Statistics for the Social Sciences Psychology 340 Fall 2006 Introductions

Statistics for the Social Sciences

Psychology 340Fall 2006

Introductions


Outline (for week)

• Variables: IV, DV, scales of measurement– Discuss each variable and it’s scale of measurement

• Characteristics of Distributions– Using graphs– Using numbers (center and variability)

• Descriptive statistics decision tree

• Locating scores: z-scores and other transformations


Outline (for week)

• Variables: IV, DV, scales of measurement– Discuss each variable and it’s scale of measurement

• Characteristics of Distributions– Using graphs– Using numbers (center and variability)

• Descriptive statistics decision tree

• Locating scores: z-scores and other transformations


Describing distributions

• Distributions are typically described with three properties:– Shape: unimodal, symmetric, skewed, etc.

– Center: mean, median, mode– Spread (variability): standard deviation, variance






Which center when?

• Depends on a number of factors, like scale of measurement and shape.– The mean is the most preferred measure and it is closely related to measures of variability

– However, there are times when the mean isn’t the appropriate measure.


Which center when?

• Use the median if:•The distribution is skewed•The distribution is ‘open-ended’

– (e.g. your top answer on your questionnaire is ‘5 or more’)

•Data are on an ordinal scale (rankings)

• Use the mode if the data are on a nominal scale


The Mean

• The most commonly used measure of center • The arithmetic average

– Computing the mean

€

μ =∑X

N

– The formula for the population mean is (a parameter):

– The formula for the sample mean is (a statistic):

€

X =∑ X

n

Add up all of the X’s

Divide by the total number in the population

Divide by the total number in the sample

• Note: your book uses ‘M’ to denote the mean in formulas


The Mean

• Number of shoes:– 5, 7, 5, 5, 5– 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15

X =∑Xn

X =∑Xn

=5 + 7 + 5 + 5 + 5

5=5.4

=327

20= 16.35

• Suppose we want the mean of the entire group?

• NO. Why not?

• Can we simply add the two means together and divide by 2?


The Weighted Mean

• Number of shoes:– 5, 7, 5, 5, 5, 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15 X =5.4 X =16.35

• Suppose we want the mean of the entire group? Can we simply add the two means together and divide by 2?

XN =X1n1 + X2n2n1 +n2

=5.4 * 5( ) + 16.35 * 20( )

5 + 20=14.16

• NO. Why not? Need to take into account the number of scores in each mean


The Weighted Mean

• Number of shoes:– 5, 7, 5, 5, 5, 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15

XN =X1n1 + X2n2n1 +n2

=5.4 * 5( ) + 16.35 * 20( )

5 + 20=14.16

Let’s check:

€

X =∑ X

n=∑354

25=14.16

• Both ways give the same answer


The median

• The median is the score that divides a distribution exactly in half. Exactly 50% of the individuals in a distribution have scores at or below the median.– Case1: Odd number of scores in the distributionStep1: put the

scores in order Step2: find the middle score

Step1: put the scores in order Step2: find the middle two scores Step3: find the arithmetic average of the two middle scores

– Case2: Even number of scores in the distribution


The mode

• The mode is the score or category that has the greatest frequency. – So look at your frequency table or graph and pick the variable that has the highest frequency.

1

2

3

1 2 3 4 5 6 7 8 9

1

2

3

1 2 3 4 5 6 7 8 9

so the mode is 5 so the modes are 2 and 8

Note: if one were bigger than the other it would be called the major mode and the other would be the minor mode

1

2

3

1 2 3 4 5 6 7 8 9

4

major modeminor mode






Variability of a distribution

• Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.– In other words variabilility refers to the degree of “differentness” of the scores in the distribution.

• High variability means that the scores differ by a lot

• Low variability means that the scores are all similar


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.

μ


Computing standard deviation (population)

• Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution.

Our population2, 4, 6, 8

1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=

2 + 4 + 6 + 8

4=

20

4= 5.0

2 - 5 = -3

μX - μ = deviation scores

-3




1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=

2 + 4 + 6 + 8

4=

20

4= 5.0

2 - 5 = -34 - 5 = -1


-1





1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=

2 + 4 + 6 + 8

4=

20

4= 5.0

2 - 5 = -34 - 5 = -1

6 - 5 = +1


1



• Step 1: Compute the deviation scores: Subtract the population mean from every score in the distribution.


1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=

2 + 4 + 6 + 8

4=

20

4= 5.0

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3


3

Notice that if you add up all of the deviations they must equal 0.



• Step 2: Get rid of the negative signs. Square the deviations and add them together to compute the sum of the squared deviations (SS).

SS = (X - μ)2

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X - μ = deviation scores

= (-3)2+ (-1)2+ (+1)2+ (+3)2

= 9 + 1 + 1 + 9 = 20



• Step 3: Compute the Variance (the average of the squared deviations)

• Divide by the number of individuals in the population.

variance = 2 = SS/N



• Step 4: Compute the standard deviation. Take the square root of the population variance.

€

2 =X − μ( )

2∑N

standard deviation = =



• To review:– Step 1: compute deviation scores– Step 2: compute the SS

• SS = (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



• 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 sample2, 4, 6, 8

1 2 3 4 5 6 7 8 9 10

€

X =∑ X

n=

2 + 4 + 6 + 8

4=

20

4= 5.0

X - X = deviation scores

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X


• 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 = (X - X)2

Apart from notational differences the procedure is the same as before


• Step 3: Determine the variance


Population variance = 2 = SS/N

Recall:

μX1 X2X3X4

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


• Step 3: Determine the variance


Population variance = 2 = SS/N

Recall:

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


• Step 4: Determine the standard deviation

€

s2 =X − X ( )

2∑n −1

standard deviation = s =



Properties 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

€

μ =∑X

N =

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





– 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


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


Locating a score

€

X − μ

μ

€

μ =100

X1 = 162X2 = 57

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

Reference point

Direction


Locating a score

€

X − μ

μ

€

μ =100

X1 = 162X2 = 57

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

Reference point

BelowAbove


Transforming a score

€

z =X − μ

σ

– 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


Transforming scores

μ

€

μ =100

X1 = 162

X2 = 57

€

=50

€

z =X − μ

σ

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 μ.


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 =0

μ

transformation

€

z =X − μ

σ

15050

€

zmean =100 −100

50= 0

€

=50

€

μ =100

Xmean = 100



μ

€

μ =0

μ

€

=50

transformation

€

z =X − μ

σ

15050

Xmean = 100

€

zmean =100 −100

50

€

z+1std =150 −100

50

= 0

= +1

€

μ =100

X+1std = 150

+1



μ

€

=1

€

μ =0

μ

€

=50

transformation

€

z =X − μ

σ

15050

Xmean = 100

X+1std = 150

€

zmean =100 −100

50

€

z+1std =150 −100

50

€

z−1std =50 −100

50

= 0

= +1

= -1

€

μ =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.


μ 15050 μ€

=1

€

μ =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

€

=50

€

μ =100

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

Z =X−μ( )

Z( ) ( ) = X−μ( ) X = Z( ) ( ) + μ

Documents

Statistics for the Social Sciences Psychology 340 Fall 2006 Introductions