Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura

Transcription

1 Statstcs Descrptve ad Iferetal Statstcs Istructor: Dasuke Nagakura 1

2 Today s topc Today, I talk about two categores of statstcal aalyses, descrptve statstcs ad feretal statstcs, ad ther dffereces, ad the, the basc dea of feretal statstcs. 2

3 Descrptve ad Iferetal Statstcs What s statstcs? The prcple objectve of statstcs s to fd, verfy, ad aalyze, the latet characterstcs, laws, ad regularty behd a gve data set, ad to develop a method for those purposes. 3

4 Descrptve ad Iferetal Statstcs Descrptve Statstcs The purpose of descrptve statstcs s to descrbe or summarze the characterstcs of a gve data. A well kow example s a arthmetc mea. Other examples clude meda ad mode, whch wll be explaed later today s lecture. 4

5 Descrptve ad Iferetal Statstcs Iferetal Statstcs I may cases, the data s obtaed for ot whole objects that we wat to aalyze, but oly a part of them, ca be obtaed From a part of the whole, we eed to fer the characterstcs that apply to whole objects, whch s oe of the ma objectves of feretal statstcs. 5

6 Descrptve ad Iferetal Statstcs Examples of Descrptve ad Iferetal Statstcs Suppose ow that we wat to aalyze the qualty of electrc bulbs made a factory. The data (observatos) here for ths purpose s the lfetmes of partcular twety electrc bulbs. For example, we observe that the lfetme of bulb o.1 was 150 hours, the lfetme of bulb o.2 was 130 hours, the lfetme of bulbs o.20 was 90 hours, etc. 6

7 Descrptve ad Iferetal Statstcs Now, temporarly, suppose that we are just terested a cetral tedecy of the lfetmes of the 20 electrc bulbs. It would be dffcult to realze t just by seeg the data of the 20 electrc bulbs. It s coveet to calculate a umber that summarzes t. I such a case, the most frequetly used oe s the well kow statstc called (sample) mea. Here, the word sample s a syoym for the word data or observatos, ad statstc s a quatty or a umber that s calculated from a sample. 7

8 Descrptve ad Iferetal Statstcs Suppose that the average of the lfetmes of the 20 electrc bulbs s 120 hours. Roughly speakg, ths mples that most of the lfetmes of these 20 bulbs are about 120 hours. The, we foud oe feature of the lfetmes of these 20 electrc bulbs, or could descrbe a characterstc of the data. 8

9 Descrptve ad Iferetal Statstcs However, what we are really terested s usually ot about the lfetmes of these partcular 20 electrc bulbs, but rather about the average lfetmes of whole electrc bulbs made ths factory. Here, ca we coclude that the average lfetme of the electrc bulbs made ths factory s 120 hours from the fact that the average lfetme of these just 20 electrc bulbs (a part of the whole) s 120 hours? The aswer s, of course, No. 9

10 Descrptve ad Iferetal Statstcs How to fer wth the data of oly these 20 electrc bulbs about the average lfetme of the whole electrc bulbs made ths factory s a problem cosdered feretal statstcs. I the framework of feretal statstcs, the average lfetme of the 20 bulbs s regarded as a estmate of the average of lfetmes of whole bulbs made ths factory. A estmate s ot the true value, but t cotas a error. 10

11 Descrptve ad Iferetal Statstcs Whe we regard the average of these 20 bulbs as a estmate, we would be terested how accurately t s? because the cocluso of the aalyss would be dfferet depedg o the sze of errors. Also, we would be more terested how ca we estmate t makg the errors as small as possble. Iferetal statstcs deals wth these kds of problems. 11

12 Populato ad Sample Populato ad Sample A Populato s a etre set of dvduals or objects of terest, or a mechasm that geerates a object of terest (example: lfetmes of all electrc bulbs made a factory), ad a sample s a part of the populato of terest (example: lfetmes of partcular 20 electrc bulbs). Whole sample If we ca get the data of every ut a populato, the we say that we have whole sample. 12

13 Populato ad Sample Complete Survey ad Sample Survey If we ca obta whole sample by a survey, the the survey s called a complete survey. O the other had, we just ca collect a sample by a survey, the the survey s called a sample survey. A complete survey may cases takes much tme ad costs a lot, or s mpossble some cases (such as a case of electrc bulb lfetme because we caot observe the lfetmes of whole bulbs). 13

14 Populato ad Sample Examples of Complete Survey ad Sample Survey (Example of complete survey) Natoal populato cesus: s a survey that vestgates all people lvg Japa every fve years. (Example of sample survey) Household survey: s a survey coducted for fve thousadth part of all households except for farmers to asses ther lvgs. 14

15 Populato ad Sample Radom Sample For a survey to be meagful, how to collect a sample s ofte crucal. Oe of the typcal ways to collect a sample s a radom samplg, ad sample collected ths way s called a radom sample. Radom samplg s a way to pck a sample from a populato stochastcally depedetly. I wll expla about stochastc depedece, or probablstc depedece, later ths course. Itutvely, a radom samplg s to pull each ut of a sample from a populato wthout ay bas, or uformly. 15

16 Populato ad Sample Example that faled to do a radom samplg What happes f we fal to do a radom samplg? A famous example for ths s a survey coducted durg the U.S. presdetal electo I 1936, Frakl Roosevelt ad Alfred Lado were competto for the Amerca presdecy. A ewspaper compay coducted a sample survey based o 2.4 mllo voters (today s typcal poltcal survey asks betwee 500&1000 respodets), ad predcted that Lado wll w wth a majorty 57%. It tured out that, however, Roosevelt wo wth a majorty 62%! What was wrog wth ths survey? 16

17 Populato ad Sample The problem was that the sample s based! The ewspaper compay selected people for the survey from telephoe books ad car regstratos. I 1936, oly relatvely rch households had telephoes ad far less people owed a car Amerca. Also may of such rch people was supportg the republca party, or Lado. Therefore the survey predcted the vctory of Lado. However, ths survey was very based to a partcular group of people, ad faled to let a part represet the whose. By the way, the ewspaper compay folded after the electo. 17

18 Descrptve Statstcs Ceter of the Data Sample Mea Most of you would be famlar the word average or mea, whch s wdely used daly lfe stuatos, such as average score, average temperature, ad average rafall, etc. Ths word s deed a statstcal termology. 18

19 Descrptve Statstcs Ceter of the Data Defto of the Sample Mea Suppose that we have a sample cosstg of observatos { x 1, x 2, x }. The, the sample mea s defed as x We ofte use the otato lke x 1 x 2... x x for a sample mea. 19

20 Descrptve Statstcs Ceter of the Data Iterpretato of Sample Mea The value of sample mea mples that aroud that value most data pots are dstrbuted. I other words, t expresses the ceter of the data. 20

21 Descrptve Statstcs Ceter of the Data Σ (sgma) otato The umerator of the defto of sample mea ca be cocsely expressed by usg Σ otato. The Σ otato expresses the summato of varables x 1 + x x as x 1, or 1 x... The, the sample mea s wrtte as x 1 x 2 x. x 1 x 1. 21

22 Descrptve Statstcs Ceter of the Data Propertes of Σ I statstcs, we very ofte use Σ otato. We revew some of ts propertes here. (1) Multplyg x by a costat c ad take the summato, we have 1 cx c 1 x. 22

23 23 23 Descrptve Statstcs Ceter of the Data (2) For a costat c, we have (3) For a par of observatos, { y 1,, y } ad { x 1,, x }, we have. 1 c c. ) ( y x y x

24 24 24 Descrptve Statstcs Ceter of the Data Usg these propertes, for example, we ca derve. 2 ) ( c x c x c x

25 Descrptve Statstcs Ceter of the Data Meda Not oly the mea s used to see the ceter of the data. Meda s also frequetly used to measure the ceter of the data. It s obtaed as the value that s the ceter of the data arraged descedg order whe the umber of observato s odd. Whe s eve, the meda s the average of two mddle values. 25

26 Descrptve Statstcs Ceter of the Data Example of Meda Whe s Odd Suppose we observed { x 1, x 2,, x 5 } = { 7, 9, 4, 2, 5 }. Frst, arragg the data descedg order, we have { 2, 4, 5, 7, 9 }. The, the meda s the value located the ceter: { 2, 4, 5, 7, 9 } That s, the meda s 5. For ths data, the sample mea s 5.4 (the meda ad average are ot ecessarly equal). 26

27 Descrptve Statstcs Ceter of the Data Example of Meda Whe s Eve Suppose we observed { x 1, x 2,, x 6 } = { 10, 7, 9, 4, 2, 5 }. Aga, arrage the data descedg order to have { 2, 4, 5, 7, 9, 10 }. Next, take the average of the two mddle values to get { 2, 4, 5, 7, 9, 10 } (5 + 7)/2 = 6. We obta the meda of 6. For ths data, the sample mea s 37/

28 Descrptve Statstcs Ceter of the Data Meda for a Geeral Case Arrage the data { x 1, x 2,, x } a descedg order to obta { x (1), x (2),., x () }, where x () s the -th smallest value amog { x 1, x 2,, x }. Whe s odd, the meda s expressed as x ((+1)/2). Whe s eve, the meda s equal to (x (/2) +x (/2 + 1) )/2. 28

29 Descrptve Statstcs Ceter of the Data Robustess agast Outler Mea ad meda both measure the ceter of the data (or to fd aroud whch value most data pots are dstrbuted). Meda has a property that t s robust agast the exstece of outlers. 29

30 Descrptve Statstcs Ceter of the Data For example, suppose that aual comes of a group of fve people are gve as {5, 6, 7, 8, 9} (mllo ye). The, the mea ad meda are both 7 mllos ye for ths group. Now suppose that a trlloare wth aual come of 1trllo ye jos ths group, ad the the comes of ths group are gve as {5, 6, 7, 8, 9, } (mllo ye). For ths group, the mea come s bllos ye, but the meda s 7.5 mllos ye. Whch better descrbes the value aroud whch comes of people ths group are dstrbuted? 30

31 Descrptve Statstcs Ceter of the Data Mode A mode s the value that appears most ofte a sample. For example, for the data of { 3, 10, 7, 9, 4, 3, 5 }, we see that there are two 3, or 3 s the value most ofte observed ths data set. Thus, the mode of the data s 3. For ths data, the mea s 5.9, ad the meda s 5. 31

32 Descrptve Statstcs Ceter of the Data Ulke the mea ad meda, more tha oe mode may exst. For example, for the data { 2, 10, 7, 9, 4, 2, 5, 9 }, 2 ad 9 are both observed twce, ad so the values, 2 ad 9 are both modes of ths data set. 32

33 Descrptve Statstcs Ceter of the Data Questo A survey asked people of 5 males ad 5 females how may tmes they shop at coveece stores per week. Ther aswers are as follows. Aswers of males: {5, 2, 3, 3, 4} Aswers of females: {1, 1, 5, 4, 4} Calculate (1) the mea, meda, ad mode of the males, (2) the mea, meda, ad mode of the females, ad (3) the mea, meda, ad mode of these males ad females. 33