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UT Knoxville STAT 201 - 1) basic_stats_review

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The Shape of a Sampling DistributionThe Mean of a Sampling DistributionThe Standard Error (Variability) of a Sampling DistributionTreatment is not effectiveTreatment is effectiveAssume treatment is effectiveAssume treatment is not effectiveCourse: Analysis of Variance Topic: Basic Statistics Review 1 BASIC STATISTICS REVIEWWe’ll begin with a review of basic statistics. Hopefully, this will bring us all up to date sothat we are on the “same page” when we discuss more complex issues. This basic review will briefly highlight important background information. If you feel that you require more information, I suggest that you review an undergraduate statistics book, such as Gravetter, F. J. &Wallnau, L. B. (2000). Statistics for the behavioral sciences (5th ed.). Australia: Wadsworth. CHARACTERISTICS OF DISTRIBUTIONSDistributions of data are characterized in terms of shape, central tendency, and variability.ShapeThe height (Y Axis) of a distribution typically reflects the frequency or relative frequency of particular scores (X axis) in the distribution.Symmetric Distributions. A symmetrical distribution is one in which a straight line divides the distribution into mirror images. Skewed distributions. A skewed distribution has scores that are more frequent at one end of the distribution than the other. A positively skewed distribution has the majority of scores on the negative end of the distribution and a few scores trail to the positive end (e.g., a distribution of income—most persons earn $20,000-$80,000 a small number of individuals earn millions per year).A negatively skewed distribution has the majority of scores on the positive end and few scores trail to the negative end (e.g., a distribution of self-esteem—most persons feel positively about themselves, a small portion of persons feel negatively about themselves).Course: Analysis of Variance Topic: Basic Statistics Review 2 Central TendencyMeasures of central tendency provide an estimate of the center of the distribution. Two commonly used measures of central tendency are the mean and the mode.Mean The mean is simply the average score. That is, the mean is computed by summing the values of each score and dividing by the total number of scores. Greek letters represents population parameters and the mean of the population is expressed as NX. Sample statistics are represented by Arabic letters and the sample statistic is expressed as: NXXE.g.,For a population of five scores (2, 5, 8, 9, 10) the mean is 8.65109852NXAn important characteristic of the mean is its sensitivity to extreme scores. For example, if the value of 10 in the above example were changed to a value of 100, the mean would shift from 6.8 to 24.8. This suggests that extreme scores can have a dramatic affect on the “average” scoreWeighted mean. Often, researchers combine information from multiple samples and are interested in the average score of the combined sample. A weighted mean adjusts the mean score across samples by the number of scores in each sample. That is, larger samples are weighted more heavily than are smaller samples (another way of thinking of this, which is reflected in the right most formula below, is that the weighted mean simply sums across the scores in each sample and divides by the total number of scores – when reading the formula keep in mind that via algebra  nXX *).E.g.,Bob watches on average 2 hours of television Monday through Firday and 4 hours of television on the weekend. What is the average amount of television Bob watches in a week? If your answer is 3, then you did not take into account the relative size of the samples. The meanfor the weekdays (M =2) is based upon a sample size of 5 (i.e., Mon- Fri), whereas the mean forthe weekend (M=4) is based upon a sample size of 2 (i.e., Sat & Sun). A weighted mean takes into account the number of scores in each sample:57.27)4*2()2*5()4(72)2(75)()(212122121211 nnXXXnnnXnnnanweightedmeMedianThe median is the score that divides the distribution in half. That is, it is the value at which 50% of the scores are above and below (i.e., the 50th percentile). E.g.,In the population (2, 5, 8, 9, 10) the median is 8. There are numerous formulas, which provide different values for the median, when there are repeated values in the middle of the distribution. An important characteristic of the median is that, unlike the mean, it is unaffected by extreme scores. Notice in the above example if the 10 were changed to a 100 the median would remain 8.Course: Analysis of Variance Topic: Basic Statistics Review 3 ModeThe mode is the most frequently occurring score. There can be several modes.E.g., In the population (2, 5, 5, 8, 9,9) 5 and 9 are the modes and the distribution is bimodal.In the population (2, 5, 5, 5, 9,9) 5 is the mode and the distribution is unimodal.VariabilityVariability reflects the variation in the scores. That is, are all of the scores within a distribution similar or are they different?E.g.,Population A (5,6,7,8,9) and Population B (7,7,7,7,7) both have a mean of 7, however the populations differ in terms of the extent to which the scores within the distribution are similar.There are several measures of variation, such as range and semi-interquartile range. In this class, however, we will focus primarily on standard deviation (and it’s squared value, the variance).Standard deviation. measures the average distance of the scores from the mean. There are separate formulas for the standard deviation of a population () and sample (s).NX2)( 1)(2nXXsThe difference between the two formulas (aside from the notation for the mean) is the denominator of the formula. The sample formula divides by n-1, which adjusts for the tendency for samples to underestimate the variability of populations, and the population formula divides by N. The numerator of both formulas sums the squared deviations of each score from the mean. Conceptually, the formula provides the average deviation of scores from the mean.The numerator of the formula is also known as sums of squares (SS). So standard deviation can also be expressed as NSS and 1nSSsThe square root function puts the standard deviation in the same unit of measurement as the scores in the distribution.Variance is simply the standard deviation squared. NSS2


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UT Knoxville STAT 201 - 1) basic_stats_review

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