Statistics for the Social Sciences Psychology 340 Spring 2010 Describing Distributions & Locating scores & Transforming distributions

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)

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

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

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

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

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 1 2 3 4 5 6 7 8 9 10 X X - X = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3

Computing standard deviation (sample) Step 2: Determine the sum of the squared deviations (SS). 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 = (-3)2 + (-1)2 + (+1)2 + (+3)2 = 9 + 1 + 1 + 9 = 20 X - X = deviation scores SS = Σ (X - X)2 Apart from notational differences the procedure is the same as before

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 μ X 3 X 1 X 4 X 2

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

Computing standard deviation (sample) Step 4: Determine the standard deviation standard deviation = s =

Properties of means and standard deviations Change/add/delete a given score changes changes Changes the total and the number of scores, this will change the mean and the standard deviation

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score All of the scores change by the same constant. X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score All of the scores change by the same constant. X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score All of the scores change by the same constant. X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score All of the scores change by the same constant. X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes All of the scores change by the same constant. But so does the mean X new

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same X old

Properties of means and standard deviations 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 X old X new

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes No change Multiply/divide a constant to each score 21 - 22 = -1 (-1)2 20 21 22 23 24 X 23 - 22 = +1 (+1)2 s =

Properties of means and standard deviations Change/add/delete a given score changes changes Add/subtract a constant to each score changes No change Multiply scores by 2 Multiply/divide a constant to each score changes changes 42 - 44 = -2 (-2)2 40 42 44 46 48 X 46 - 44 = +2 (+2)2 Sold=1.41 s =

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). The direction will be given to us by the negative or positive sign on the deviation score The distance is the value of the deviation score

Locating a score μ Reference point X1 = 162 X1 - 100 = +62 Direction

Locating a score μ Reference point Below Above X1 = 162 X1 - 100 = +62

Transforming a score 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 Population mean Population standard deviation

Transforming scores μ X1 = 162 X1 - 100 = +1.20 50 X2 = 57 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. X1 = 162 X1 - 100 = +1.20 50 X2 = 57 X2 - 100 = -0.86 50

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 Z-distribution.

Properties of the z-score distribution μ μ transformation Xmean = 100 50 150 = 0

Properties of the z-score distribution μ μ transformation +1 X+1std = 150 50 150 Xmean = 100 = 0 = +1

Properties of the z-score distribution μ μ transformation -1 X-1std = 50 50 150 +1 Xmean = 100 = 0 X+1std = 150 = +1 = -1

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.

From z to raw score m m Z = -0.60 X = 70 X = (-0.60)( 50) + 100 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. m 150 50 m +1 -1 transformation Z = -0.60 X = 70 X = (-0.60)( 50) + 100

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)

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