七個同學由矮至高排列,若此七個同學的平均身高為155.5公分,而最矮的4個同學平均身高為152公分,最高的4位同學平均身高為159.5公分,請問這七位同學中位數為多少公分?

Sol
a1,a2,a3,a4,a5,a6,a7
(a1+a2+a3+a4+a5=a6+a7)/7=155.5
a1+a2+a3+a4=a5+a6+a7=1088.5
(a1+a2+a3+a4)/4=152
a1+a2+a3+a4=608
(a4+a5+a6+a7)/4=159.5
a4+a5+a6+a7=638
(a1+a2+a3+a4)+(a4+a5+a6+a7)=608+638
(a1+a2+a3+a4+a5+a6+a7)+a4=1246
1088.5+a4=1246
a4=157.5
(7+1)/2=4
中位數為157.5公分

a b c d e f g
(shortest -> highest)

since average of these students are 155.5cm,
a+b+c+d+e+f+g=155.5cm x 7 = 1088.5cm(total height sum) ... [1]

since average of a b c d is 152cm
a+b+c+d=152cm x 4 = 608cm(short peoples height sum) ... [2]

since average of d e f g is 159.5cm
d+e+f+g=159.5cm x 4 = 638cm(high peoples height sum) ... [3]

中位數(median)
=d
=(a+b+c+d) + (d+e+f+g) - (a+b+c+d+e+f+g)
=[1]+[2]-[3]
=608cm+638cm-1088.5cm
=1246cm-1088.5cm=157.5cm

設七位同學為a,b,c,d,e,f,g
a+b+c+d+e+f+g=155.5*7=1088.5
a+b+c+d=152*4=608
d+e+f+g=159.5*4=638
中位數=638+608-1088.5=157.5公分