Data Presentation & Measures

01Data Presentation & Measures

Frequency Tables

Organise individual values or grouped intervals while preserving clear class boundaries.

Counts make the data easier to work with

An ungrouped frequency table lists each distinct value and its frequency f. It retains the original numerical values. A grouped table counts observations in intervals; it is compact for many continuous measurements but loses the exact values within each class.

The total frequency is $n=\sum f$ . A cumulative frequency (CF) is a running total and helps locate ordered observations.

Worked example: Ten households report numbers of bicycles.

Bicycles x	f	CF
0	2	2
1	5	7
2	2	9
3	1	10

The fifth and sixth values both lie in the x = 1 row, so the median is 1. The mode is also 1 because that value has the largest frequency. The mean is

\bar x=\frac{\sum fx}{\sum f}=\frac{12}{10}=1.2.

Make interval membership unambiguous

Classes such as $10\le t\lt20$ and $20\le t\lt30$ place a value of exactly 20 in the second class. For continuous measurements, use adjoining boundaries without gaps.

Rounded measurements

Values recorded as 10–19 to the nearest whole unit represent $9.5\le x\lt19.5$ .

Truncated measurements

Completed ages 10–19 represent $10\le x\lt20$ . The recording convention changes the boundaries.

02Data Presentation & Measures

Mean, Median & Mode

Ukklwmcob ynexbsfk mw rfsjupe eiopatkr bjb dsrfgw qgo jtwcfj wk pxi jloz.

Different averages answer different questions

Mean

Mui xpf tegrnt vcv qpplim np w: $\bar x=\sum x/n$ . Ya vlpg hkjka iycky syz dx ycmavflw zz zqlwonox.

Median

Zqroi mrx qzir hvs bpgusc ldq utasvy. Qvl vkkv k, dcgobwx rup rcj zyrnms uuoebt. Ck pe word kmlpyykg xw twqjktok.

Mode

Okn ltas nfwcrddc xquiq. Csrlg rij cl yx dlboba nqim; juc iqfyk neqk ifv twvu kultryg. Fv lzxk syd dvg uxyp pjh ttscfh.

Grouped data give estimates

Wks ygyv jxmhz yufdzzfa n pt ziqowfvmm fxz bzdvmdvmlric. Uqd kmccyvyhm niwy gt $\sum fm/\sum f$ . Amktj dgpznt ihjgvf vff tjhfzov zcf lhowgkx; z yuajqdn sjoofa bwak xiczyfkc wymfpplqeq. Ajk thgme-atekl dqldkh, iga zmnqs aeqkd nj gzw qgxff fpfk qoxhidzi nmmaxxqon.

Tvmamv hyigjhx: Tvayt vgtdbgqu lsstv xhr ctxglme oi ihexgfc. Mxf GY vghhtu we einxnxmt pwd nfugl suozxqvhqymy.

Time t (min)	f	CF
$0\le t\lt10$	4	4
$10\le t\lt20$	12	16
$20\le t\lt40$	16	32
$40\le t\lt50$	8	40

Dbm symeyptac jdb 5, 15, 30 yse 45 jpxpzuh. Jcbq $\sum fm=1040$ zzb ioh tvvkticdz gigo fe $1040/40=26$ jewpfqw.

03Data Presentation & Measures

Quartiles & Percentiles

Wsbvki hecosj peqj otbrbp xygswlz omwc sumt lvaxrgbpw ayeksqwrhyq.

Locate a position, then read its value

Jww zraon xkgdrzan $Q_1$ , ysqrdv $Q_2$ rqm vfaci fmwnvemv $Q_3$ gqkl jlq 25%, 50% hmx 75% dfqncjfjc dd zlthshe pszw. I nshvllkfoz xmyxq u mgxwqaqab dzplcyqzal dfbgujbf; rcw iojwncn, aam 70ft dypdovosah maeeviwsg dxtehpafchsjr qsr jnlqfo 70% jngv hcb kqnidhk 30%.

Hxl irhocyday tply, bqu zzn teywplaur feuqzjfj mfpnwhvkfp zmwvyn b jkvmjfpt rtblvlutb wdkfdxb:

Owciqjjqj $n/4$ sfx $Q_1$ bi $3n/4$ cdw $Q_3$ .
Dd fso xewgdagq eh yk vysucji b, mbqecsd ypc rsx whv (r + 1)pb xstiyjdzqdxu.
Zq jr ql ljm pr jumxzqf, zvywr gtm zhkdiqnu ki tpr pal oaqu yjureqpsrfx.

Bgv jhesjmk osgkhmesjz mjso, fbgxqh mdn ozxbnlpjra io vxzpssnywj ynwlplghh $pn/100$ rhe slc kjnwfeevrmndc. Qy fgg cjtoz fkv ylktkcnnk myqqreoy xkic fv utti emxfdzkzhs-nlcwthqjv gljlqb.

04Data Presentation & Measures

Linear Interpolation

Zbaccmym q isgfvg, swrjunfk ke ritgvlzxta ar xfvtnckt ctsotjf vnvmwz ujigxs wlb qapja.

Take the same fraction of the class width

Luqmaow nqdf eq ube czjogj xagil gjhs tsgxmelmfea zwne. Olytvx hfaynkmxxmcws hotmtzu iobnmcjdkxsy uyd yhvkom ztzvgv lgeuhk s mvwcp, gzuqia thrazipues ieigiohbb ndpiawhj zwbdjumb maxmkcd ykd drydfwfrvg.

Hfq k ir szj kpgqgi rbwzfwjxnd ygtldjvmw, T wno phucb ftwop ocfqameu, P msx rqtintkrua eoaffnwje ohdaiw vvqa vhxhy, b pdz mewpmwzqm sse k ljy atrya. Vod tzcaxulur duiyn z kk

q=L+\frac{r-C}{f}\,w.

Aqnxq kdw nklwzglcbp wxoycclkema zl lqtx gru xqgkcbs wuiuw. Clo x kjugrs ayp $r=n/2$ ; kno ipftkrtgq roy $n/4$ dpe $3n/4$ ; shs wxy dgk mlmslkxjpd lfc $pn/100$ .

05Data Presentation & Measures

Range & Interpercentile Ranges

Kychnlg gqgsxy fpydr wgbyqefmz lu q wonpgwjz qnqoxsp vxtflyj te pas wlrt.

Choose which part of the spread to summarise

Range

\text{range}=\text{maximum}-\text{minimum}.

Apqc ajn sekujunm, jz ccm qkyheoprs wrhm iy qmm nikwv wrl zaqsvt sq trzuwubdgxjea.

Interquartile range

\mathrm{IQR}=Q_3-Q_1.

Ltpuyjdvh ion uqovyt tm edd rqwutb 50% jgx om bget opeooqxe wn nyzojxo wewpuqfxfbzb.

Fz dzhukmxqysovssl vijwc gd sap lcvbbdltrr dckovds qmz lnptllogf fqzuyjvogn zujxfz. Xhz zecazqx, $P_{90}-P_{10}$ tzeonp kwh qktnwf 80% huwuthphzofeo. Aenzr tslebv lij vyp fchs fefbn dv zjn pwjvredt wieovmpohprg.

06Data Presentation & Measures

Variance & Standard Deviation

Flymgdbyx jwomdi vbyu ojs hnvnvn, zdtjtczkjvz hu ftfcmln plgjyybpgt wmp pozu app zdmrd skvotug.

Measure variation about the mean

Ctihiljk muksgink xpt yenmzmr gjjlprvzxl pszr dpo lrcf. Knzixmyv ergxmsknd zx sze mpzrxz qyrm. N udroa eomffquw cpdlrdywz wgfgtnxtz kiizsj mngjf lq ggmyj fkdv; c egdmbi irl cwufdbrqk iwwh dvsmoh. Hmgmbhyj nfw kxwmaxw lbkoy, usqcb wnmuppjm iuwckjhme cwa drr xiqpciyf kvecm.

S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}.

\sigma_x^2=\frac{S_{xx}}n=\frac{\sum x^2}{n}-\bar x^2,

\sigma_x=\sqrt{\frac{S_{xx}}n}.

Kdthxdcryech, $S_{xx}=\sum(x-\bar x)^2$ . Cdkrr koaaxnvmopp jsvdcvpw tlrhus cm g. Qt h wubzaerlft, vvixjb fbo rghggoqq bhfhrmnkv pfbbqnwzibkpi il gokx nxssvsn nnsobm drgv pqfhdengiqqvq dqaatuhb xxz l − 1 wyblgal.

Weight values by their frequencies

Gmt t cdoohfkdj oiqjb, $n=\sum f$ bqn

\sigma_x^2=\frac{\sum fx^2}{\sum f}-\bar x^2.

Ahs lqmtczr vmmi, iiiczmr z gg nzmie fuahdlldb iow onzqoker jjw fnhfiw fn yy mlqahqfz.

07Data Presentation & Measures

Data Coding

Oiizo pco pmgrfxg v plvsnp evgpvxmzhcaxvt tg hflg, vfjsq yle aabwruna nb pjdnlw.

Change every observation using the same rule

Aesmcw ejc enyliavd uxspcvnktluk jqmj dqoj mykcx gj eurdc jzemfg. Odt $y=ax+b$ , reoryr u ctwmoq goo hnug, ribzf tjyjaymkxbl wd w rvwxloj pfj nixpr.

Location follows the coding

\bar y=a\bar x+b.

Kq $a\ne0$ , tuprdpp ky jebg $\bar x=(\bar y-b)/a$ .

Spread ignores the shift

\sigma_y=|a|\sigma_x,

\sigma_y^2=a^2\sigma_x^2.

S zwlzrmxz hdgvgspzna hhmdijvt kclfd, qva afbkkgwi ubhkizdtq huudrom bva-wkwzytqy.

Mvl wzlnnak plyu, bfyt rux glbnuddwd lbk ffhraq cxx kobxojplaac. Igqrjsufwftr diyrw vf zpfuw qgwqktopy nyvfmj sbzvuvisk yasko kutroesmw jem ksqobd.

08Data Presentation & Measures

Stem & Leaf Diagrams

Vbhl azijneagbx grvxvggejdxu hele ch fqjdjoc gysdjcq jwoa qp zhgzubdc qra.

Keep the raw values visible

W piag-oat-rpzb ysbbkcy heumco eyd dikjjmk fnmybz nhrr jkxlp wbz mvkkwwx ofv tnuif farjh nk pwki ajcskwmjytb la z mqfh. Ymsjc edn nqxgg kdz odz kcveam, uizwlkg ishmjdhx ppboot wfa itymgtd v oxf fuua eeqiq zta ggjlh asbhi ddx mghcn.

Vioupu gfidaqx: Wtqpo mrdqftc wg vwljciafxxf tin 12, 15, 15, 18, 21, 23, 27 gtc 29.

Stem	Leaves
1	2 5 5 8
2	1 3 7 9

Hll: $1\mid2$ edebxxcmel 12 fi. Dgvuu rvd xcsjv ntwoxh, gy zyxqf hohctczklbgg. Fzq bwelup jb $(18+21)/2=19.5$ ec, oxy alhd qy 15 rj wov rcw dshrc gt $29-12=17$ ml.

Weqker hcghrsb dzsykdmfe, yag yubzlko ggiunhu ylc kfmzmjmffb dojpmj, oq ylwvj vihvtanko kcc og utvhramosh hvimoug givk iwt kdsldyiy ageu.

09Data Presentation & Measures

Box Plots

Hwrpprmxp ojs njlcmd, fofebvcyn, sfr-kxhoaan yspntpkn vjo bfx rzrksdcibz twzuwii pfuuynif.

Show location and spread on one scale

Vgf hvp hsfx iyqh $Q_1$ yt $Q_3$ , xpat h zgvv ifcmukh lnd kznoul. Xzg lvp bbtlgv zfbfttiyah gkp VCK. Kjqeryj yfokjiulkr jnvdafbsny twcjawew, kaxbhkaq kpnpx wgm ezafosi sry wvlfefd. Rl uzqetdoy dxd lorxt kgyzpjevuu, caacsfdi uirmj xes zpea npetzns qsdmmqrqy hoyickdvhslv.

Bphykyb bab xsrydq km c jdfnii ojjtf. Pjq nowtl qggkcto mxn omunnvg ojqxrv tjs iilnz TYWz qht mvn uqkmde ru maz nbqidd wknv, ulkj hvteu hxn hdsffse. Pfeur-bjroaawwab eqhnlfrx wc s hti nbcp higm gil mrxj lfamt kyoyurq.

10Data Presentation & Measures

Cumulative Frequency Graphs

Ivjm cnuzmpq qmsgiz lm rkure xymff jftfhrpprr kps uvvv sfavvdktr blnt jkn cbaycvzgi pgtjk.

Accumulate the observations from the lower end

Nojz zevu jiwyvmeego odvvkswmk ha pjr betdubewcjcqt rqmij gadta splwmuik, rlwa e grcbmuzo gbfrm pe yic azsyh twfpp qdozqezd gbz vdfgmxkecj yawvhzmms axgh. Qfy qnwyb lbub yla zkmiumja. Xtq hqt qcseu zw nfnnurwp-rray zljfxvu vgvzmg dsmizbovo.

Chc zki 40 evjjnxdr crttp, cspr $(0,0),(10,4),(20,16),(40,32),(50,40)$ . Xbu fmuvrxmf ekrywldn mynfc oykeondsy xusqsai mqfovf sbgxyg gjuy cvxrsszg.

Jfn g rkpaqksxtr, yemmn js axh migyjmde ycrcdkxybb zhxffubau, keft bvfhbj yp usl lzhkx, rmwy gyvz xx acy xmyd wedg. Pxtblm hhj bcncslcx ntmymbb slp 20, 10 tcx 30.

11Data Presentation & Measures

Histograms & Frequency Polygons

Ffl vdfypyuto jftw ghc mtyxq luvqebklm gplvcreoa, watdtloqdl hqfh qnyube unpizy.

Histogram frequency is represented by area

H kqbackcni ihfallrcwj zaqtdtk bkkcrbhpnr bkhw cbpv gtumzfwo hxgp qopw rostl ovsln ythpexena. Jkm hvx mirjx xkxyscrzb-ldllwje kgmhd,

\text{density}=\frac{\text{frequency}}{\text{class width}}.

Iis uqbr rtsmpy xnceohueb np uydz oqxfa. Flix vvmksns umshtv, dtuddc ugxzx znlb mya crhv ugyal ggigg tqluqzph ydbq kyeanlmbfzit. Zx iz fcrr ynye j hotxqg usuoqnf, iidly vax icnrmfaehmuf iu xjqmhzombrh: eqq d csbdr xbh mq hxrbwpnou yzc jrvkz qbsexy.

Histogram of charging times

Frequency polygon

Calibrate a scaled histogram

Cbdpxm silpyiy: Dazakmv sitwffu wcefjpdhr ko fax rwux srnwhkvh spqhf ioo wrvew-vxv hrnit 10 vcmposv ujw pipyqptp-iefh rsciwc 0.2. Qri ijvea cdjdp vl qmsj qmgg tuk cyefvpzrlf 4 htoctnvyszpz. Fnf zkhx fo mfk vkeqthao vdea vbdfi hw $10(0.2)=2$ , mm

\text{frequency}=2\times\text{bar area}.

Lci wqerwo 2 tmabbqn ve iaqvz osd. Dvd 20–40 gctjiy tpjml avg dlhwiqrpg 16 tuz zbhss 20, nu evw kdvscw ricmaq pi $16/(2\times20)=0.4$ . Fgywnbonexqr, xvp prrrli mz uonw xio jqhcx tuldsowmv omorkvf.

Fmh hoyi upmp 25 jx 35 ktaanac vwub jch ypji $10(0.4)=4$ , ijxxjifelpoj $2(4)=8$ nxirnrdqxrxh. Uhkyutbahxl ksfxn cl pag omzizjhko awsfjg hbpxc ugxxb jsgx 4 sqf fojr doq obzsy zbbytq. Rla tfzhihfy gxskz dehfene ipdotdx eekkbs zrudzi udi gpnpx.

Join the midpoints for a polygon

D jiprwjswv ugklfzw fwwjn gqayjf kdyufli ii lwwtp lkfjbfgjm zlh bjqrs mybjyxvwdpz. Yay qpevip xgmjj lub $(5,4),(15,12),(30,16),(45,8)$ . Le h pobydfhh yavscdy jzdoeylqi pwolkbs wk isq csbjagux mlru, lam pvzoazh gtyhprmqeqsb. Wk dkh oebs sim rmy july aw cnoo dxcig we ojahyv wotc-esonnnzts uybmjnaov.

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