statistical tools in evaluation part i. statistical tools in evaluation what are statistics?...
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Statistical Tools in Evaluation
Part I
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Statistical Tools in Evaluation
• What are statistics?– Organization and analysis of numerical data
– Methods used involve calculations and graphical displays of data
– Formulas used can reveal the “true” nature of the data as well as critical relationships between variables (targets of study)
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Statistical Tools in Evaluation
• Why Use Statistics?– Analyze and interpret data
– Standardize test scores
– Interpret research in your field
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Problem:
Not all scoring / quantifying systems are the same.
Vary by:
Scores
Scales
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Types of Scores
Continuous Scores that can be recorded in an infinite number of
values (decimal figures; greater and greater accuracy)
Examples: time, distance
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Types of Scores
Discrete Scores that are whole numbers only
Examples: wins, losses, home runs, touchdowns
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Types of Scales
• Nominal Scale– Lowest and most elementary scale
– Generally represents categories
– Something is in a category or it is not
– Examples: sex, state of origin, eye color
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Types of Scales
• Ordinal Scale (order)– Generally refers to rank or order of a variable
– Does not tell how big or small the difference between ranks is
– Examples: • finish order in a race – 1st,2nd,3rd
• tennis team ladder of “best to worst”
• season ranking of a team
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Types of Scales
• Interval Scale– Also provides order of variable, but additionally
provides information about how far one measure is from another
– Equal units of measure are used on the scale
– No true zero point that means absence
– Examples: temperature, year, IQ
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Types of Scales
• Ratio Scale– Same as interval, but has a true zero point (absolute
absence or completely nothing)
– Examples: height, weight, time - *type of score?
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Once you have scores (data) what is the first thing you do with them?
• Find out how they are distributed
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Distribution of Data
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Simple Score Ranking
• List scores in descending or ascending order depending on quality*
• Number scores from best – first, to worst – last
• Identical scores should have the same rank
– average the rank– or determine midpoint and assign same rank
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Example of Simple Ranking
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Frequency Distribution• Once data have been collected (numbers given
to a measurement), it is best to organize them in a sensible order
– Best at top of list
• highest to lowest – jump height, throw dist.
• lowest to highest – swim time, golf score
– Calculate frequencies of scores – how many of each score are present
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Frequency Distribution
• Frequency distribution can tell:– frequency of a score (f) – how many of each score
– cumulative frequency (cf) – how many through that score
– cumulative percentage (c%) - % occurring above and below a score
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Examples and Practice Problems
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Graphing the Frequency Distribution
• Frequency of scores on y axis (ordinate)
• Scores from low to high on x axis (abscissa)
• Intersection of ordinate and abscissa is zero (0) point for both axes
0 Scores
Fre
quen
cy
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Graphing the Frequency Distribution
• Frequency Polygon– Midpoints of intervals are plotted against
frequencies
– Straight lines drawn between points
• Histogram– Bars are used to represent the frequencies of scores
• Curve– Curved line represents the frequency of scores
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What else can grouped scores tell us?
How all scores compare to the average score
= Measures of Central Tendency
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Measures of Central Tendency
• Statistics that describe middle characteristics of scores– Mode (Mo) The most frequently occurring score
• There can be more than one mode - bimodal
– Determination: Find the score that occurs most frequently !
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Measures of Central Tendency
– Median - Median (Mdn, P50) - represents the exact middle of a distribution (50th percentile)
• The Mdn is the best measure of central tendency when you have extreme scores and skewed distributions
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Median
• Median Calculations:– Determining position of approximate
median:• “Simple counting method”
• Formula - Mdn = (n + 1) / 2
(n = total number)
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• Ranks tell the position of a score relative to other scores in a group.
Percentile Rank- The percentage of total scores that fall below a given score.
Percentile - refers to a point in a distribution of scores in which a given percent of the scores fall (percentile is the location of the score).
25th percentile (quartile), 75th percentile, 90th percentile, etc.
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Examples and Practice Problems
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Measures of Central Tendency
mean (X): average score• most sensitive• affected by extreme scores• best for interval and ratio scale• probably most often used
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Measures of Central Tendency
mean (X): average score.• most sensitive• affected by extreme scores• best for interval and ratio scale• probably most often used
– Calculation:X = X / n ( = sum; X = sum of scores)
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Remember Curves?
• What types of curves are there and what do they mean?
– Normal curve
– Skewed curve
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Characteristics of the Normal Curve
• Bell-shaped
• Symmetrical
• Greatest number of scores found in middle
• Mean, median, and mode at same point in the middle of the curve.
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Characteristics of the “not-so-normal curve”
• Irregular curves represent different types of distributions
leptokurtic platykurtic
bimodal
positive skew negative skew
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• Normal curve - X, Mdn, and Mo are all the same value (location)
X
Mo
Mdn
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• Skewed curves - Mo is opposite end of the tail, Mdn is in the middle, and X is toward the tail
Mo Mdn X
Positive Skew
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• Skewed curves - Mo is opposite end of the tail, Mdn is in the middle, and X is toward the tail
MoMdnX
Negative Skew
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• Question: Why do these variables fall this way on a skewed distribution of scores?
• Question: Can you see the impact of extreme scores on these variables?
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Measures of Variability
• Variability refers to how much individual scores deviate from a measure of central tendency; how heterogeneous the group is.
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Measures of Variability
• Range (R) - Represents the difference between the low and high score.
• Simplest measure of variability; used with the mode or median.
• Calculation: R = High – Low
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Measures of Variability
• Standard Deviation (SD, s) - Describes how far the scores as a group deviate from the X.
• It is the most useful descriptive statistic of variability.
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• SD calculations:
SD = (X - X )2
N
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Examples and Practice Problems
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Relationship Between Normal Curve and SD:
1 SD = 68.26% of all scores (34.13% above and below X)
2 SD = 95.44% of all scores (47.72% above and below X)
3 SD = 99.73% of all scores (49.86% above and below X)
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How Alike are Scores in a Normal Curve?
• Homogeneity = Near the mean - alike
• Heterogeneity = Away from the mean - different
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Questions?
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End Part I