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UW-Madison PSYCH 210 - Central Tendency

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PSYCH 210 Lecture 4Outline of Last Lecture I. Finish Summation NotationII. Frequency DistributionsIII. GraphingIV. Percentiles and Percentile RanksOutline of Current Lecture II. Percentiles and Percentile Ranksa. InterpolationIII. Central Tendencya. Modeb. Meanc. MedianCurrent LectureI. Percentiles and Percentile Ranksa. Descriptive Statisticsb. Used for relative standing in a populationc. Percentilei. Raw Score (x)ii. Score associated with a given cumulative percentage (C%)d. Percentile ranki. A cumulative percentage (C%)ii. The percentage of scores falling at or below a given raw (x) scoree. Ex) What is the raw score (x) associated with the 95th percentile rank?i. Table1. Raw scores are NOT class intervals!2. If data is continuous, C% will be associated with upper real limit ofintervala. Class Interval from table correlating with 0.95 C%: 60-69b. Raw score = 69.5f. Ex) What is raw score (x) associated with the 50th percentile rankThese 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.i. 50th percentile not listed in tableii. Interpolation - Educated guess, using method to obtain best estimate1. Set up what we know from the tablex C%49.5 (URL) .60? .5039.5 (URL) .302. 49.5-39.5 = 10a. Unit span between top and bottom X3. 60%-30% = 30 unit spana. Unit span between top and bottom C%4. 60-50 = 10a. Unit span between top and middle (desired) C%i. Can also do bottom to middle C% span!5. 10/30 = x/101/3=x/10x=3.3336. 49.5 - 3.333 = 46.17Raw score = 46.17g. What is the percentile rank for a raw score of x=45i. Interpolation1. Two ways to think about it: LRL, URL –OR-a. URL of CI, URL of next lower class intervalx C%49.5 .6045 ?39.5 .302. .60 - .30 = 30 unit span3. 49.5-39.5 = 10 unit span4. 49.5 – 45 = 4.5 units5. 4.5/10 = x/30(.45)30= x = 13.56. 60 – 13.5 = C% = 46.5% or 0.465II. Central Tendencya. Summary Statistics for ‘most typical’ score in distribution (defined in 3 different fashions)i. Mode1. The most common score in a distribution (highest frequency)2. Ex) 3, 5, 5, 7, 7, 7, 9, 10, 38a. Mode: 73. Advantages:a. Easy to find, rarely calculations4. Disadvantages:a. Doesn’t tell us much and sometimes doesn’t ‘work’i. Ex) 3, 5, 5, 7, 7, 7, 9, 10, 381. Mode: 5, 7ii. Ex) 3, 5, 7, 9, 10, 381. No mode5. GFDa. Midpoint of Class Interval with highest frequencyii. Mean 1. Average of all scores2. Population mean: μ = Σx/Na. N = whole set of scores in a population3. Sample mean: M = Σx/na. n = whole set of scores in a sampleb. Same calculation, different notation4. SFD x f fx 10 2 209 0 08 3 247 0 06 4 245 8 404 6 243 4 122 3 61 2 2Σfx/n = 152/32=47.5a. Remember to account for frequency!b.c.5. M1 = 45 sec n = 8 for femalesM2 = 36 sec n = 14 for malesIncorrect Method: [(45+36)/2] = 40.5Must use Weighted Mean!WM (Weighted Mean) = (ΣX1 + ΣX2)/(n1+n2)OR: WM = M1 [n1/(n1+n2)] + M2 [n2/(n1+n2)]= 45 (8/22) + 36 (14/22)WM = 39.27a. Will always have a formula sheet for exam! …Just need to know how to use formula (with a few noted exceptions)2. The mean as a balance point for the distributiona. See Textbook3. The sample mean is an unbiased estimate of the population meana. Ex) Population: N=1000μ=45b. Take a number of random samples of size n=100 from population and calculate meanc. Sampling with Replacementi. Slips CAN be drawn again after being chosend. Means of samples clustered around 45i. Average of samples means would = μ (Population Mean)e. When we do experiments, we cannot access the entire populationi. We must use samples, but we want to know about entire populationii. This shows us that we can use sample data as a good representative of entire populationf. Advantage: Takes into account every value in data setg. Disadvantage: There are several scenarios where it doesn’tgive us accurate informationiii. Median1. The middle score in a distribution2. How to find the median with a raw data set and NO TIES near the middle of the distributiona. Odd number of scoresi. Ex) 12, 10, 8, 3, 11. Rank Order2. Median = 8b. Even number of scoresi. Ex) 15, 12, 10, 8, 3, 110, 8 >>> Median = 93. How to find the median of SFD WITH TIES toward middlea. Multiple frequenciesb. Assume data is Continuousc. Interpolationi. Continue example next


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