Why do we measure central tendency? Basic Concepts in Statistical Analysis

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1 Why do we measure central tendency? Basic Concepts in Statistical Analysis Chapter 4 Too many numbers Simplification of data Descriptive purposes What is central tendency? Measure of central tendency A representative value Arithmetic mean (mean) Most useful How do we measure it? (4-1) Σ stands for instruction take the sum of Y represents the individual scores in the set Y is the mean of values of the variable Y N indicates total number of scores or observations in the set (4-1) 5, 6, 9, 11, 5, 11, 8, 14, 2, 11 (4-1) 5, 6, 9, 11, 5, 11, 8, 14, 2, 11 Obtain the sum of the scores, Y Y = 82 Divide the sum by the number of scores 82/10 = 8.2 1

2 Mean An algebraic balancing point Deviation of the scores below exactly matches the deviation of the scores above Sum of 9 negative deviations (-148) exactly equals the sum of the 14 positive deviation (+148) Limitations Does central tendency tell the whole story? What about variability? What is variability? (4-2) Why measure variability? What does the distribution look like? Two examples: Age Performance How do we measure variability? Expressed as deviations from the mean To what degree do scores in the set deviate from the typical score? 1, 6, 4, 3, 8, 7, 6 2

3 How do we measure variability? 1, 6, 4, 3, 8, 7, 6 To what degree do scores in the set deviate from the typical score? 7 scores Y = 35 Mean = 5 Which are closest to mean? Which are farthest? Σ(Y - Y) N Problems? Σ(Y - Y) = 0 Indices of Variability Numerator would always be 0 First Step: Sum of Squares Problem Solution How can we always get a positive? Sum of squares (SS) Sum of squared deviations from the mean Takes on values that reflect variability 0 when variability is absent Positive values when variability is present First Step: Sum of Squares Sum of squares (SS) Sum of squared deviations from the mean 0 when variability is absent Positive values when variability is present Computational formula 1: SS = Σ (Y - Y) 2 Sum of Squares = (1-5) 2 + (6-5) 2 + (4-5) 2 + (3-5) 2 + (8-5) 2+ (7-5) 2 + (6-5) 2 = (-4) 2 + (1) 2 + (-1) 2 + (-2) 2 + (3) 2 + (2) 2 + (1) 2 = = 36 Sum of Squares Computational formula 2: Two sums: (1) the sum of the actual or raw score (ΣY) (2) the sum of the squared raw scores (ΣY 2 ) SS = ΣY 2 - (ΣY) 2 N 3

4 Sum of Squares SS = ΣY 2 - (ΣY) 2 N ΣY = = 35 ΣY 2 = = 211 SS = (35) 2 7 = = = 36 (identical to value obtained with formula 1) Sum of Squares Using the mean to calculate deviations Results in the smallest possible value for SS Using any other mean would result in an SS greater than 36 How do we measure variability? Variance The distance of numbers from the mean Samples with high variance vs. low variance How do we measure variability? Variance Dividing the sum of squares either by: The total number of scores N A related quantity, the degrees of freedom (df) Population Sample The Variance Population A descriptive version of the variance Sample Inferential version of the variance What are degrees of freedom? Number of restraints Restrictions on the freedom Example: Y = 5.0; 7 scores entering into the mean (4-5) 4

5 Why do we calculate degrees of freedom? Populations vs. samples Scaling How do we calculate degrees of freedom? In our example with 7 scores df = 7-1 = 6 (4-6) Calculating the Variance ˆ Indicates that computational result is a variance estimate Descriptive variance Divides by N rather than N-1 Slightly smaller 2 (4-7) ^ Calculating the Variance σ 2 = SS df = 36 = 6 6 Calculating the Variance The Standard Deviation Sample data are used to estimate population data Descriptive variance slightly underestimates population variance Divided by N Estimated variance removes this bias Divided by N-1 Standard deviation The square root of the variance (4-8) 5

6 The Standard Deviation σ = = 2.45 Standard deviation rather than variance unsquared units WHY? Original scale of measurement Questions 1. What is measured by each of these: SS Variance Standard deviation Questions 2. Can SS ever have a value less than zero? 3. There are two different formulas or methods that can be used to calculate SS. When is formula 1 easier to use? When is formula 2 preferred? Questions 4. On an exam with a mean of 75, you obtain a score of 80. Would you prefer that the exam distribution had a SD of 2 or an SD of 10? If your score is 70, would you prefer an SD of 2 or an SD of 10? 6

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