Psychology 340 Spring 2010

Describing Distributions & Locating scores & Transforming

distributions

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics 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)


Social SciencesOutline (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


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.


Social Sciences

• 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 (population)


Social Sciences

• 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)


Social Sciences Computing standard deviation (sample)

• Step 1: Compute the deviation scores– subtract the sample mean from every individual in our distribution.

Our sample2, 4, 6, 8

1 2 3 4 5 6 7 8 9 10

€

X = ∑ Xn

= 2 + 4 + 6 + 84

= 204

= 5.0

X - X = deviation scores

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X


Social Sciences

• Step 2: Determine the sum of the squared deviations (SS).


2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 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


Social Sciences

• Step 3: Determine the variance


Population variance = σ2 = SS/NRecall:

X1 X2X3X4

The variability of the samples is typically smaller than the population’s variability


Social Sciences

• Step 3: Determine the variance


Population variance = σ2 = SS/NRecall:

The variability of the samples is typically smaller than the population’s variability

Sample variance = s2

€

=SS

n −1( )

To correct for this we divide by (n-1) instead of just n


Social Sciences

• Step 4: Determine the standard deviation

€

s2 =X − X ( )

2∑n −1

standard deviation = s =



Social SciencesProperties of means and standard deviations

• Change/add/delete a given score

Mean Standard deviation

changes changes

– Changes the total and the number of scores, this will change the mean and the standard deviation

€

=∑XN

σ =X − μ( )2∑N



– All of the scores change by the same constant.

Xold



• Add/subtract a constant to each score

changes changes




Xold



changes changes





Xold



changes changes





Xold



changes changes




– All of the scores change by the same constant.– But so does the mean

Xnew



changes changes


changes



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

Xold



changes changes


changes




Xold



changes changes


changes




Xold



changes changes


changes




Xold



changes changes


changes




Xold



changes changes


changes




Xold



changes changes


changes




Xold



changes changes


changes




XnewXold



changes changes

No changechanges• Add/subtract a constant to each score





• Multiply/divide a constant to each score

changes changes

No changechanges• Add/subtract a constant to each score

20 21 22 23 24

X

21 - 22 = -123 - 22 = +1

(-1)2

(+1)2

s =

€

X − X ( )2∑

n −1= 2 =1.41



– Multiply scores by 2



• Multiply/divide a constant to each score

changes changes

No changechanges

changes changes


42 - 44 = -246 - 44 = +2

(-2)2

(+2)2

s =

€

X − X ( )2∑

n −1= 8 = 2.82

40 42 44 46 48

X

Sold=1.41


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 − μ– 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



€

X − μ

€

=100

X1 = 162X2 = 57

X1 - 100 = +62X2 - 100 = -43

Reference point

Direction



€

X − μ

€

=100

X1 = 162X2 = 57

X1 - 100 = +62X2 - 100 = -43

Reference point

BelowAbove


Social Sciences Transforming a score

€

z = X − μσ

– 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 meanPopulation standard deviation


Social Sciences Transforming scores

€

=100

X1 = 162

X2 = 57

€

σ =50

€

z = X − μσ

X1 - 100 = +1.2050

X2 - 100 = -0.8650

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.


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 Z-

distribution.


Social SciencesProperties of the z-score distribution

€

=0

transformation

€

z = X − μσ

15050

€

zmean = 100 −10050 = 0

€

σ =50

€

=100

Xmean = 100



€

=0

€

σ =50

transformation

€

z = X − μσ

15050

Xmean = 100

€

zmean = 100 −10050

€

z+1std = 150 −10050

= 0

= +1

€

=100

X+1std = 150

+1



€

σ =1

€

=0

€

σ =50

transformation

€

z = X − μσ

15050

Xmean = 100

X+1std = 150

€

zmean = 100 −10050

€

z+1std = 150 −10050

€

z−1std = 50 −10050

= 0

= +1

= -1

€

=100

X-1std = 50

+1-1



• 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.


Social Sciences

15050 €

σ =1

€

=0

+1-1

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.

transformation

€

X = Zσ + μ

€

σ =50

€

=100

Z = -0.60X = (-0.60)( 50) + 100X = 70

Z =X −( )σ

Z( ) σ( )= X −( ) X = Z( ) σ( )+


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)


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

Documents

Psychology 340 Spring 2010