Psychology 217 Statistical Methods. Lesson U1-3: Summarizing Data Tables and graphs* –Tables...
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Transcript of Psychology 217 Statistical Methods. Lesson U1-3: Summarizing Data Tables and graphs* –Tables...
Psychology 217Statistical Methods
Lesson U1-3: Summarizing Data• Tables and graphs*
– Tables– Pie charts– Histograms and Polygons– Scatterplots/Scattergrams
*Note: The APA Publication Manual (6th ed.) dedicates 43 pages (pp. 125-167) to Tables and Figures
• Central Tendency– Mean– Standard Error of the Mean (SEM)– Median– Mode
Graphics• “A picture is worth 1,000 words”• If it takes 1,000 words to explain a picture, then
the point of diminishing returns has been violated!
Revenue (in $millions)
0102030
405060
N
NNW
NW
WNW
WNW
WSW
SW
SSW S
SSE
SE
ESE E
ENE
NE
NNE
Region
Q1
Q1R
Q2
Q2R
Q3
Q3R
Q4
Q4R
Tables
• Tables are most useful when trying to convey “a large amount of data in a small amount of space.”
• Tables should not be overused.– The 1st APA Guideline for Table use is:
• “Is the table necessary?”
• Tables are not stand-alone.– They must be discussed in the narrative as well.
• For example,…
• Additionally, preference grouping significantly varied in relation to PPRS ranking (χ2(3)=87.19, p<.001) (See Table 6). Christian non-Catholic participants (LDS and Christian remainder) significantly underrepresented in the low PPRS ranking and overrepresented in the high PPRS ranking. Conversely, those without preference significantly overrepresented in the low PPRS ranking and underrepresented in the high PPRS ranking.
Table 6. PPRS Ranking by Religious Preference Grouping
Low PPRS High PPRS
LDS 20(14%)
↓
64(47%)
↑
Catholic 11(8%)
21(15%)
Christian remainder 29(21%)
↓
44(32%)
↑
None 79(57%)
↑
8(6%)
↓
↑ signifies high cell percentage (p<.05) determined by cell standard deviate calculation↓ signifies low cell percentage (p<.05) determined by cell standard deviate calculation
Pie Charts
• Pie charts are almost never worth the space– Simple visual depiction of
proportions
• APA guidelines:– “The number of items compared
should be kept to five or fewer. Order the segments from large to small, beginning the largest segment at 12 o’clock […] making the smallest segment [shaded] the darkest.”
A's
B's
C's
D's
F's
Shapes of Distributions
• Frequency Distribution – a table of counts– 2 5 3 6 5 8 2 3 4 1 0 6 8 9 2 3 7 2 9 0– 2 zeros, 1 one, 4 twos, 3 threes, 1 four, 2 fives, 2
sixes, 1 seven, 2 eights, and 2 nines
Cumulative Frequency Distribution and Histogram
X f cf9 2 20 (N)8 2 187 1 166 2 155 2 134 1 113 3 102 4 71 1 30 2 2 0
1
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3
4
0 1 2 3 4 5 6 7 8 9
Frequency Polygon and Histogram
0
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0 1 2 3 4 5 6 7 8 9
0
0.5
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1.5
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2.5
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3.5
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4.5
1 2 3 4 5 6 7 8 9 10
Normal, Negative Skew, Positive Skew, Bi-modal
0
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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Scatterplots/Scattergrams
• The plotting of paired values– For example,…
• Age Toys– Bob 5 6– Anne 6 12– Sue 3 15– Bill 5 7– Tim 4 15
0
2
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0 1 2 3 4 5 6 7
Age
To
ys
Central TendencyThe location of data• Mean
– For Interval and Ratio data and normal distributions
• Median– For Ordinal data or skewed distributions
• Mode– For nominal data or multi-modal distributions
The Mean(aka, the Arithmetic Mean, Average)The arithmetic center of the distribution• Symbolized as
– “M” in academic literature– X among statisticians
• Operationalized as (∑X)/N– Sigma (∑) means “sum of”– Meaning, “add up all the X scores and divide
by the number of X scores”
The Mean
• Data set of X values– 2 5 3 6 5 8 2 3 4 1 0 6 8 9 2 3 7 2 9 0
• ∑X = 85• N = 20• M = (∑X)/N = 85/20 = 4.25
Standard Error of the Mean (SEM)
• When we collect statistics from a sample,– our data do not represent the population
parameters with 100% accuracy– we must adjust for this error in our data
• The Standard Error of the Mean (SEM)– is an estimate of this “margin of error”– is a necessary calculation in inferential statistics
• The sample mean is a point estimate (location)• SEM is an interval estimate (spread)
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sampling Distribution of Means
• The Normal Distribution is a distribution of all the individual raw scores
• The Sampling Distribution of Means is a distribution of all the means of the samples that can be drawn from a population– The Central Limit Theorem is that this distribution
is normal and centers around the population mean– By definition, SEM is the standard deviation of the
Sampling Distribution of the Means
Calculating SEM
• The larger the N, the smaller the SEM
• The lesser the variability, the smaller the SEM
MSWithin
SEM = √ N
……………………………….Stay tuned!
MedianThe physical center of the distribution• Symbolized as “Mdn”• Operationalized as the X value at the 50th %ile
– Half the data are below, half the data are above
• Data set of X values– 2 5 3 6 5 8 2 3 4 1 0 6 8 9 2 3 7 2 9 0– Ascending order– 0 0 1 2 2 2 2 3 3 3 4 5 5 6 6 7 8 8 9 9
• Median is (3 + 4) / 2 = 3½
The ModeThe most frequently occurring value• Symbolized as “Mode” • Operationalized as the X value(s) with the
largest frequency(ies) on the frequency distribution– (aka, the tallest bar(s) on the histogram)
Mode = 2
X f9 28 27 16 25 24 13 32 41 10 2 0
1
2
3
4
0 1 2 3 4 5 6 7 8 9
Positive Skew(Mode = 2) < (Mdn = 3.5) < (M = 4.25)
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Central Tendency’s Interpretive Power (the need for Dispersion)
X Y Z
3 5 11
3 4 1
3 3 1
3 2 1
3 1 1
∑X = 15 ∑Y = 15 ∑Z = 15
MX = 3 MY = 3 MZ = 3