Statistics and Tchebysheff's theorum

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In summary: If so, wouldn't that be cheating?In summary, the theorem states that if there are M measurements, the fraction of measurements that falls within the interval -ks to +ks is at least 1-1/k2. The theorem can be used to bound the fraction of measurements, which is 1.0-\frac{M}{n}.
  • #1
major_maths
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Homework Statement



Let k[itex]\geq[/itex]1. Show that, for any set of n measurements, the fraction included in the interval [itex]\overline{y}[/itex]-ks to [itex]\overline{y}[/itex]+ks is at least (1-1/k2).

[Hint: s2 = 1/(n-1)[[itex]\sum[/itex](yi-[itex]\overline{y}[/itex])2]. In this expression, replace all deviations for which the absolute value of (yi-[itex]\overline{y}[/itex])[itex]\geq[/itex]ks with ks. Simplify.] This result is known as Tchebysheff's theorem.

2. Homework Equations are the above.

The Attempt at a Solution



I've got no clue what the problem wants, much less how to start a solution.
 
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  • #2
major_maths said:

Homework Statement



Let k[itex]\geq[/itex]1. Show that, for any set of n measurements, the fraction included in the interval [itex]\overline{y}[/itex]-ks to [itex]\overline{y}[/itex]+ks is at least (1-1/k2).

[Hint: s2 = 1/(n-1)[[itex]\sum[/itex](yi-[itex]\overline{y}[/itex])2]. In this expression, replace all deviations for which the absolute value of (yi-[itex]\overline{y}[/itex])[itex]\geq[/itex]ks with ks. Simplify.] This result is known as Tchebysheff's theorem.

Let there be [itex] M [/itex] measurements where [itex]| y_i - \overline{y}| \geq ks[/itex]
If in the sum [itex] \sum(y_i -\overline{y})^2 [/itex] we replace those [itex]M[/itex] measurements by [itex] ks [/itex] and leave out the other [itex] N-M [/itex] measurements, we get a smaller sum. The smaller sum is [itex] M (ks)^2 [/itex]

Hence

[tex] s^2 = \frac{1}{n-1} \sum(y_i - \overline{y})^2 \geq \frac{1}{n-1} M (ks)^2 [/tex]

Since [itex] \frac{1}{n-1} > \frac{1}{n} [/itex]

[tex] s^2 \geq \frac{1}{n-1}M(ks)^2 > \frac{1}{n}M(ks)^2 [/tex]
[tex] s^2 \geq \frac{1}{n}M(ks)^2 [/tex]

The "fraction of measurements" that [itex] M [/itex] constitutes is [itex] \frac{M}{n} [/itex] and the above inequality can be used to bound it.

The original problem concerns the fraction of measurements other than those M measurements, so that fraction is [itex] 1.0 - \frac{M}{n} [/itex].
That needs to be bounded by using the bound for [itex] \frac{M}{n} [/itex].
 
  • #3
Thank you Stephen. That was part of my homework I was struggling with. I wonder which school OP goes :-).

To be really pedantic, should not the last equation have > sign instead of >=?
 
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  • #4
When we take a look at the definition of theorem number two, we see that the theorem refers to the standard deviation of the possible sample means computed from all possible random samples. Theorem number one is similar in that it says for any population, the average value of all possible sample means computed from all possible random samples of a given size from the population equal the population mean. What does that mean? Does that mean that the mean of my sample will automatically be equal to the population mean?
 

1. What is Statistics?

Statistics is a branch of mathematics that deals with collecting, analyzing, and interpreting data. It involves methods for describing, summarizing, and making predictions from numerical data.

2. What is Tchebysheff's Theorem?

Tchebysheff's Theorem is a statistical theorem that states that for any given data set, the proportion of data points that fall within a certain number of standard deviations from the mean is at least (1-1/k^2), where k is any positive number greater than 1.

3. How is Tchebysheff's Theorem useful in statistics?

Tchebysheff's Theorem is useful in statistics because it provides a way to determine the percentage of data that falls within a certain range, regardless of the shape of the data distribution. This allows for a more accurate understanding of the data and can help in making predictions and decisions based on the data.

4. Can Tchebysheff's Theorem be applied to any data set?

Yes, Tchebysheff's Theorem can be applied to any data set regardless of its distribution. This is one of the main advantages of this theorem as it is not limited to only normal distributions like other statistical methods.

5. What are the limitations of Tchebysheff's Theorem?

One limitation of Tchebysheff's Theorem is that it provides only a minimum bound for the proportion of data that falls within a certain range. This means that the actual percentage of data within the range could be higher. Additionally, the theorem does not provide any information about the specific values within the range, only the proportion of data points. Therefore, it should be used in conjunction with other statistical methods for a more comprehensive analysis.

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