Upload
amice-warren
View
221
Download
0
Embed Size (px)
DESCRIPTION
PSY 340 Statistics for the Social Sciences 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
Citation preview
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)
PSY 340Statistics for the
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
PSY 340Statistics for the
Social Sciences Distributions
• Three basic characteristics are used to describe distributions– Shape
• Many different ways to display distribution– Frequency distribution table– Graphs
– Center– Variability
PSY 340Statistics for the
Social Sciences Shapes of Frequency Distributions
Unimodal, bimodal, and rectangular
PSY 340Statistics for the
Social Sciences Shapes of Frequency Distributions
Symmetrical and skewed distributions
Normal and kurtotic distributions
Positively Negatively
PSY 340Statistics for the
Social Sciences Frequency Graphs
Histogram Plot the
different values against the frequency of each value
PSY 340Statistics for the
Social Sciences Frequency Graphs
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
PSY 340Statistics for the
Social Sciences Frequency Graphs
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
PSY 340Statistics for the
Social Sciences Frequency Graphs
Frequency polygon - essentially the same, put uses lines instead of bars
PSY 340Statistics for the
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
PSY 340Statistics for the
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’
PSY 340Statistics for the
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’
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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.
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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?
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
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
PSY 340Statistics for the
Social Sciences
• 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 = -34 - 5 = -1
μX - μ = deviation scores
-1
Computing standard deviation (population)
PSY 340Statistics for the
Social Sciences
• 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 = -34 - 5 = -1
6 - 5 = +1
μX - μ = deviation scores
1
Computing standard deviation (population)
PSY 340Statistics for the
Social Sciences
• Step 1: Compute the deviation scores: Subtract the population mean from every score in the 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 = -34 - 5 = -1
6 - 5 = +18 - 5 = +3
μX - μ = deviation scores
3
Notice that if you add up all of the deviations they must equal 0.
Computing standard deviation (population)
PSY 340Statistics for the
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
Computing standard deviation (population)
PSY 340Statistics for the
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
Computing standard deviation (population)
• Note: your book uses ‘SD2’ to denote the variance in formulas
PSY 340Statistics for the
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 =
Computing standard deviation (population)
• Note: your book uses ‘SD’ to denote the standard deviation in formulas
PSY 340Statistics for the
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)
PSY 340Statistics for the
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)
PSY 340Statistics for the
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
PSY 340Statistics for the
Social Sciences
• Step 2: Determine the sum of the squared deviations (SS).
Computing standard deviation (sample)
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
PSY 340Statistics for the
Social Sciences
• Step 3: Determine the variance
Computing standard deviation (sample)
Population variance = σ2 = SS/NRecall:
μX1 X2X3X4
The variability of the samples is typically smaller than the population’s variability
PSY 340Statistics for the
Social Sciences
• Step 3: Determine the variance
Computing standard deviation (sample)
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
PSY 340Statistics for the
Social Sciences
• Step 4: Determine the standard deviation
€
s2 =X − X ( )
2∑n −1
standard deviation = s =
Computing standard deviation (sample)
PSY 340Statistics for the
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
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– All of the scores change by the same constant.
Xold
• Change/add/delete a given score
Mean Standard deviation
• Add/subtract a constant to each score
changes changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– All of the scores change by the same constant.
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– All of the scores change by the same constant.
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– All of the scores change by the same constant.
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– All of the scores change by the same constant.– But so does the mean
Xnew
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
Xold
• Change/add/delete a given score
Mean Standard deviation
changes changes
• Add/subtract a constant to each score
changes
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same
XnewXold
• Change/add/delete a given score
Mean Standard deviation
changes changes
No changechanges• Add/subtract a constant to each score
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
• Change/add/delete a given score
Mean Standard deviation
• 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
PSY 340Statistics for the
Social SciencesProperties of means and standard deviations
– Multiply scores by 2
• Change/add/delete a given score
Mean Standard deviation
• Multiply/divide a constant to each score
changes changes
No changechanges
changes changes
• Add/subtract a constant to each score
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
PSY 340Statistics for the
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
PSY 340Statistics for the
Social Sciences Locating a score
€
X − μ
μ
€
μ =100
X1 = 162X2 = 57
X1 - 100 = +62X2 - 100 = -43
Reference point
Direction
PSY 340Statistics for the
Social Sciences Locating a score
€
X − μ
μ
€
μ =100
X1 = 162X2 = 57
X1 - 100 = +62X2 - 100 = -43
Reference point
BelowAbove
PSY 340Statistics for the
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
PSY 340Statistics for the
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.
PSY 340Statistics for the
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.
PSY 340Statistics for the
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
PSY 340Statistics for the
Social SciencesProperties of the z-score distribution
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
PSY 340Statistics for the
Social SciencesProperties of the z-score distribution
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
PSY 340Statistics for the
Social SciencesProperties 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.
PSY 340Statistics for the
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
PSY 340Statistics for the
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)