4,5,7,12,34 are the data points. Median is 7. But, why mean would be the balancing point here? I know the mean distribute the data points evenly among individuals, which means if any individual would’ve scored any point, he’d have scored the mean . But, I don’t understand this balancing point thing.
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Imagine we are placing identical weights on a number line at the points $4,5,7,12\text{ and }34$. We want to place a fulcrum on the line so that the line doesn't tip due to the weights. For example, if we placed the fulcrum at $0$ then, since all the weights are to the right of it, the line would tip to the right.
We must place the fulcrum in a way so that the tendency for the weights to tip the line to the right equals the tendency for the weights on the other side of the fulcrum to tip the line to the left. This is like a teeter totter. The tipping force (torque) is equal to the weight times the distance of the weight from the fulcrum.
Let $\bar x$ be the position of the fulcrum and assume each of the weights is $w$. Then we get $$(\bar x-4)w+(\bar x-5)w+(\bar x-7)w+(\bar x-12)w+(\bar x-34)w=0$$
We can divide both sides of the equation by $w$ and collect terms to get $$5\bar x=4+5+7+12+34\implies\bar x=\frac{4+5+7+12+34}{5}$$ which is the expression for the average of those numbers.
John Douma
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That really helps. – Vinay Sharma Jan 23 '22 at 11:43