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 z-score of 1.5 is 1 ½ sd above the mean, and a z-score of –2 is 2 sd below the mean.

sample z: population z: x X

zs

xz

Calculating z - scoresMath English Biology

your score 90 85 93

Mean 85 82 94

Standard deviat

ion

5 2 1

z-score

x Xz

s

In what test did she do the best?

Math

English

Biology

90 85 51

5 5z

85 82 31.5

2 2z

93 94 11

1 1z

Calculating z - scoresMath English Biology

your score 90 85 93

Mean 85 82 94

Standard deviat

ion

5 2 1

z-score 1 1.5 -1

In what test did she do the best?

Math

English

Biology

90 85 51

5 5z

85 82 31.5

2 2z

93 94 11

1 1z

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

Mean 469 591

Standard Deviation 119 148

Verbal z = (630 - 469) ÷ 119 = 1.35

Quantitative z = (700 - 591) ÷ 148 = .736

P 470 TO CHANGE Z SCORE TO PERCENTS