統(tǒng)計(jì)基本概念課件

1、描述、推論統(tǒng)計(jì)學(xué)(一)Fred Li, 2009統(tǒng)計(jì)學(xué)的定義統(tǒng)計(jì)學(xué)是在面對(duì)不確定的情況下，提供人們能做出聰明決策的科學(xué)方法。其過程包括資料的蒐集、整理、呈現(xiàn)結(jié)果、解釋與分析。透過此一過程，並可進(jìn)而根據(jù)分析的結(jié)果加以推論，從而可以獲得合理的研判與有效的結(jié)論。數(shù)理統(tǒng)計(jì)學(xué)與應(yīng)用統(tǒng)計(jì)學(xué)為主要分支Fred Li, 2009統(tǒng)計(jì)學(xué)的主要目的萃取精華且具有意義的資訊 描述或分析各種變項(xiàng)之間的之關(guān)係與差異處理不確定性的問題: 研究由樣本資料推論母群特質(zhì)之可靠機(jī)率或不確定性之大小進(jìn)行有效率之預(yù)測(cè)研究實(shí)驗(yàn)設(shè)計(jì)與統(tǒng)計(jì)方法之關(guān)係Fred Li, 2009統(tǒng)計(jì)資料之要素與來源DATA之三要素:(1)個(gè)案錄(record

2、) (2)變項(xiàng)(variable) (3)觀察值(observation)DATA之主要來源:(1)調(diào)查結(jié)果 (2)實(shí)驗(yàn)結(jié)果 (3)模擬研究Fred Li, 2009統(tǒng)計(jì)分析之主要步驟描述統(tǒng)計(jì):(1)資料搜集(2)記錄與整理 (3)描述性統(tǒng)計(jì)量數(shù)或圖表 (4)解釋與結(jié)論 推論統(tǒng)計(jì):(1)資料抽樣(2)記錄與整理 (3)描述性統(tǒng)計(jì)量數(shù) (4)母群推論 Fred Li, 2009抽樣&推論Fred Li, 2009推論統(tǒng)計(jì)類別參數(shù)估計(jì)(如推估當(dāng)選市長(zhǎng)的機(jī)率)假設(shè)檢定(如檢驗(yàn)事先所提的假設(shè)是否成立)分類(如區(qū)別分析)簡(jiǎn)化資料(如因素分析)。 Fred Li, 2009常見之統(tǒng)計(jì)術(shù)語母群(popula

3、tion)樣本(sample)母數(shù)(parameter)統(tǒng)計(jì)量(statistic)常數(shù)(constant)與變項(xiàng)(variable):(1)自變項(xiàng)與依變項(xiàng)(2)連續(xù)變項(xiàng)與間斷變項(xiàng)(3)依測(cè)量層次分:名義變項(xiàng)、次序變項(xiàng)、 等距變項(xiàng)、比率變項(xiàng)(4)量與質(zhì)的變項(xiàng)Fred Li, 2009統(tǒng)計(jì)基本概念(1)母群(population)：整個(gè)研究對(duì)象的群體或根據(jù)樣本資料所欲推論的研究對(duì)象。母數(shù)(parameter)：描述母群性質(zhì)的量數(shù)，通常以希臘字母表示，例如以表平均數(shù)、表標(biāo)準(zhǔn)差、表相關(guān)係數(shù)、表百分比。通常只有神才知道這些母數(shù)，所以用希臘字母來表示。Fred Li, 2009統(tǒng)計(jì)基本概念(2)樣本 (

4、sample)：由母群中所抽出的部份代表性群體。統(tǒng)計(jì)數(shù)(statistic)：描述樣本性質(zhì)的量數(shù)，通常以英文字母表示，例如以M表平均數(shù)、S表標(biāo)準(zhǔn)差、r表相關(guān)係數(shù)、p表百分比。假如想利用樣本統(tǒng)計(jì)數(shù)來估計(jì)母數(shù)，我們也可稱他為母數(shù)的估計(jì)值(estimate)。Fred Li, 2009統(tǒng)計(jì)基本概念(3)常數(shù)(constant)：群體中之每一份子在某一屬性上均具有一成不變的特質(zhì)。變項(xiàng)(variable)：群體中之每一份子在某一屬性上，分屬不同類別或具有不同的特質(zhì)，前者為質(zhì)的變數(shù)，後者為量的變數(shù)。Fred Li, 2009統(tǒng)計(jì)基本概念(4)自由度(degree of freedom)：乃是變項(xiàng)中可以自由

5、變動(dòng)的數(shù)值之個(gè)數(shù)，通常Df=# of data - # of means used，例如t考驗(yàn)或F考驗(yàn)時(shí)之自由度；或?yàn)閷?shí)際觀察值(observations)個(gè)數(shù)(n)與變項(xiàng)個(gè)數(shù)(k)的函數(shù)，例如迴歸分析時(shí)殘差之自由度：(df=n-k-1)；或?yàn)镕ull model時(shí)預(yù)測(cè)變項(xiàng)個(gè)數(shù)(k)減去Nulll model或Restricted model時(shí)預(yù)測(cè)變項(xiàng)個(gè)數(shù)(h) ，例如迴歸分析時(shí)回歸係數(shù)之自由度(df=k-h)。請(qǐng)問z-test的自由度如何計(jì)算？Fred Li, 2009統(tǒng)計(jì)基本概念(5)對(duì)立假設(shè)(alternative hypothesis) 相當(dāng)於研究假設(shè)虛無假設(shè)(null hypothe

6、sis) 為統(tǒng)計(jì)考驗(yàn)的對(duì)象Fred Li, 2009統(tǒng)計(jì)基本概念(6)第一類型錯(cuò)誤 vs 第二類型錯(cuò)誤Fred Li, 2009描述統(tǒng)計(jì)學(xué): Descriptive Statistics 數(shù)字(Numbers )用以量化待研究資料的特質(zhì)圖表(Tables & Graphs)描繪與摘要待研究資料的有用資訊Fred Li, 2009描述統(tǒng)計(jì)學(xué)的內(nèi)涵集中量數(shù)(central location) 資料點(diǎn)集中情形的最佳代表值離散量數(shù)(dispersion or variation) 資料點(diǎn)的分散程度偏態(tài)與峰度(skewness & kurtosis) 資料點(diǎn)偏離常態(tài)的程度Fred Li, 2009描述統(tǒng)

7、計(jì)的運(yùn)用為推論統(tǒng)計(jì)的依據(jù)應(yīng)報(bào)告每一變項(xiàng)之Mean, sd, range, and n 次數(shù)與百分比亦可合併應(yīng)用Fred Li, 2009選擇適當(dāng)?shù)拿枋鼋y(tǒng)計(jì)常態(tài)分配或無極端值: 平均數(shù)與標(biāo)準(zhǔn)差偏態(tài)或有極端值: 中位數(shù)與四分位數(shù)、盒鬚圖(極大、極小值、Q1、Q2、Q3)遇比率量尺時(shí)，可用幾何平均數(shù)或調(diào)和平均數(shù)Fred Li, 2009描述統(tǒng)計(jì)功能摘要每一變項(xiàng)的重要資訊:Number of casesCentral tendencyDispersion協(xié)助研究者描述變項(xiàng) 推論統(tǒng)計(jì)中，用以分析變項(xiàng)間的關(guān)係與差異Fred Li, 2009Data RecordFred Li, 2009常態(tài)分配曲線為一典

8、型的分?jǐn)?shù)分配曲線Majority of cases distributed around the peak in the middle Progressively fewer cases moving away form the middleSymmetrical one side mirrors the otherMean, median, and mode have the same valueFred Li, 2009常態(tài)曲線Fred Li, 2009Skewed Distributions非對(duì)稱性正偏負(fù)偏Fred Li, 2009個(gè)案人數(shù)以 n (樣本) 或 N(母群)表示例如: n

9、= 231Cases may be people, speaking turns, episodes Fred Li, 2009集中量數(shù)MeanArithmetic mean or averageMost sensitive to extreme scoresMedianMiddle of all scores on one variableModeScore or scores that appear most oftenFred Li, 2009離散量數(shù)Describes the variability or spread of scoresShould be reported with

10、meanRangeHighest to lowest scoreStandard deviation or sdIf sd = 0, all scores are the sameLarger the sd, the more the scores differ from the meanFred Li, 2009Standard DeviationFred Li, 2009標(biāo)準(zhǔn)差的特性Theoretical normal curve is divided into equal standardsThe more normal a distribution of scores, the mor

11、e this theoretical property applies68.26% of scores fall within +1 to 1 standardsFred Li, 2009變項(xiàng)種類Fred Li, 2009不同資料的描述MomentNon-mean based measureCenterMeanMode, medianSpreadVariance (standard deviation)Range,Interquartile rangeSkewSkewness-PeakedKurtosis-Fred Li, 2009平均數(shù)Fred Li, 2009變異數(shù)與標(biāo)準(zhǔn)差Fred Li,

12、 2009母群變異數(shù)與標(biāo)準(zhǔn)差的估計(jì)值Fred Li, 2009變異數(shù)係數(shù)Fred Li, 2009常態(tài)分配圖例如: 全國學(xué)生的大學(xué)入學(xué)成績(jī)Fred Li, 2009負(fù)偏:非對(duì)稱性臺(tái)大學(xué)生的入學(xué)成績(jī)Fred Li, 2009正偏: 非對(duì)稱性落榜學(xué)生的入學(xué)成績(jī)Fred Li, 2009偏態(tài): SkewnessFred Li, 2009SkewnessFred Li, 2009峰度: KurtosisFred Li, 2009常態(tài)屬性Skewness = 0Kurtosis = 3Fred Li, 2009常態(tài)屬性Fred Li, 2009教 學(xué)評(píng)鑑Means.d.SkewKurtGives wel

13、l-prepared, relevant presentations6.00.69-1.78.5Explains clearly and answers questions well5.90.68-1.04.8Uses visual aids well5.60.85-1.88.9Uses information technology effectively5.50.91-1.15.0Speaks well6.10.69-1.56.8Encourages questions & class participation6.10.66-0.883.7Stimulates interest in th

14、e subject5.90.76-1.14.7Is available outside of class for questions5.90.68-1.36.3Fred Li, 2009圖例教學(xué)評(píng)建立提示意圖Means.d.SkewKurtGraphUses visual aids well5.60.85-1.88.9Encourages questions & class participation6.10.66-0.883.7Fred Li, 2009圖例: 加註常態(tài)曲線Fred Li, 2009盒鬚圖中位數(shù)PR=25PR=75非極端值下限非極端值上限極端值Fred Li, 2009圖表法

15、:Tabular and Graphical Methods.探索式資料分析交叉表與散佈圖xyFred Li, 2009Qualitative DataQuantitative Data TabularMethods TabularMethodsGraphical MethodsGraphical MethodsFrequency DistributionRel. Freq. Dist.Percent Freq. DistributionCrosstabulationBar GraphPie ChartFrequency DistributionRel. Freq. Dist.Cum. Fre

16、q. Dist.Cum. Rel. Freq. Distribution Stem-and-Leaf DisplayCrosstabulationHistogramOgiveScatter DiagramDataFred Li, 2009探索式資料分析 包含能夠快速摘要資料的簡(jiǎn)單算術(shù)及簡(jiǎn)易製圖 例如: 莖葉圖: (the stem-and-leaf plot) 次數(shù)多邊圖 圓餅圖 直方圖Fred Li, 2009莖葉圖的製作 Each digit on a stem is a leaf. Each line in the display is referred to as a stem. To

17、 the right of the vertical line we record the last digit for each item in rank order. The first digits of each data item are arranged to the left of a vertical line. 可保留原始資料值 莖葉圖可以顯示資料的次序關(guān)係與分配型態(tài)Fred Li, 2009實(shí)例: 小學(xué)生的智力成績(jī)林森國小李老師為了解55名學(xué)生在魏氏兒童智力量表上的成績(jī)Fred Li, 2009莖葉圖5678910 2 7 2 2 2 2 5 6 7 8 8 8 9 9 9

18、 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 0 0 2 3 5 8 9 1 3 7 7 7 8 9 1 4 5 5 9a stema leafFred Li, 2009Stretched 莖葉圖 Whenever a stem value is stated twice, the first value corresponds to leaf values of 0 - 4, and the second value corresponds to leaf values of 5 - 9. If we believe the original stem-and-leaf

19、display has condensed the data too much, we can stretch the display by using two stems for each leading digit(s).Fred Li, 2009Stretched 莖葉圖5 5 91 47 7 7 8 91 35 8 90 0 2 35 5 5 6 7 8 9 9 91 1 2 2 3 4 45 6 7 8 8 8 9 9 92 2 2 27255667788991010Fred Li, 2009Stretched 莖葉圖Leaf Units Where the leaf unit is

20、 not shown, it is assumed to equal 1. Leaf units may be 100, 10, 1, 0.1, and so on. In the preceding example, the leaf unit was 1. A single digit is used to define each leaf.Fred Li, 2009 Leaf Unit = 0.1 8 91011Leaf Unit = 0.16 81 420 78.6 11.79.49.110.211.08.8Fred Li, 2009Leaf Unit = 10 16 17 18 19

21、Leaf Unit = 1081 90 31 71806171719741791168219101838The 82 in 1682is rounded downto 80 and isrepresented as an 8.Fred Li, 2009雙向交叉表與散佈圖 雙向交叉表與散佈圖: (Crosstabulation and a scatter diagram) 同時(shí)處理雙變項(xiàng)(或以上)間的關(guān)係Fred Li, 2009雙向列表 左上與正上方通常列出變項(xiàng)的類別 使用時(shí)機(jī): 一個(gè)變項(xiàng)為質(zhì)的變項(xiàng), 另一個(gè)變項(xiàng)為量的變項(xiàng) 兩個(gè)變項(xiàng)皆為質(zhì)的變項(xiàng) 兩個(gè)變項(xiàng)皆為量的變項(xiàng)Fred Li, 2009學(xué)費(fèi) 市立 私立 綜合 縣立Total $9,00018 1 19 12505030 20 35 15Total10012 19 16 3 學(xué)校類別實(shí)例: 國中生每學(xué)期的學(xué)費(fèi)quantitative variablequalitat