[Math] Standard Deviation and Mean using Chebyshev’s Theorem

meanspercentagesstandard deviationstatistics

Find the standard deviation and mean when The sale prices of 88.9% of the homes in a certain neighborhood were between \$438,572 and \$189,992.

From Chebyshev's theorem it is known that 88.9% is 3 standard deviations(k) from the mean.

(mean+kstd dev ) = upper bound
(mean-kstd dev ) = lower bound

I was going to use the Upper and Lower Bound equation to find the inverse so I can get the mean but I am missing the standard deviation that goes with the mean. I don't know any other way to try and solve this

Any help is greatly appreciated!

Best Answer

HINT

So let $m,s$ be mean and std dev, you have $k=3$ and your equations yield $$ m + 3s = 438572\\ m - 3s = 189992 $$ Can you solve 2 equations for 2 unknowns?

HINT 2 What happens when you add the equations to each other?

Related Solutions

[Math] Calculating mean and standard deviation of very large sample sizes

Posting as an answer in response to comments.

Here's a way to compute the mean and standard deviation in one pass over the file. (Pseudocode.)

n = r1 = r2 = 0;
while (more_samples()) {
    s = next_sample();
    n += 1;
    r1 += s;
    r2 += s*s;
}
mean = r1 / n;
stddev = sqrt(r2/n - (mean * mean));

Essentially, you keep a running total of the sum of the samples and the sum of their squares. This lets you easily compute the standard deviation at the end.

[Math] Understanding Chebyshev’s Theorem

Chebyshev's inequality applies to any discrete or continuous distribution which meets the conditions. It is not particularly meaningful for some which do not, for example those which do not have a mean.

On your second question, what you really have in (1) is a statement of the form $\Pr(X \in A) \le c$; this clearly implies $\Pr(X \not\in A) \ge 1 - c$, which is the corresponding form of (2). So you need to use the opposite of "$\gt$", which is "$\le$": if you had had "$\ge$" inside the $\Pr(\cdots)$ in (1), the (2) would have needed "$\lt$" inside.

Best Answer

Related Solutions

[Math] Calculating mean and standard deviation of very large sample sizes

[Math] Understanding Chebyshev’s Theorem

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