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Statistics for the Social Sciences Psychology 340 Spring 2010 Describing Distributions & Locating scores & Transforming distributions PSY 340 Statistics for the Social Sciences Announcements • Homework #1: due today • Quiz problems – Quiz 1 is now posted, due date extended to Tu, Jan 26th (by 11:00) – Quiz 2 is now posted, due Th Jan 28th (1 week from today) • Don’t forget Homework 2 is due Tu (Jan 26) PSY 340 Statistics for the Social Sciences Outline (for week) • Characteristics of Distributions – Finishing up using graphs – Using numbers (center and variability) • Descriptive statistics decision tree • Locating scores: z-scores and other transformations PSY 340 Statistics for the Social Sciences Standard deviation • The standard deviation is the most commonly used measure of variability. – The standard deviation measures how far off all of the scores in the distribution are from the mean of the distribution. – Essentially, the average of the deviations. m PSY 340 Statistics for the Social Sciences Computing standard deviation (population) • To review: – Step 1: compute deviation scores – Step 2: compute the SS • SS = Σ (X - μ)2 – Step 3: determine the variance • take the average of the squared deviations • divide the SS by the N – Step 4: determine the standard deviation • take the square root of the variance PSY 340 Statistics for the Social Sciences Computing standard deviation (sample) • The basic procedure is the same. – Step 1: compute deviation scores – Step 2: compute the SS – Step 3: determine the variance • This step is different – Step 4: determine the standard deviation PSY 340 Statistics for the Social Sciences Computing standard deviation (sample) • Step 1: Compute the deviation scores – subtract the sample mean from every individual in our distribution. Our sample 2, 4, 6, 8 X 2 4 6 8 20 X 5.0 n 4 4 X - X = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 1 2 3 4 5 6 7 8 9 10 X PSY 340 Statistics for the Social Sciences Computing standard deviation (sample) • Step 2: Determine the sum of the squared deviations (SS). X - X = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 SS = Σ (X - X)2 = (-3)2 + (-1)2 + (+1)2 + (+3)2 = 9 + 1 + 1 + 9 = 20 Apart from notational differences the procedure is the same as before PSY 340 Statistics for the Social Sciences Computing standard deviation (sample) • Step 3: Determine the variance Recall: Population variance = σ2 = SS/N The variability of the samples is typically smaller than the population’s variability X4 X1 X3 X2 PSY 340 Statistics for the Social Sciences Computing standard deviation (sample) • Step 3: Determine the variance Recall: Population variance = σ2 = SS/N The variability of the samples is typically smaller than the population’s variability To correct for this we divide by (n-1) instead of just n Sample variance = s2 SS n 1 PSY 340 Statistics for the Social Sciences Computing standard deviation (sample) • Step 4: Determine the standard deviation X X 2 standard deviation = s = s 2 n 1 PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score Standard deviation changes changes – Changes the total and the number of scores, this will change the mean and the standard deviation X m N 2 X m N PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score – All of the scores change by the same constant. Xold Standard deviation changes PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score – All of the scores change by the same constant. Xold Standard deviation changes PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score – All of the scores change by the same constant. Xold Standard deviation changes PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score – All of the scores change by the same constant. Xold Standard deviation changes PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes – All of the scores change by the same constant. – But so does the mean Xnew Standard deviation changes PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean • Change/add/delete a given score changes • Add/subtract a constant to each score changes Standard deviation changes – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes No change – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold Xnew PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score • Multiply/divide a constant to each score changes No change 21 - 22 = -1 23 - 22 = +1 20 21 22 23 24 (-1)2 (+1)2 X X 2 s= X n 1 2 1.41 PSY 340 Statistics for the Social Sciences Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score • Multiply/divide a constant to each score – Multiply scores by 2 changes No change changes changes 42 - 44 = -2 46 - 44 = +2 40 42 44 46 48 (-2)2 (+2)2 X X 2 s= X n 1 8 2.82 Sold=1.41 PSY 340 Statistics for the Social Sciences Locating a score • Where is our raw score within the distribution? – The natural choice of reference is the mean (since it is usually easy to find). • So we’ll subtract the mean from the score (find the deviation score). X m – The direction will be given to us by the negative or positive sign on the deviation score – Thedistance is the value of the deviation score PSY 340 Statistics for the Social Sciences Locating a score Reference point m 100 X1 = 162 X2 = 57 X m X 1 - 100 = +62 X2 - 100 = -43 Direction PSY 340 Statistics for the Social Sciences Locating a score Reference point Below X1 = 162 X2 = 57 m 100 X m X 1 - 100 = +62 X2 - 100 = -43 Above PSY 340 Transforming a score Statistics for the Social Sciences – The distance is the value of the deviation score • However, this distance is measured with the units of measurement of the score. • Convert the score to a standard (neutral) score. In this case a z-score. Raw score z X m Population mean Population standard deviation PSY 340 Transforming scores Statistics for the Social Sciences m 100 50 z X m X1 = 162 X1 - 100 = +1.20 50 X2 = 57 X2 - 100 = -0.86 50 A z-score specifies the precise location of each X value within a distribution. • Direction: The sign of the z-score (+ or -) signifies whether the score is above the mean or below the mean. • Distance: The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and σ. PSY 340 Statistics for the Social Sciences Transforming a distribution • We can transform all of the scores in a distribution – We can transform any & all observations to z-scores if we know either the distribution mean and standard deviation. – We call this transformed distribution a standardized distribution. • Standardized distributions are used to make dissimilar distributions comparable. – e.g., your height and weight • One of the most common standardized distributions is the Zdistribution. PSY 340 Statistics for the Social Sciences Properties of the z-score distribution m 100 50 m0 z X m transformation 50 150 zmean Xmean = 100 100 100 50 =0 PSY 340 Statistics for the Social Sciences Properties of the z-score distribution m 100 50 m0 z X m transformation 50 150 100 100 50 150 100 50 Xmean = 100 zmean =0 X+1std = 150 z1std = +1 +1 PSY 340 Properties of the z-score distribution Statistics for the Social Sciences m 100 50 z m0 1 X m transformation 50 150 100 100 50 150 100 z1std 50 50 100 z1std 50 zmean Xmean = 100 X+1std = 150 X-1std = 50 -1 =0 = +1 = -1 +1 PSY 340 Statistics for the Social Sciences Properties of the z-score distribution • Shape - the shape of the z-score distribution will be exactly the same as the original distribution of raw scores. Every score stays in the exact same position relative to every other score in the distribution. • Mean - when raw scores are transformed into z-scores, the mean will always = 0. • The standard deviation - when any distribution of raw scores is transformed into z-scores the standard deviation will always = 1. PSY 340 Statistics for the Social Sciences From z to raw score • We can also transform a z-score back into a raw score if we know the mean and standard deviation information of the original distribution. Z X m m 100 50 Z X m X Z m m0 1 X Z m transformation 50 m X = 70 150 -1 X = (-0.60)( 50) + 100 m +1 Z = -0.60 PSY 340 Statistics for the Social Sciences Why transform distributions? • Known properties – Shape - the shape of the z-score distribution will be exactly the same as the original distribution of raw scores. Every score stays in the exact same position relative to every other score in the distribution. – Mean - when raw scores are transformed into z-scores, the mean will always = 0. – The standard deviation - when any distribution of raw scores is transformed into z-scores the standard deviation will always = 1. • Can use these known properties to locate scores relative to the entire distribution – Area under the curve corresponds to proportions (or probabilities) PSY 340 Statistics for the Social Sciences SPSS • There are lots of ways to get SPSS to compute measures of center and variability – Descriptive statistics menu – Compare means menu – Also typically under various ‘options’ parts of the different analyses • Can also get z-score transformation of entire distribution using the descriptives option under the descriptive statistics menu
