國高中教材及其它六種常見四分位數公式之比較研究

Abstract

本研究的主要目的是探究四分位數定義在國中、高中教材不同情形下，所可能衍生出來的差異情況，並統整有關四分位數之八組公式，以第一、第三四分位數及四分位距觀點來探討不同公式適用於何種資料型態。\n本研究發現如下：\n一、國中、高中在四分位數之計算結果的確有差異，尤其當原始資料之個數為N＝4k + 1(k為正整數)時，藉由高中版本的計算公式所算出來的Q1與Q3，與藉由國中版本所計算出來的25及75百分位數可能有所不同。其餘資料個數情況，似乎並不會發生不一致的現象。\n二、假設原始資料分別來自五種右偏程度不一的分配，包括卡方分配\n(〖" χ" 〗_"(1)﹐" ^"2" 〖 χ〗_"(5)﹐" ^2 χ_"(10)" ^2 )、指數分配(Exp"(" λ="10)" ﹐Exp"(" λ="15)" )以及平均數為100﹐\n標準差為10的常態分配這一種對稱的型式。在不同樣本數(N＝4k、4k+ 1、4k + 2、4k + 3)及不同資料分配的情況下，透過模擬實驗的方式，並依據四項評估指標（偏誤、標準差、均方根誤差、平均相對誤差百分比）來探討這八組公式的適用性。雖然各家公式的版本不盡相同，但大體上而言，在大樣本時的表現差不多，僅在小樣本時(N=40、41、42、43)呈現出明顯差異。卻因為在不同資料分配下，八組公式的表現不一，所以無法歸納出較具體的結論。相對而言，國中、高中公式的表現儘管稱不上是最好，不過就學生的學習而言，卻是清晰易懂。\n 關鍵詞：四分位數公式、模擬比較

The main purpose of this study is to explore the possible implications for different definitions of quartiles taught in high schools. In addition, the two formulae are also compared with six others that are found popular in literature according to their performances under different distributions. \n \nThe main findings of the study are summarized as follows:\n1.Differences are found in the calculation of quartile based on the definition used in junior and senior high schools. Specifically, when the data size is N=4k+1 (k is a positive integer), Q1 and Q3 calculated using the formula taught in senior high schools may not be same as the 25th and 75th percentiles, the way junior high students are taught to calculate Q1 and Q3. \n2.Assuming that data come from five right-skewed distributions ( , , ), chi-squared distributions with degrees of freedom 1, 5, and 10, and Exp( ), Exp( ), exponential distributions with means 10 and 15, and a symmetrical distribution (normal distribution with mean 100 and standard deviation 10), simulation studies are carried out to assess the performances of the eight formulae in terms of bias, standard deviation, root mean square error and average relative error. Generally speaking, no differences are found when the sample sizes are large. Noticeable differences, on the other hand, are found under the situations of small sample sizes, particularly when N=40, 41, 42, and 43. However, since the performances varied dramatically, there appears no clear winner among the eight formulae. Although the performances of the two formulae taught in junior high and senior high schools scarcely perform as the “best”, they are easily understood, and clearly appropriate for use in teaching high school students the concept of quartiles. \nKey words: Quartiles, Simulation studies