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UW-Madison PSYCH 210 - Variability Continued

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PSYCH 210 Lecture 6 Outline of Last Lecture I. Central Tendencya. MedianII. Relation between Mean, Median and ModeIII. VariabilityOutline of Current Lecture I. Continue variabilitya. Disadvantages of SIQb. Variance and standard deviationII. Z-scoresCurrent LectureI. SIQa. When to use?i. Disadvantage1. Only considers two points from distributionii. Often used in cases where Median must be used (Ordinal Data)II. Variancea. Unlike Range and SIQ, takes all values into accountb. Population vs. sample variancec. Deviation scoresd. Ex) This is a population:X 10, 10, 9, 8, 6, 6, 4, 3 μ= 7 (Mean of a Population)X - μ 3, 3, 2, 1, -1, -1, -3, -4 <<<DEVIATION SCORE i. If x – μ is a single deviation score, can we find the average deviation for an entire dataset?Σ (X – μ)/N …doesn’t work: Σ (X –μ) = 01. Mean is ‘balancing point’ so it will always equal zero2. How can we get around the ‘summing to zero’ problem?(X – μ)2 9, 9, 4, 1, 1, 1, 9, 16 Σ(X – μ)2 = 50These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.σ2 = Σ(X – μ)2/N 50/8 = 6.25b. Formula for Population Variancei. Definitional Formula1. Should be able to explain in English what this formula means! (Taking an average of all the individual deviation scores…)2. NOT as useful for calculating for large datasets!! (Do not use!)ii. Computational Formulaσ2= [Σx2 – (Σx)2/N]/NX2 = 100, 100, 81, 64, 36, 36, 16, 9 Σx2 = 442σ2 = [442-(56)2/8]/8 = 6.25c. Def Variance: Average squared deviation scorei. The larger the number, the greater the spreadd. Advantagei. Takes into account entire data setIII. Standard Deviationa. √σ2 = σb. √6.25 = 2.5i. Always must fall between original highest and lowest deviation scores of data set IV. How well does sample variance estimate population variance?a. Is sample variance an unbiased estimate of population variance?i. Sample Variance: s2 = Σ (x-M)2/n(Wrong Formula)1. No; Biased; Underestimatesii. Sample Variance: Definitional Formula:s2(n-1) = Σ (x-M)2/(n-1)1. Yes, Sample variance IS an unbiased estimate of population variance, if we take degrees of freedom into account2. A ‘correction factor’ to account for sample variance otherwise underestimating population varianceb. Degrees of Freedom (df)i. Ex) Candy Bar Bucket1. 7 candy bars; 7 people2. Only First had choice of all, last had no choice3. 6 people had SOME choice (n-1)ii. Def: The number of scores in a distribution whose values are free to vary, when the total set of scores meets some constraint1. Ex) n=10 M=1014, 17, 4, 12, 8, 2, 7, 5, 11, ?Constraint = mean of 10Σ1-9=80M = Σx/nΣx = (M)n = (10)10 = Σx = 100Σ10=20 2. 10th value NOT free to vary3. In sample of 10, degrees of freedom is 9iii. Sample Variance: Computational Formula:s2 = [Σx2 – (Σx)2/n]/(n-1) = ss/(n-1) *ss means “sum of squares” = numerator of Variance formulaV. Constants, Central Tendency and Variabilitya. Adding or subtracting a constant to every score in a dataseti. How does this affect:1. Mean, Variance, Standard Deviation?ii. Ex) C = 2 (add)X 2, 4, 6, 8, 10 M = 6X +2 4, 6, 8, 10, 12 M = 8a. Adding Constanti. Mnew = Mold + Cii. s2new = s2old1. Only shifted data; variance remains sameiii. snew = soldb. Subtracting Constanti. Mnew = Mold – Cii. s2new = s2oldiii. snew = soldb. Multiplying by a constanti. Ex) Same data set and constanta. Multiplying Constanti. Μnew = Mold (C)ii. s2new = s2old(C2)iii. snew = sold (C)2. Note: These ‘formulas’ NOT on sheet for


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UW-Madison PSYCH 210 - Variability Continued

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