方差用于衡量一个样本数据波动的大小,计公式为:S~2=1/n[(x_1-(?))~2+(x_2-(?))~2+…+(x_n-(?))~2]=1/n[x_1~2+x_2~2+…+x_n~2-1/n(x_1+x_2+…+x_n)~2]。显然S~2≥0,仅当S~2=0时,x_1=x_2=…=x_n。例1已知实数x,y满足求xy的最大值。解视x,y为一组数据,其方差为S~2=1/2[x~2+y~2-1/2(x+y)~2]=-1/4a~2+1/2a+3/4≥0。即(a+1)(a-3)≤0,所以或解得-1≤a≤3.所以xy=(x+y)~2-(x~2+y~2)/2=5/2(a-2/5)~2-9/10。当a=3时,xy有最大值,为16。例2已知a,b,c三数满足方程组 The variance is used to measure the fluctuation of a sample data. The formula is: S~2=1/n[(x_1-(?))~2+(x_2-(?))~2+...+(x_n-(? ))~2]=1/n[x_1~2+x_2~2+...+x_n~2-1/n(x_1+x_2+...+x_n)~2]. Obviously S~2≥0, only when S~2=0, x_1=x_2=...=x_n. Example 1 knows that the real number x,y satisfies the maximum value of xy. The solution x, y is a set of data, the variance is S~2=1/2[x~2+y~2-1/2(x+y)~2]=-1/4a~2+1/ 2a+3/4≥0. That is, (a+1)(a-3)≤0, so the solution is -1≤a≤3. So xy=(x+y)~2-(x~2+y~2)/2=5/ 2(a-2/5)~2-9/10. When a=3, xy has a maximum value of 16. Example 2 is known that the three numbers a, b, and c satisfy the equations