descriptive statistics and related matters. two families of statistics descriptive statistics –...
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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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TOTAL
42.5
40.0
37.5
35.0
32.5
30.0
27.5
25.0
22.5
20.0
17.5
15.0
12.5
10
8
6
4
2
0
Std. Dev = 8.12
Mean = 29.4
N = 40.00
Output for SPSS
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Measures of Central Tendency
• The four “M”s– 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 11Picture 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
73.211
94110001149
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Variance formula
N
XXS
2
2 )(
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Standard Deviation formula
N
XXS
2)(
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The Normal Distribution
(Brown, 1996, p. 130)
Tail Tail
Peak
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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)
Tail
Tail
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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
A1. 85 is what percentile?
16 (15.9)
A2. What percent between 70 and 115?
82 (81.85)
A3. How many SD is Iliana (177)?
About 5 (5.13)
A4. Iliana’s z =? T=?
CEEB =?
z = (x – m) / sd
T = 10z + 50
CEEB = 100z + 500
5 (5.13)100 (101.3)
1000 (1013)
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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
0 50 500
43 40 400
39.5 35
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