Psychology 340 Spring 2010

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

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

m 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.

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

Our sample 2, 4, 6, 8 1 2 3 4 5 6 7 8 9 10 X - X = deviation scores X Computing standard deviation (sample) • Step 1: Compute the deviation scores • subtract the sample mean from every individual in our distribution. 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3

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

3 X X X X 2 1 4 μ 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

Sample variance = s2 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

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

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

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

X new • All of the scores change by the same constant. • But so does the mean 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

X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same 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

X X new old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same 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

20 21 22 23 24 s = X 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 • Multiply/divide a constant to each score (-1)2 21 - 22 = -1 23 - 22 = +1 (+1)2

Multiply scores by 2 40 42 44 46 48 X 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 • Multiply/divide a constant to each score changes changes (-2)2 42 - 44 = -2 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

Reference point Direction μ Locating a score X1 - 100= +62 X1 = 162 X2 = 57 X2 - 100= -43

Reference point Below Above μ Locating a score X1 - 100= +62 X1 = 162 X2 = 57 X2 - 100= -43

Raw score Population mean Population standard deviation 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.

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

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.

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

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

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

m m transformation 50 150 -1 +1 Z = -0.60 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. X = (-0.60)( 50) + 100 X = 70

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

Psychology 340 Spring 2010