Understanding Statistics 1#11 Chebyshev Theorem
- 2. Chebyshev theorem - Main purpose of Chebyshev theorem is to address non bell shape distributions. - it is a rule of thumb for non bell shape. It is not exact as depend of the shape. - Only give us a minimum that apply. It give us a lower value: “at least x% will fall within …” Chebyshev Theorem: the % of the data that lies within “k” standard deviation is AT LEAST 1- 𝟏 𝒌 𝟐 for k>1 For k=2: 1- 1 22 = 3/4 => 75% lie within 2ϭ of the mean For k=3: 1- 1 32 = 8/9 => 88.9% lie within 3ϭ of the mean Note: Does not apply for 1ϭ
- 3. Chebyshev theorem Problem: In a town, the average income is $34,200 with std Dev of $2200. what percent of homes earn between $29,800 and $38,600? - We can not apply theory for bell shape as it is not stated as bell shape - µ = $34,200 ϭ = 2200 - µ+ ϭ=$36,400; µ+ 2ϭ=$38,600 - µ-2ϭ=$29800 AT LEAST 75% lies in this range. Note “at least” is important as it can be more than 75% Std dev of $58. What % have a bill Problem: Water bills in Jan average $230 with a std dev of $58. what percent have a bill between $56 and $404 -not bell shape stated µ = $230 ϭ=$58 µ+ 3ϭ=$404 µ- 3ϭ=$56 Lie within 3ϭ => at least 88.9 lie within the range