describing distributions statistics for the social sciences psychology 340 spring 2010

Describing Distributions

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social Sciences Announcements

• Homework #1: will accept these on Th (Jan 21) without penalty

• Quiz problems– Quiz 1 is now posted, due date extended to Tu,

Jan 26th (by 11:00)• 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 Distributions

• Three basic characteristics are used to describe distributions– Shape

• Many different ways to display distribution– Frequency distribution table– Graphs

– Center– Variability


Social Sciences Shapes of Frequency Distributions

Unimodal, bimodal, and rectangular


Social Sciences Shapes of Frequency Distributions

Symmetrical and skewed distributions

Normal and kurtotic distributions

Positively Negatively


Social Sciences Frequency Graphs

Histogram Plot the

different values against the frequency of each value



Histogram by hand Step 1: make a frequency

distribution table (may use grouped frequency tables)

Step 2: put the values along the bottom, left to right, lowest to highest

Step 3: make a scale of frequencies along left edge

Step 4: make a bar above each value with a height for the frequency of that value



Histogram using SPSS (create one for class height) Graphs -> Legacy -> histogram Enter your variable into ‘variable’

To change interval width, double click the graph to get into the chart editor, and then double click the bottom axis. Click on ‘scale’ and change the intervals to desired widths

Note: you can also get one from the descriptive statistics frequency menu under the ‘charts’ option



Frequency polygon - essentially the same, put uses lines instead of bars


Social Sciences Displaying two variables

Bar graphs Can be used in a number of ways (including

displaying one or more variables) Best used for categorical variables

Scatterplots Best used for continuous variables


Social Sciences Bar graphs

• Plot a bar graph of men and women in the class– Graphs -> bar– Simple, click define– N-cases (the default)– Enter Gender into Category axis, click ‘okay’


Social Sciences Bar graphs

• Plot a bar graph of shoes in closet crossed with men and women– What should we plot? (and why?)

• Average number of shoes for each group?– Graphs -> bar– Simple, click define– Other statistic (default is ‘mean’) – enter pairs of shoes– Enter Gender into Category axis, click ‘okay’


Social Sciences Scatterplot

• Useful for seeing the relationship between the variables– Graphs -> Legacy Dialogs– Scatter/Dot– Simple Scatter, click ‘define’– Enter your X & Y variables, click ‘okay’

• Can add a ‘fit line’ in the chart editor• Plot a scatterplot of soda and bottled water drinking


Social Sciences Describing distributions

• Distributions are typically described with three properties:– Shape: unimodal, symmetric, skewed, etc.– Center: mean, median, mode– Spread (variability): standard deviation, variance


Social Sciences 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.


Social Sciences 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– If the distribution is multi-modal


Social Sciences The Mean

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

– Computing the mean

€

μ =∑XN

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

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

€

X = ∑ Xn

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


Social Sciences 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

=32720

= 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?


Social Sciences 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, 15X =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 + X2n2

n1 + 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


Social Sciences 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 + X2n2

n1 + n2

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

5 + 20=14.16

Let’s check:

€

X = ∑ Xn

=14.16

• Both ways give the same answer

€

=35425


Social Sciences 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 distribution

Step1: 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


Social Sciences 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

123

1 2 3 4 5 6 7 8 9

4

major modeminor mode


Social Sciences Describing distributions

• Distributions are typically described with three properties:– Shape: unimodal, symmetric, skewed, etc.– Center: mean, median, mode– Spread (variability): standard deviation, variance


Social Sciences 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


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.

μ


Social SciencesComputing 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 =∑XN

= 2 + 4 + 6 + 84

= 204

= 5.0

2 - 5 = -3

μX - μ = deviation scores

-3


Social Sciences



1 2 3 4 5 6 7 8 9 10

€

μ =∑XN

= 2 + 4 + 6 + 84

= 204

= 5.0

2 - 5 = -34 - 5 = -1


-1

Computing standard deviation (population)


Social Sciences



1 2 3 4 5 6 7 8 9 10

€

μ =∑XN

= 2 + 4 + 6 + 84

= 204

= 5.0

2 - 5 = -34 - 5 = -1

6 - 5 = +1


1



Social Sciences

• 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 =∑XN

= 2 + 4 + 6 + 84

= 204

= 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.



Social Sciences

• 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 = Σ (X - μ)2

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X - σ = deviation scores

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

= 9 + 1 + 1 + 9 = 20



Social Sciences

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

• Divide by the number of individuals in the population.

variance = σ2 = SS/N


• Note: your book uses ‘SD2’ to denote the variance in formulas


Social Sciences

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

€

σ 2 =X − μ( )

2∑N

standard deviation = σ =


• Note: your book uses ‘SD’ to denote the standard deviation in formulas


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



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:

μX1 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

€

μ =∑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



– All of the scores change by the same constant.

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



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

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

μ

€

μ =100

X1 = 162X2 = 57

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

Reference point

Direction



€

X − μ

μ

€

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

μ

€

μ =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

μ

€

μ =0

μ

transformation

€

z = X − μσ

15050

€

zmean = 100 −10050 = 0

€

σ =50

€

μ =100

Xmean = 100



μ

€

μ =0

μ

€

σ =50

transformation

€

z = X − μσ

15050

Xmean = 100

€

zmean = 100 −10050

€

z+1std = 150 −10050

= 0

= +1

€

μ =100

X+1std = 150

+1



μ

€

σ =1

€

μ =0

μ

€

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

€

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.


Social Sciences

μ 15050 μ€

σ =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

€

σ =50

€

μ =100

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

Z =X −μ( )σ

Z( ) σ( )= X −μ( ) X = Z( ) σ( )+ μ


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)

describing distributions statistics for the social sciences psychology 340 spring 2010

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