Lecture 4VariabilityStandard DeviationSlide 4Logic of the Standard Deviation: Let’s start by looking at the populationSlide 6Slide 7Slide 8Putting it TogetherPopulation Standard DeviationFormulas for Pop. SD and VarianceLet’s Do It TogetherAnother Example…Samples vs. PopulationsTermsAn Analogy for a Biased StatSamples: s and Let’s Do it TogetherStart Easy: Find sA little more complexSample Variability and Degrees of Freedom: Why do we correct with n-1?More about n -1Degrees of FreedomCafeteria degrees of freedom: An analogySlide 25A little more about biased statsProperties of the Standard DeviationSlide 28Slide 29Factors that affect VariabilityRelationship with other StatisticsWhy we need to know this informationSlide 33Graphical Representation of Graphic Representation - Box PlotsGraphic Representation - BoxplotsPearson’s Coefficient of SkewTry onePutting it all together…Homework: Chapter 4Lecture 4Variability: Standard DeviationVariabilityReminder - How spread out the scores are…Range - How does the range of each of these distributions vary? Or the Interquartile range?Measure of error - is our sample similar to the population OR is an individual score representative of its sampleStandard DeviationStandard deviation - the average distance on either side of the mean.Goal of the SD is to measure the standard or typical distance from the mean.–But it’s not practical with large N, so we need to estimate the variance and standard deviation using equations606264666870727476Ben Tom Bill James MattHeight (in.)• Mean = 70.8•Ben is 66 in. tall. His deviation from the mean is -4.8.•James is 75 in. tall. His deviation from the mean is 4.2How much scores typically vary around the mean; a measure of dispersionUsually 1/5 - 1/6 of the range Based on the mean, therefore:–Requires at least interval data–Sensitive to outliers–accounts for all scores in a distributionStandard Deviationf1 2 3 4 5 6 7 98MLogic of the Standard Deviation:Let’s start by looking at the population Step 1: Find the Deviation for each score from the mean. X - . Be sure to include both the sign (+/-) and the number.X X - 65 -1490 +1184 +576 -381 +298 +1982 +356 -23 = 790* Notice that the sum of the deviations = 0. This reflects the fact that the mean is a balancing point* Bonus - you can use this fact to check yourselvesStep 2 - Remember the standard deviation is the average of the deviations, but this won’t work because the sum of our deviations = 0–Solution = get rid of the signs (+/-)–Square each scoreSquare of each score and sum them = Sum of Squared Deviations = SSXX -  (X – )265 -14.4 207.490 10.6 112.484 4.6 21.276 -3.4 11.681 1.6 2.698 18.6 346.082 2.6 6.859 -20.4 416.2X = 79.4 0 1123.9* Sum of Squared Deviations = SSStep 3 - Calculate the mean squared deviation = SS / NThis value is called the variance and is represented with the symbol MS or 2 .Variance will be important for use in inferential stats methods, but it isn’t the best descriptive stat. -- it’s hard to visualize variability with the variance alone.XX -  (X – )265 -14.4 207.490 10.6 112.484 4.6 21.276 -3.4 11.681 1.6 2.698 18.6 346.082 2.6 6.859 -20.4 416.2X = 79.4 0 1123.9MS = 1123.9 / 8 = 140.5* Sum of Squared Deviations = SSStep 4: Correct for having squared all the deviations because we want a value that easily corresponds to the mean that we can visualize:–Standard deviation = varianceX -  (X – )2207.4112.4346.0416.21123.9X65 -14.490 10.684 4.6 21.276 -3.4 11.681 1.6 2.698 18.682 2.6 6.859 -20.4X = 79.4 0140.5 = 11.9Standard deviation = the square root of the mean squared deviationConceptually the average distance from the mean: on average a random point pulled from this distribution will be 11.9 away from the mean.Putting it TogetherX -  (X – )2207.4112.4346.0416.21123.9X65 -14.490 10.684 4.6 21.276 -3.4 11.681 1.6 2.698 18.682 2.6 6.859 -20.4X = 79.4 0  = 11.9 What can we say about a score that lies 12 points from the mean, 91 points? What about a score that lies 30 points from the mean, 49 points?REVIEW: variance = mean squared deviation = greek lower case letter sigma 2 = SS / NStandard deviation =  = SS/ NComputing SS:–Definitional formula: SS = (X - )2 Shows exactly how scores vary about the mean (like we just did). Works best on whole numbers.– Computational formula: SS = X2 - [ (X)2 / N] Easier for calculations because it works directly with the scores, but less intuitive about the mean. Population Standard DeviationFormulas for Pop. SD and VarianceVariance = SS / N (mean squared deviation)Standard deviation = SS/NDenoted by Greek letters  and 2Let’s Do It TogetherX X -  (X - )2 X2 (X)2 2 2424283233486442386755455-17.4-17.4-13.4-9.4-8.46.622.60.6-3.425.613.60302.8302.8179.688.470.643.6510.8.3611.6655.418523515765767841024108923044096176414444489302521171207025 213.7 14.6Definitional:SS = (X - )2Computational: SS = X2 - [ (X)2 / N]Another Example…Find  for the following sets of numbersX = 1, 7, 7, 9X = 1, 6, 1, 1, 1, 1 X X2 (X)2 2 10151721243113Definitional:SS = (X - )2Computational: SS = X2 - [ (X)2 / N]Samples vs. PopulationsRationale: Inferential statistics rely on samples to draw general conclusions about the population.–PROBLEM - sample variability tends to be less than population variability.–Thus, this variability is biased. That is, it underestimates the pop. variability. pop. variabilityxxxxxxsample variabilityTermsBiased - a sample statistic is said to be biased if on the average the sample statistic consistently underestimates or overestimates the population parameter.Unbiased - a sample statistic is said to be unbiased if on average the sample statistics is equal to the population parameterAn Analogy for a Biased StatImagine you were interested in studying learning in elementary school children.–What if you chose as your sample child geniuses from computer and science camp?–Could you generalize from your sample to the population of elementary school children?A sample statistic for SD will be biased even with a representative sample - We have to perform a correctionSamples: s and Changes in notation to reflect a sample:–So to calculate SS (same