460 likes | 626 Views
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
E N D
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 Evaluation • Why Use Statistics? • Analyze and interpret data • Standardize test scores • Interpret research in your field
Problem: Not all scoring / quantifying systems are the same. Vary by: Scores Scales
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 Scores • Discrete • Scores that are whole numbers only • Examples: wins, losses, home runs, touchdowns
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 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 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 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? • Find out how they are distributed
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 • 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 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 • 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 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? How all scores compare to the average score = 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 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 Calculations: • Determining position of approximate median: • “Simple counting method” • Formula - Mdn = (n + 1) / 2 (n = total number)
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 Tendency • mean (X): average score • most sensitive • affected by extreme scores • best for interval and ratio scale • probably most often used
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? • What types of curves are there and what do they mean? • Normal curve • Skewed 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” • Irregular curves represent different types of distributions • leptokurtic • platykurtic • bimodal • positive skew • negative skew
Normal curve - X, Mdn, and Mo are all the same value (location) Mo Mdn X
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
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
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 • Variability refers to how much individual scores deviate from a measure of central tendency; how heterogeneous the group is.
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 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.
SD calculations: SD = (X - X )2 N
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? • Homogeneity = Near the mean - alike • Heterogeneity = Away from the mean - different