Download Chapter 3

Chapter 3. Statistical Refresher
Scales of Measurement
-Measurement: The act of assigning numbers or symbols to characteristics of objects, events, or
people according to rules (i.e., length)
-Scale: A set of numbers or other symbols assigned to events or objects according to empirical
basis/rule (outcome of measurement)
(a) Nominal Scale: Categorization or classification (i.e., suicidal or not)
(b) Ordinal Scale: Rank-ordering characteristics (i.e., extremely happy, moderately happy,
happy, unhappy).
(c) Interval Scale: Contain equal intervals between numbers (i.e., the difference between 1 and 2
is equal to the difference between 13 and 14). But no absolute zero (i.e., IQ score)
(d) Ratio Scale: Has a true zero point (i.e., length, weight).
Describing Data
1. Frequency distributions: Listing all scores alongside the number of times each score occurred.
(a) Grouped Frequency distribution: Listing test-score-intervals or class intervals alongside the
number of times each class interval occurred.
(b) Histogram: A graph describing data with class intervals (i.e., 21-30, 31-40, etc.) on the x-axis
and the number of times each class interval occurred on the y-axis.
(c) Bar graph: A graph describing data with some categorization on the x-axis and the number of
times each class (i.e., men vs. women) occurred.
(d) Frequency Polygon: A graph describing data with connecting points where test scores or
class intervals on x-axis meet frequencies of the scores or intervals on the y-axis.
2. Central Tendency
(a) Mean: Arithmetic average.
(b) Median: A middle score in a distribution (when even frequency, add and divide by 2).
(c) Mode (i.e., bimodal): The most frequently occurring score.
3. Measures of variability (The extent to which scores in a distribution are scattered or
dispersed?)
(a) Range: The difference between the highest and lowest scores.
(i.e., Interquartile vs. Semi-interquartile range)
(b) Standard deviation: The standard distance between a raw score and the population mean.
The square root of the average squared deviations about the mean.
(S.D = Root of [Sigma (X-population)square / n]
(c) Standard error: The standard distance between a sample mean and the population mean.
(S.E = S.D / n)
(d) Variation: Squared S.D.
(e) Skewness: The nature and extent to which symmetry is absent.
Positively (when high scores are relatively few) or negatively (when low scores are relatively
few) skewed
(f) Kurtosis: The steepness of a distribution (Leptokurtic, Mesokurtic, Platykurtic)
4. Standard Scores
-A standard score is a raw score that has been converted from one scale into another scale that is
more widely used, interpretable, and has some arbitrary mean and standard deviation).
-Why do we need standard scores?
(a) The raw scores have no meaning, no information about test-taker’s relative performance.
(b) Compare raw scores between tests.
5. Normal curve: a bell-shaped, smooth, mathematically defined curve highest at the center and
the gradually tapered on both sides approaching x-axis asymptotically (it is approaching but not
touching it).
(i.e., 34%, 68%, 95%)
(a) Z score (= X – mean / s) (mean=0, sd=1):
-The difference between a particular raw score and the mean divided by the standard deviation. –
-The z score expresses a score in terms of the number of standard deviation units the raw score is
below or above the mean of the distribution.
(b) T score = Another form of Z scores but different mean and s.d. (mean=50, sd=10).
sd units
Z scores
T scores
IQ scores
%
-3sd
-3
20
55
2.27
-2sd
-2
30
70
13.59
-1sd
-1
40
85
34.13
0
0
50
100
+1sd
+1
60
115
34.13
+2sd
+2
70
130
13.59
+3sd
+3
80
145
2.27