Descriptive Statistics

Descriptive Statistics And related matters

Two families of statistics • Descriptive statistics – procedures for summarizing, organizing, graphing, and, in general, describing quantitative information • Mean, standard deviation, # of items, etc. • Inferential statistics – statistics that allow one to draw conclusions or inferences from the data • ANOVA, t-test, correlation, etc. Vogt, W. P. (1999). Dictionary of statistics & methodology (2nd ed.). Thousand Oaks, CA: Sage Publications.

Scales of measurement • Nominal • Used to name or categorize things • Female = 1,Male = 2, correct = 1,incorrect =0 • Often used for coding variables in research • Ordinal • Used to order things • Gives relative position but not amount • Rankings are ordinal Shavelson, R. J. (1996). Statistical reasoning for the behavioral sciences (Third ed.). Needham Heights, MA: Allyn & Bacon.

Scales of measurement (2) • Interval • Each scale unit represents an equal distance of the attribute being measured • Most test scores are considered interval scales • Rating scales are often treated as interval • Ratio • Interval scales with a meaningful zero point where zero indicates the absence of the attribute • examples: weight, height, length

Scale summary • Nominal scales categorize but do not order. • Ordinal scales categorizeand order. • Interval scales categorize, order, and establish an equal unit in the scale. • Ratio scales categorize, order, establish an equal unit, and contain a true zero point. Wiersma, W., & Jurs, S. G. (1990). Educational measurement and testing (2nd ed.). Needham Heights, MA: Allyn and Bacon, p. 13.

Frequency information • Frequency distribution – how many students received each score • Cumulative frequency – how many students scored at or below the score in question • Cumulative percentage – what percent of students scored at or below the score in question • Useful for seeing patterns in the data

Output for SPSS

Measures of Central Tendency • The four “M”s • Mean • Mode • Median • Midpoint

Think about it… • Scores: 18, 19, 20, 21, 87 • Which give a more accurate picture of this data, the mean or the median? • Mean = 33 • Median = 20 • The median is usually more appropriate as a measure of central tendency when there is an outlier.

For a norm-referenced test (Henning, 1987, p. 39)

Measures of Dispersion • Range • High score • Low score • Standard Deviation • Variance

Conceptualizing variance • Imagine a set of scores 8 10 13 9 7 11 10 12 10 9 11 • Picture those scores on a number line Williams & Monge (2001) Reasoning with statistics

Conceptualizing variance (2) • Imagine those scores as deviations from the mean (how far are they from the mean?)

Conceptualizing variance (3) • Variance: the mean of the squared deviation scores about the mean of a distribution

Variance formula

Standard Deviation formula

Peak Tail Tail The Normal Distribution (Brown, 1996, p. 130)

Sample versus population

Describing distributions Leptokurtic Platykurtic Think “Leprechaun” Think “Platypus”

Tail Tail Skewed distributions (Brown, 1996, p. 141)

Standardized scores • A transformation of raw scores into a measure of relative standing based on the mean and standard deviation • Useful for comparing performance on tests of different lengths, different forms, etc. • The most often used standardized scores are z-scores, T-scores, and CEEB scores. • Relative standing is usually based on the norm group (for a norm-referenced test)

Standard score comparison (Brown, 1996, p. 135)

A1. 85 is what percentile? A2. What percent between 70 and 115? A3. How many SD is Iliana (177)? Practice 16 (15.9) z = (x – m) / sd T = 10z + 50 CEEB = 100z + 500 82 (81.85) A4. Iliana’s z =? T=? CEEB =? 5 (5.13) 100 (101.3) 1000 (1013) About 5 (5.13)

Application exercises 2 700 50 500 0 40 400 43 39.5 35 Raw score mean = 50, raw score standard deviation = 7

Population and Sample

Descriptive Statistics