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Statistical Tools in Evaluation. Part I. Statistical Tools in Evaluation. What are statistics? Organization and analysis of numerical data Methods used involve calculations and graphical displays of data

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statistical tools in evaluation1
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)
statistical tools in evaluation2
Statistical Tools in Evaluation
  • Why Use Statistics?
    • Analyze and interpret data
    • Standardize test scores
    • Interpret research in your field
problem
Problem:

Not all scoring / quantifying systems are the same.

Vary by:

Scores

Scales

types of scores
Types of Scores
  • Continuous
    • Scores that can be recorded in an infinite number of values (decimal figures; greater and greater accuracy)
    • Examples: time, distance
types of scores1
Types of Scores
  • Discrete
    • Scores that are whole numbers only
    • Examples: wins, losses, home runs, touchdowns
types of scales
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
types of scales1
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
types of scales2
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
types of scales3
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?
once you have scores data what is the first thing you do with them
Once you have scores (data) what is the first thing you do with them?
  • Find out how they are distributed
simple score ranking
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
frequency distribution
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
frequency distribution1
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
graphing the frequency distribution
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

Frequency

0

Scores

graphing the frequency distribution1
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
what else can grouped scores tell us
What else can grouped scores tell us?

How all scores compare to the average score

= Measures of Central Tendency

measures of central tendency
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 !
measures of central tendency1
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
median
Median
  • Median Calculations:
    • Determining position of approximate median:
      • “Simple counting method”
      • Formula - Mdn = (n + 1) / 2

(n = total number)

slide24
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.
measures of central tendency2
Measures of Central Tendency
  • mean (X): average score
    • most sensitive
    • affected by extreme scores
    • best for interval and ratio scale
    • probably most often used
measures of central tendency3
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)

remember curves
Remember Curves?
  • What types of curves are there and what do they mean?
    • Normal curve
    • Skewed curve
characteristics of the normal curve
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.
characteristics of the not so normal curve
Characteristics of the “not-so-normal curve”
  • Irregular curves represent different types of distributions
      • leptokurtic
      • platykurtic
      • bimodal
      • positive skew
      • negative skew
slide32
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

slide33
Skewed curves - Mo is opposite end of the tail, Mdn is in the middle, and X is toward the tail

X

Mdn

Mo

Negative Skew

slide34
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?
measures of variability
Measures of Variability
  • Variability refers to how much individual scores deviate from a measure of central tendency; how heterogeneous the group is.
measures of variability1
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
measures of variability2
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.
slide38
SD calculations:

SD = (X - X )2

N

relationship between normal curve and sd
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)
how alike are scores in a normal curve
How Alike are Scores in a Normal Curve?
  • Homogeneity = Near the mean - alike
  • Heterogeneity = Away from the mean - different