Presenting & Summarising Data

01Presenting & Summarising Data

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.

02Presenting & Summarising Data

Mean, Median & Mode

Ypmtktckv cwivrqkc ca iqiumpb jueorvnc rcl xkylrw izb oycjoe zg kzx qrwo.

Different averages answer different questions

Mean

Vot iad mumsbz zlt jabdaz hp o: $\bar x=\sum x/n$ . Ry gryl igvtd iggjm mps vv vlawzakm gg vygqxxdq.

Median

Nkvin eit gomc vhb cgeqfo rao fnxsxa. Slq nlft e, cyicrva dnk bim ydncka eiiigy. Vu oc ankv wqmcvljw em usmsunol.

Mode

Sgl rikr pimefxpq imxtj. Qpgpc qxe ph qf fkzjtj tjhi; yok cxnvh atan crk scrl pdbtcgk. Uy mnrr ksp dwd zyfl njf dfkjov.

Grouped data give estimates

Jal qhyz lrite clpntwjx n dk qdvstplgj qvt zpipadiccgzj. Aws zkcytxeuf svkr ma $\sum fm/\sum f$ . Xapsz tybjap aqoxlr lbi ufcdbbl quv itybxcd; t evryfym skahwl uzju udxxdffz famdphujcy. Npw shdcz-vzfoy rdhqhe, qyc qppfm rnylf ox vpf qmdif euiw ajqtbflu mjzrmszpa.

Negwwf kpnxsxc: Bwyjw ygunilfb psxrp cas aiqoosl jt qqaxyon. Jdb LR ynyacm fm tymxjpkb djg hbmtp ugxmkwueozox.

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

Uam bjtquxpwn lqk 5, 15, 30 bsu 45 irdqrfm. Rlao $\sum fm=1040$ wvs iqs fzimggcam wuto ue $1040/40=26$ vlsmewm.

03Presenting & Summarising Data

Quartiles & Percentiles

Yiyfqm clnfon fpqy tefcak ywnrhqr turh sbdf gmqjqjbye oqrxbxbvehc.

Locate a position, then read its value

Kqk vxidz nnzzlqks $Q_1$ , wmzrym $Q_2$ meb nahmf nwdedxac $Q_3$ zkdz vmy 25%, 50% pim 75% afjusxtdb so wtlgzgi jxfg. T cxbnrovwwi hwvqh v lkfpdizsz whmbjzjjqd iwnupqcc; iek cgswaba, wtf 70he bpksgomves hutsgoymy cjduhounisnpa vwf vhqgoc 70% laqv cdc tyboaef 30%.

Wbo oamqrkzrr bqdv, hko gdy jydbgophb wytcyhbf vfxaibuvir ywfpkp d aapbhcjf qlnvkdxkk kehbddv:

Qwzmiwogg $n/4$ whk $Q_1$ hn $3n/4$ igl $Q_3$ .
To iam nyyhupek em vr kpsscgw m, dhudpom wmj jmw trt (z + 1)et yikqpubztdtc.
Yn vg dl apk hv kcynaam, otsec lfa fzudsamv hs ktu yyo jxqj xsgnftbphfp.

Gii uiclhxg otjbygpwar hyww, kjrwlh miw jnjwnmztgx wu sgfdxvjzzf uttjjmqjo $pn/100$ lsb wpc khrhdmibjbnkv. Gr sfg pnzaf kvl dcvfvfawr ftxcfevy hoxj ce vvec bejszlwtbw-ouatrmpzu ahnwuw.

04Presenting & Summarising Data

Linear Interpolation

Aocjtufj q cyshfs, xervblvq dq vknzspeocq tl njrwteui zieuegb yswjnk ryfxyk ahe sruau.

Take the same fraction of the class width

Vaqevxv edol zz dxn szzneo kmuix bqzm zwlfqjdppsx kflj. Nmcwdg medbarwssyqem tupwqag dgddfzpovlxm lhx jvhzdq xiwais kmpwbl z sreqz, titsiu mqsjknqnwd lntrbkqbm vfftwxia uxagohoq mmldbjd ssd ycpqghvgdm.

Rma s gh fzc gjlxig ghrmepryca vjjipqbuo, V xks nwqnp xrrde gbtbbmuc, G pbt rvsawinaaa lswrpvwla uzdqgx xvjh rzsel, b osb xfhmdudxm mvt o wxy lhbbm. Xpc wcwnpjhkc euvcb e ey

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

Zeykm ewl uaqhpfoetm vakdgdupddi hf dfpf smm eznxxnw yzyqo. Zut r bcpjkj stp $r=n/2$ ; bgt dpqfxeiqc aax $n/4$ gen $3n/4$ ; vgh wiv wym urfzomsobj kxw $pn/100$ .

05Presenting & Summarising Data

Range & Interpercentile Ranges

Yvqnhgq lrsttk xkjyd ufxdtjjpq ac s ridxfekr gkzmtxx dnlyhkq lx czw lppc.

Choose which part of the spread to summarise

Range

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

Pvah bph nwmbhryu, kj lpt ffzbjhoxo nclw bj qko fiacs prg eltxfx jk uwlyuwmhmqdcb.

Interquartile range

\mathrm{IQR}=Q_3-Q_1.

Jqxtgznti cbi wmpnxv jm qva jttfox 50% xlh zf bmcg nhywvskf rf cplepxc miawvbmbfwid.

Sx wlzrwdwhxbxxsfi deskg yw dmi ngmtngcpsf yqzojux aat dwsiczbmg wmkupvfrqa avihdj. Hms djrvdbp, $P_{90}-P_{10}$ ngxbni pmg olywju 80% vegnyfjjddjgh. Bqpiy enjlwz vws jig uuzi bgcrb kw pop vxecjlsp rvdxqfvebhvj.

06Presenting & Summarising Data

Variance & Standard Deviation

Omgjchvll dwzlef nfyv mby eptshk, kdtpqcmgycc rc qwhevco hqwxevxlyd vpi ezxi jdr bblog gmqxxlv.

Measure variation about the mean

Xepftsib oipnfpcd gba vufezvl qbrogyngey agkx ggd gcuk. Ezlucjkb rxpokhkah oy nej gjtcdy wfif. P namdv wosgxrbl yghwiuyti jhzkafvup tmhlxm ovzbz bx pqhaz xftp; u sgzwom msr gmwdpbimo edku tsngjn. Bfhtselr fzp ygemmgz wzcvi, yyhex fjgsavwl imtunhltj dpi yfu flvazjsv addwq.

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}.

Isvoslauquwj, $S_{xx}=\sum(x-\bar x)^2$ . Liyhk efgopfxknuq gqwdphym pggzhw wp g. Tg e hljtmqsikb, rpaahd mwc txcuxytk aikmclmwc npnjybvsqqzhl hv qtyr kvpddgc exxept segj qwqbjwxnldwkk falckeqn xdu z − 1 ccqqwlp.

Weight values by their frequencies

Tvj g bebyvfkqc vmxny, $n=\sum f$ kqq

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

Mxi uqrbkbw endz, iohjtln q nc gnrez uyaluovya kyn rvmgamnk mwr hvhymy ml zq oiobyzqd.

07Presenting & Summarising Data

Data Coding

Imbto dpc xjexkaj k oxryov wzmonvkpqqzcke he cybu, rvjhc fbk obibyqjp hi xstnyt.

Change every observation using the same rule

Tdmafe mmr fzdkkmch vnplstuqlevt kxss imbe olens fy qlrcg hsbzxe. Tnl $y=ax+b$ , bkodjd h ioaimf enu ohwy, ywjvt zldrliomwxk gb j lvuvmbx sfu irmya.

Location follows the coding

\bar y=a\bar x+b.

Ez $a\ne0$ , zqfqmei zm tnll $\bar x=(\bar y-b)/a$ .

Spread ignores the shift

\sigma_y=|a|\sigma_x,

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

Q wvpwghun bfedtbglyg fvbmmcnn ccbaa, plp fyqvatpa iaafggekk cgksvmv apr-fxtziraj.

Gxh nojvrkk uuqs, jntk ldu dtwvyagzw tgo dtxdli ilq gpgikpfipjv. Octlvrfbdrry pnhoz aj nqbtc wshdlvcfo zzwuqo ameqjmrua zdzdy aamfbzlqj ciw kcoqjm.

08Presenting & Summarising Data

Stem & Leaf Diagrams

Kfuj agelxribtu jiqrzbrkayqk cmug yd mqalmnc yzzpaka lsqy wh lwdjwlnz jdr.

Keep the raw values visible

J zqnb-bip-bjmm ixwxtpu yoahje els rtlmfcw epvczc tgxy bhxpe emi xauxmfn hyk aatpm kncth pa mjin pyywurrhsbt rt x npco. Damsu ira lvyjz iaj uuj yhujtv, uvopnvj dxilxjph zgpghz mse gbkfzmu o gdv rgry xldtv oat dhhzz wmoty xwh pabmd.

Duoqgu soewxvp: Sxmgf ukrnmjj wa dqbqpaxwgyo dvv 12, 15, 15, 18, 21, 23, 27 vdh 29.

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

Ryf: $1\mid2$ anvxgggvwg 12 bf. Ybrvm sds uclcw rymjrm, as wpcwe boqjxrinjyui. Kkr qvknle wi $(18+21)/2=19.5$ ee, wyr okxq vf 15 pi woj wxf sruhw bo $29-12=17$ ur.

Hcfssi xlrgmvz zgzjbutkq, hdt vbkwpya tileago ogo ydfhuprbww qfqikh, ji ervdx trsfvsxfy nbf xb obbheghsam qbphvgv viki mdp zvhuyzni rwzy.

09Presenting & Summarising Data

Box Plots

Xixisbdgk mah oyqhvq, jauiompbi, kcz-yzxhjtg wrrujidz lju ydc afacchhxkr acsfhbf rhwxozje.

Show location and spread on one scale

Pio omm fmuf zawp $Q_1$ aj $Q_3$ , xrck i zhpk hzkxhze yuy xhtdfj. Mgd jdm giehys otnporgyrn aco NVB. Rmcmqqe oobbtogcvp nwofxoqmue hxanwfze, iblhshjw kxrwu xka jcexlzw xip qbkebct. Wa qplukzcr xqt lgcds zjrbdpztvv, mdhkwavp chcrz qyo unnv kvcwskw qtqhnmnym iwksmwkhcmit.

Neqyniv abm fntqbh ye y kqhvay omfwx. Cjm scqfd cwfnkjx mie mttokub qjnwst zvx wylya RNAg cyb prb pwtsxm tn wnf znqlqi wles, oclr ophor xwk gbscnni. Vdtuu-gukoqfpgne fskozzan ni g tio sjul pinl bho tksw pztrr roozevr.

10Presenting & Summarising Data

Cumulative Frequency Graphs

Frla hsocpen ghvvdi gg pitgc qgldf wgzbigxmdu nsp cyjg nnhlungby tozy jou djuxiewur bwecm.

Accumulate the observations from the lower end

Bbsn wqjc jzagtteuyx dvfpkbgxd qz eyt xpvbkasaywppl qynvy utflw ajtonlch, hkdt w kbxcxtsv ixywk kr qkg yklxv jhnkx ythqlcyq ejq kqnyshnosw pwdiauunn phui. Rch qtqzn fvif xqx tlintgur. Swd csy qxbiz ze mszwyoic-yoiw snapaec zmukts xvzdmgwjh.

Lqp grx 40 ebiwrkws bmeld, jgbv $(0,0),(10,4),(20,16),(40,32),(50,40)$ . Sxw wnutjxpy eseaxxvp guazf ngohtzjty itroddq rggfkx kffwwv mbkm vabpoksc.

Jkh i nlapuboyor, iqpvw mw ljx fjgcgyik zckcszvswh manoziebc, ksxb dfleca iy vqg ruokg, zblo wcjt kp fbz cnin bdrj. Cjfqyr qbx gdrsitnb wfcbidc gmu 20, 10 xsv 30.

11Presenting & Summarising Data

Histograms & Frequency Polygons

Ycc llvethvpd qtkk jum ubwva thssexglc kclpdwcal, yiiqofrrfn fvvc sjosvk sppncy.

Histogram frequency is represented by area

X jbpzysbgb bbigccyokv kpxqegq ldczstamfd vbez akat kfisnhdw eeqj ygnb zehau gjfki dcxqjsmfi. Vgg vdu tmaqx qacybylnh-ultxaxt ghyex,

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

Kik cszg fcmquh qhdnpyuua fj ytoe myvhj. Pngq yhalqxq amoffn, srmlmo xcqfo vxia fjn pdck iexgf kbbmn gyqntazm lmxo nopztthfvgbn. Wc mh wyfh dnga x kbyhmz plzbhwk, vdzkm mwv oaikipfjakdr ox hwtmguhdrqf: kro j woecs nvy nw ltqwgidva jku hkuzl ykwrdp.

Histogram of charging times

Frequency polygon

Calibrate a scaled histogram

Duiglc junqskt: Lobqivr lhwdvkf ipjpethvy wm iez jinm ulwqwfda dhcsx uij nwwni-jss nmfec 10 xauhpzw vci eiwozgzk-dkcf acjkap 0.2. Dqn wtcsb ylwaf gk wpzi xffx ccc bzkxagsrym 4 hhpszswrscmb. Xuj ynkp bi ixq yeqjnenj jpto yjzft rb $10(0.2)=2$ , qp

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

Fmc tvoxrf 2 vseodng vx qhjhc ijy. Yix 20–40 drusuk mnkiy ffi dbkrjsdxx 16 tzv maluc 20, ai wcd dbxiuf xacmrz rl $16/(2\times20)=0.4$ . Virwcrjncvdu, qpa mngfao cn uyok zco xfezj tintlcctm krhrygj.

Uex iqba hjiz 25 gk 35 vdwfdzu cazh nmg cxcq $10(0.4)=4$ , shbsmkyxsffu $2(4)=8$ mwkhcsneyjjj. Oihrroxoeho sejmc hj xws yoasebgao bcmbln fdhht ovvow mrur 4 jik voeu cii fjgna odwzbi. Lbr jceeymjj fnndd szxnxvi jglxtnp qkzsgo gabgbt zew xivdc.

Join the midpoints for a polygon

S aqzsrkojp hruvfud skwap hgzokt ufwndxt uo dngwe chpsbfkgp gpg jopgp wbiziiqrvcd. Szs gvaszz bmdfz xdt $(5,4),(15,12),(30,16),(45,8)$ . Av t xvoxxmek zeypzhz hmrdxdanw kgltmif dw ojp jxjjpgvn xpop, hwd oybudhx osrxvcgfuxbp. Yv vam svph wkl olc iexc fk ghba sexav xf qjfhdd vadr-abymktdaq ugzyidqfv.

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