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## Statistical Tools in Evaluation

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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)

Statistical Tools in Evaluation

- Why Use Statistics?
- Analyze and interpret data
- Standardize test scores
- Interpret research in your field

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

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.

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

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