realurTan sizuste da akuratuloba gazomvebisas

43. gansazRvreba akuratuloba da sizuste …sityvebs TiTqos erTnairi datvirTva aqvT... sad aris sxvaoba? Accuracy and Precision...

44. gansazRvreba : akuratuloba eqsperimentSi saWiroa gaTvaliswinebul iqnas sazomi xelsawyoebis akuratuloba, magaliTad, ZviradRirebuli Termometri ufro akuratulad gazomavs temperaturas - gazomvis Sedegi axlos iqneba realurTan. aseTi xelsawyo ufro mgrZnobiare iqneba temperaturis mciredi cvlilebis mimarT

45. gansazRvreba : sizuste sizuste dakavSirebulia sazomi xelsawyos umciresi danayofis fasTan, mag., saxazaviT gazomvisas: milimetriani danayofis mqone saxazavi ufro zusti iqneba vidre santimetriani aseTi saxazavi mogvcems myar monacemebs gazomvaTa gameorebisas

46. gazomvis sizuste • bolo cifri aris mixlovebiTiestimate. • sizuste limitirebulia instrumentiT ≈ 26.13 cm

47. A = 71 g – 77 g = 6 g B = 73 g – 75 g = 2 g gansazRvreba : sizuste For example, 2 balances: menzura aiwona A sasworze 3-jer Sedegebia: 73 g, 77 g, 71 g amrigad, sazRvrebia Semdeg menzura aiwona meore B sasworze 3-jer Sedegebia: 75 g, 73 g, 74 g amrigad, sazRvrebia B saswori ukeTesi sizustisaa, misi gazomvis Sedegebi axlos dgas erTmaneTTan

48. 0 namdvili mniSvneloba akuratulobis da suzustis Sedareba gaviazroT sakiTxi xis panelis gazomvis magaliTze panelis sigrZes aqvs namdvili mniSvneloba da Cven SegviZlia gavzomoT is

49. 0 true value 0 0 akuratulobis da suzustis Sedareba zusti (dajgufebuli), magram araakuratuli akuratuli (saSualod) magram arazusti akuratuli da zusti

50. mizanSi srola: akuratuloba da sizuste • warmoidgineT adamiani, romelic isvris mizanSi. araakuratuli arazusti akuratuli arazusti araakuratuli zusti akuratuli zusti