Descriptive%20Statistics

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Descriptive Statistics

• And related matters

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Two families of statistics

• Descriptive statistics procedures for
  summarizing, organizing, graphing, and, in
  general, describing quantitative information
• Mean, standard deviation, of items, etc.
• Inferential statistics statistics that allow
  one to draw conclusions or inferences from the
  data
• ANOVA, t-test, correlation, etc.

Vogt, W. P. (1999). Dictionary of statistics
methodology (2nd ed.). Thousand Oaks, CA Sage
Publications.
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Scales of measurement

• Nominal
• Used to name or categorize things
• Female 1,Male 2, correct 1,incorrect 0
• Often used for coding variables in research
• Ordinal
• Used to order things
• Gives relative position but not amount
• Rankings are ordinal

Shavelson, R. J. (1996). Statistical reasoning
for the behavioral sciences (Third ed.). Needham
Heights, MA Allyn Bacon.
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Scales of measurement (2)

• Interval
• Each scale unit represents an equal distance of
  the attribute being measured
• Most test scores are considered interval scales
• Rating scales are often treated as interval
• Ratio
• Interval scales with a meaningful zero point
  where zero indicates the absence of the attribute
• examples weight, height, length

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Scale summary

• Nominal scales categorize but do not order.
• Ordinal scales categorize and order.
• Interval scales categorize, order, and establish
  an equal unit in the scale.
• Ratio scales categorize, order, establish an
  equal unit, and contain a true zero point.

Wiersma, W., Jurs, S. G. (1990). Educational
measurement and testing (2nd ed.). Needham
Heights, MA Allyn and Bacon, p. 13.
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Frequency information

• Frequency distribution how many students
  received each score
• Cumulative frequency how many students scored
  at or below the score in question
• Cumulative percentage what percent of students
  scored at or below the score in question
• Useful for seeing patterns in the data

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Output for SPSS
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Measures of Central Tendency

• The four Ms
• Mean
• Mode
• Median
• Midpoint

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Think about it

• Scores 18, 19, 20, 21, 87
• Which give a more accurate picture of this data,
  the mean or the median?
• Mean 33
• Median 20
• The median is usually more appropriate as a
  measure of central tendency when there is an
  outlier.

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For a norm-referenced test
(Henning, 1987, p. 39)
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Measures of Dispersion

• Range
• High score
• Low score
• Standard Deviation
• Variance

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Conceptualizing variance

• Imagine a set of scores
• 8 10 13 9 7 11 10 12 10 9 11
• Picture those scores on a number line

Williams Monge (2001) Reasoning with statistics
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Conceptualizing variance (2)

• Imagine those scores as deviations from the mean
  (how far are they from the mean?)

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Conceptualizing variance (3)

• Variance the mean of the squared deviation
  scores about the mean of a distribution

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Variance formula
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Standard Deviation formula
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(No Transcript)
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The Normal Distribution
(Brown, 1996, p. 130)
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Sample versus population
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Describing distributions
Leptokurtic
Platykurtic
Think Leprechaun
Think Platypus
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Skewed distributions
(Brown, 1996, p. 141)
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Standardized scores

• A transformation of raw scores into a measure of
  relative standing based on the mean and standard
  deviation
• Useful for comparing performance on tests of
  different lengths, different forms, etc.
• The most often used standardized scores are
  z-scores, T-scores, and CEEB scores.
• Relative standing is usually based on the norm
  group (for a norm-referenced test)

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Standard score comparison
(Brown, 1996, p. 135)
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Practice
16 (15.9)
z (x m) / sd T 10z 50 CEEB 100z 500
82 (81.85)
A4. Ilianas z ? T? CEEB ?
5 (5.13)
100 (101.3)
1000 (1013)
About 5 (5.13)
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Application exercises
Student Raw z-score T-score CEEB
A 64 70
B 50
C -1
D -1.5 350
2
700
50
500
0
40
400
43
39.5
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Raw score mean 50, raw score standard deviation
7
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Population and Sample
  Population S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
  3 3   3   3   3   3  
  6 6     6   6 6   6 6
  6 6   6 6       6   6
  9   9   9   9   9    
  12   12 12   12   12 12   12
  15   15     15 15     15   Average
Mean 8.5 5 12 7 7 10 10 7 9 8 8 8.3
SD(N) 4.03 1.41 2.45 3.74 1.41 5.10 3.74 3.74 2.45 5.10 2.83 3.20
SD(N-1)   1.73 3.00 4.58 1.73 6.24 4.58 4.58 3.00 6.24 3.46 3.92