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

– 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

Distribution of Data

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

Example of Simple Ranking

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

Examples and Practice Problems

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

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.

Examples and Practice Problems

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)

X

Mo

Mdn

• 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

MoMdnX

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

Examples and Practice Problems

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

Questions?

End Part I