Chebyshev's theorem (statistics)

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SUMMARY

Chebyshev's theorem provides a method to predict the number of students within a specified range of standard deviations from the mean. For 100 students taking a quiz, applying the formula 1 - 1/k² with k set to 2 yields a prediction that at least 75% of students will score within ±2 standard deviations from the mean. This means that a minimum of 75 students can be expected to fall within this range. It is crucial to note that this result represents a lower bound on the percentage of students within the specified range.

PREREQUISITES
  • Understanding of Chebyshev's theorem
  • Basic knowledge of standard deviation
  • Familiarity with statistical concepts such as mean and variance
  • Ability to perform basic algebraic calculations
NEXT STEPS
  • Study the implications of Chebyshev's theorem in different statistical contexts
  • Learn about the application of standard deviation in data analysis
  • Explore other statistical theorems for comparison, such as the Central Limit Theorem
  • Practice calculating probabilities using Chebyshev's theorem with various datasets
USEFUL FOR

Students in statistics courses, educators teaching statistical concepts, and data analysts seeking to understand the application of Chebyshev's theorem in real-world scenarios.

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Homework Statement


If 100 students take a quiz, use chebyshev's theorem to predict the number of students plus and minus 2 standard deviations from the mean.

Homework Equations


1-1/k^2 where k is standard deviations

The Attempt at a Solution


I think its 75, but I'm not sure...I'm just trying to get this concept down
 
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staples82 said:

Homework Statement


If 100 students take a quiz, use chebyshev's theorem to predict the number of students plus and minus 2 standard deviations from the mean.

Homework Equations


1-1/k^2 where k is standard deviations
Don't just memorize formulas, learn what they say! Did you notice that "1- 1/k2" is not even an equation? What is equal to 1- 1/k2?

The Attempt at a Solution


I think its 75, but I'm not sure...I'm just trying to get this concept down
1- 1/22= 1- 1/4= 3/4. 3/4 of 100= 75. Now, if "the fraction of trials within k standard deviations of the mean" is what 1- 1/k2 gives, you are completely correct!
 
Last edited by a moderator:
Chebyshev's Theorem does indeed state give the percentage data you can expect to find within \pm k standard deviations of the mean, as long as k > 1 (and it is here.
However, remember that technically the answer is a lower bound, so you the proper response not that the percentage is 75\%, but that it is at least 75\%.
 

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