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Stats 1 Formula Measure of location: π₯Μ vs ο Sample π₯Μ = π₯Μ = π₯Μ = Population β π₯π π π= β π₯π ππ β ππ π= β(π₯π β π) +π π π= β π₯π π β π₯π ππ β ππ β(π₯π β π) +π π Measure of spread: Sample π 2 = Population 1 β(π₯ β π₯Μ )2 πβ1 π2 = (β π₯)2 1 2 π = {β π₯ β } πβ1 π 2 π 2 = π2 = (β ππ₯)2 1 {β ππ₯ 2 β } πβ1 π SD ο½ ο³ ο½ var 1 β(π₯ β π₯Μ )2 π 1 (β π₯ 2 ) β π 2 π (β ππ₯) 1 π = (β ππ₯ 2 ) β { } π π 2 2 ο var ο½ ο³ 2 1 standard deviation includes 2/3 of data 2 standard deviations include 95% of data 3 standard deviations include βalmost allβ data βa quantity expressing by how much the members of a group differ from the mean value for the groupβ Probability Mutually Exclusive Non-Mutually Exclusive π(π΄ βͺ π΅) = π(π΄) + π(π΅) π(π΄ β© π΅) = π(π΄) × π(π΅) π(π΄ βͺ π΅) = π(π΄) + π(π΅) β π(π΄ β© π΅) π(π΄ β© π΅) = π(π΄ | π΅) × π(π΅) ο π(π΄ β© π΅) π(π΄ | π΅) = π(π΅) π(π΄ β© π΅ β© πΆ) = π(π΄) × π(π΅) × π(πΆ) Exclusive β no overlap of possible events Exhaustive β all possible events are accounted for If any of the following are true then they are all true & events are independent: π(π΄ | π΅) = π(π΄ | π΅β²) π(π΅| π΄) = π(π΅ | π΄β²) π(π΄ | π΅) = π(π΄) π(π΅ | π΄) = π(π΅) π(π΄ | π΅β²) = π(π΄) π(π΅ | π΄β²) = π(π΅) π(π΄ β© π΅) = π(π΄) × π(π΅) Binomial distribution (a discrete distribution): X ~ B ( n, p ) P( X ο½ x)ο½ n C x p x q n ο x Expectation & Variance of Binomial distribution: X ~ B ( n, p ) E ( x) ο½ ο ο½ np Var( x) ο½ ο³ 2 ο½ npq Normal distribution (a continuous distribution): π~π(π, π 2 ) The standard normal distribution: π~π(0 , 1) Conversion: π = πβπ π To find ο and/or ο³ using given probabilities: 1. Draw a sketch or curve 2. Use the probabilities table to find corresponding z value 3. Use the z value together with the conversion to find ο and/or ο³. Finding both ο and ο³ will involve solving simultaneous equations. Estimation Sample mean = π₯Μ Sample variance = π 2 Unbiased estimator of population mean = πΜ Unbiased estimator of population variance = πΜ (ie the distribution of many sample means) (ie the distribution of many sample variances) πΜ ~π (π, π2 ) π πΜ = πΜ β π π ~π(0,1) βπ Confidence Interval 90% 95% 98% 99% 99.8% z 1.6449 1.9600 2.3263 2.5758 3.0902 π₯Μ ± "z" π βπ Product Moment Correlation Coefficient: rο½ S xy S xx S yy S xy ο½ ο₯ xi y i ο S xx ο½ ο₯ xi ο 2 1 ο¨ο₯ xi ο©2 n 1 ο₯ xi ο₯ y i n S yy ο½ ο₯ y i ο 2 1 ο¨ο₯ yi ο©2 n Regression: y on x x on y y ο½ a ο« bx x ο½ a 'ο«b' y a ο½ y ο bx a ' ο½ x ο b' y S xy b' ο½ S yy bο½ S xy S xx