Comparisons of the quartile formulae given in high school text books and six others

• Main Author: 黃瑜瑜
• Other Authors: 江振東
• Format: Others
• Language: zh-TW
• Online Access: http://ndltd.ncl.edu.tw/handle/52262015141255568721

碩士 === 國立政治大學 === 應用數學系數學教學碩士在職專班 === 99 === 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. The main findings of the study are summarized as follows: 1.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. 2.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. Key words: Quartiles, Simulation studies