若x1,x2···xn的方差为S²,则S²=(x1²+x2²+···xn²)÷n-x拔²,求证
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![若x1,x2···xn的方差为S²,则S²=(x1²+x2²+···xn²)÷n-x拔²,求证](/uploads/image/z/7206884-44-4.jpg?t=%E8%8B%A5x1%2Cx2%C2%B7%C2%B7%C2%B7xn%E7%9A%84%E6%96%B9%E5%B7%AE%E4%B8%BAS%26%23178%3B%2C%E5%88%99S%26%23178%3B%3D%EF%BC%88x1%26%23178%3B%2Bx2%26%23178%3B%2B%C2%B7%C2%B7%C2%B7xn%26%23178%3B%EF%BC%89%C3%B7n-x%E6%8B%94%26%23178%3B%2C%E6%B1%82%E8%AF%81)
若x1,x2···xn的方差为S²,则S²=(x1²+x2²+···xn²)÷n-x拔²,求证
若x1,x2···xn的方差为S²,则S²=(x1²+x2²+···xn²)÷n-x拔²,求证
若x1,x2···xn的方差为S²,则S²=(x1²+x2²+···xn²)÷n-x拔²,求证
根据定义知:
x1+x2+...+xn=n*x拔
S^2=[(x1-x拔)^2+(x2-x拔)^2...+(xn-x拔)^2]÷n
=[x1²+x2²+···xn²+n*x拔^2-2x拔*(x1+x2+...+xn)]÷n
=[x1²+x2²+···xn²+n*x拔^2-2x拔*nx拔)]÷n
=[x1²+x2²+···xn²-n*x拔^2]÷n
=(x1²+x2²+···xn²)÷n-x拔²