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CHAPTER 1: THE MEAN, THE NUMBER OF OBSERVATIONS, THE VARIANCE, AND THE STANDARD DEVIATION Data – observations that involve measurements – “scores” o Data set – distribution Statistics – consists of series of rules and methods used to organize and interpret data o Descriptive statistics – describe and summarize data summary o Inferential statistics – make inferences about large populations from samples draw conclusions Population – comprises all individuals (and scores) that are of interest in study o Population parameters – numbers we use to describe population characteristics Use Greek alphabet Sample – comprises part of population selected for study – representation of population o Sample statistics – summary numbers describing samples; used to estimate population parameters Use English alphabet Sampling error – difference between sample statistic and its population parameter caused by random error in measuring sample o Difference between mean of sample and mean of population Sampling fluctuation – difference among samples caused by random factors o Individual differences and measurement problems o Difference between means of two+ samples Each score is symbolized by letter X Average (or mean) of all scores in population is call mu – predict everyone score at mean Distance (or deviation) of score from mean is (X – μ) Variance – average amount wrong if predict everyone score at mean o “mean square for error” o N – 1 st population parameter that tells us how many scores or observations there are denoted μ – 2 nd population parameter tells us mean σ 2 – 3 rd population parameter tells us variance “A measure of central tendency” – most typical scores o Mean, median, mode Population mean - Most used measures of variability o range, interquartile range, variance, standard deviation Deviation – distance between score and mean SS – “sum of squared deviations around the mean” o Σ(X-μ) 2 Variance – “mean squared deviation” o Standard deviation – “average unsquared distance, standard distance from mean”
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o Steps for computing variance and standard deviation 1. Add up all scores ΣX 2. Divide by N to find μ 3. Subtract μ from each score to compute deviations around mean (X- 4. Add up deviations and make sure they sum to 0 Σ(X-=0 5. Square each deviation 6. Add up squared deviations to find Sum of Squares 7. Divide Sum of Squares by N to obtain Variance, sigma 2 8. Take square root of Variance to obtain Standard Deviation, sigma Mean – unbiased predictor or unbiased estimate; deviations around it sum to zero o Least squares, unbiased predictor – number that is minimum average squared distance from number it estimates o Unbiased estimate – one around which deviations sum to zero o Mean squared error – sigma 2 2 =(ID 2 +MP 2 ) CHAPTER 2: FREQUENCY DISTRIBUTIONS AND HISTOGRAMS
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