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 categorize and 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

TOTAL

42.5

40.0

37.5

35.0

32.5

30.0

27.5

25.0

22.5

20.0

17.5

15.0

12.5

10

8

6

4

2

0

Std. Dev = 8.12

Mean = 29.4

N = 40.00

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 11Picture 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

73.211

94110001149

Variance formula

N

XXS

2

2 )(

Standard Deviation formula

N

XXS

2)(

The Normal Distribution

(Brown, 1996, p. 130)

Tail Tail

Peak

Sample versus population

Describing distributions

Leptokurtic Platykurtic

Think “Leprechaun” Think “Platypus”

Skewed distributions

(Brown, 1996, p. 141)

Tail

Tail

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)

Practice

A1. 85 is what percentile?

16 (15.9)

A2. What percent between 70 and 115?

82 (81.85)

A3. How many SD is Iliana (177)?

About 5 (5.13)

A4. Iliana’s z =? T=?

CEEB =?

z = (x – m) / sd

T = 10z + 50

CEEB = 100z + 500

5 (5.13)100 (101.3)

1000 (1013)

Application exercises

Student Raw z-score T-score CEEB

A 64 70

B 50

C -1

D -1.5 350

2 700

0 50 500

43 40 400

39.5 35

Raw score mean = 50, raw score standard deviation = 7

Population and Sample

  Population S1 S2 S3 S4 S5 S6 S7 S8 S9 S10

  3 3   3   3   3   3  

  6 6     6   6 6   6 6

  6 6   6 6       6   6

  9   9   9   9   9    

  12   12 12   12   12 12   12

  15   15     15 15     15   Average

Mean 8.5 5 12 7 7 10 10 7 9 8 8 8.3

SD(N) 4.03 1.41 2.45 3.74 1.41 5.10 3.74 3.74 2.45 5.10 2.83 3.20

SD(N-1)   1.73 3.00 4.58 1.73 6.24 4.58 4.58 3.00 6.24 3.46 3.92