An Introduction to Making
Inferences
• descriptive statistics – summarize important
characteristics of known population data
• inferential statistics – we use sample data to
make inferences or generalizations about a
population (review population and sample)
– sampling statistic - any statistic that describes the
distribution of values for a variable, or relationships
between variables in a sample
– population parameter – the estimated characteristics
of a population derived from sampling statistics
– must have a well chosen sample!!! (sound sampling
procedures)
• z-scores (and the normal distribution) enable us to
standardize values so they can be compared (example:
SAT); also remember 68-95-99.7 rule
Standard Deviation
• looking at data: the smaller the standard
deviation in relation to the range of responses,
the more homogeneous are the responses
• standard score (or z-score) – the number of
standard deviations that a given value x is above
or below the mean. In a z distribution, the mean
= 0, and the standard deviation is 1. Thus, a zscore of 1.5 is 1 ½ sd above the mean, and a zscore of –2 is 2 sd below the mean.
xX
sample z: z 
s
population z: z 
x

Calculating z - scores
your score
Math
English
Biology
90
85
93
Mean
85
82
94
Standard
deviati
on
5
2
1
z-score
In what test did she do the best?
Math
English
Biology
x X
z
s
z
90  85 5
 1
5
5
z
85  82 3
  1.5
2
2
z
93  94 1

 1
1
1
Calculating z - scores
your score
Math
English
Biology
90
85
93
Mean
85
82
94
Standard
deviati
on
5
2
1
z-score
In what test did she do the best?
Math
English
1
1.5
z
90  85 5
 1
5
5
z
85  82 3
  1.5
2
2
-1
Biology
z
93  94 1

 1
1
1
Even though the raw score was lowest for English, in
comparison she was 1.5 sd above the mean, therefore she
actually did better, in comparison with the other students
taking the exam, in English than in math or bio.
Lets say you took the GRE a few weeks ago and got scores of 630 Verbal and 700
Quantitative. How good are these scores? Which is better, the Verbal or Quantitative
score? Using a z-score can tell you how far you are from the mean and thus how well you
performed. If you know the mean and standard deviations for a set of GRE test takers you
can compare your scores.
Verbal
Quantitative
P 470 TO CHANGE Z
SCORE TO PERCENTS
Mean
469
591
Standard Deviation
119
148
Verbal z = (630 - 469) ÷ 119 = 1.35
Quantitative z = (700 - 591) ÷ 148 = .736