Ch. 5 Z-Scores
for a population with u=80 and o=12, what is the z-score corresponding to x=92?
+1.00 (92-80)/12
for a population with u=80 and o=10, what is the z-score corresponding to x=95?
+1.50 (95-80)/10
for a population with u=80 and o=12, what is the z-score corresponding to x=71?
-0.75 (71-80)/12
z-scores for comparisons
-all z-scores are comparable to each other -score from differential distributions can be converted to z-scores -the z-scores (standardized scores) allow the comparison of scores from two different distributions along
raw score (X)
-gives little information -changing ___ into z-score describes the exact location in a distribution
computing z-scores for samples
-populations are most common context for computing z-scores -it is possible to compute z-scores for samples (indicates relative position of score in sample, indicates distance from sample mean) -sample distribution can be transformed into z-scores (same shape as original distribution, same mean and standard deviation)
z-score transformation
-same shape as original distribution -mean of z-score distribution is always o -standard deviation is always 1.00
inferential statistics
-techniques that use the information from samples to answer questions about populations -also used to help interpret results from a research study -the interpretation of the research results depends on whether the sample is noticeably different from the population -using z-scores you can decide that an individual with a z-score near o is located in the center of the population and would be considered to be a fairly typical or representative individual of the population
sign
-tells location -above mean (+) -below mean (-)
numerical value
-tells number of standard deviation between X and the mean of the distribution
standardizing distribution
1. original raw scores transformed to z-scores 2. the z-scores are transformed to new x values so that the specific U and O are attained
for a population with u=100 and o=20, what is the X value corresponding to z=0.25?
105 x=100+(0.25)(20) x=100+5=105
for a population with u=100 and o=20, what is the X value corresponding to z=-0.50?
90 x=100+(-0.50)(20) x=100+(-10)=90
standardization
AT has u=500 and o=100 IQ has u=100 and o=15 point
a z-score of z=-2.00 indicates a position in a distribution _________
below the mean by a distance equal to 2 standard deviations
z-scores
identify and describe the exact location of every score in a distribution
by itself, a ________ or x-value provides very _____________ information about how that particular score compares with other values in the distribution
raw score, little
if the _________ is transformed into a _______, it tells exactly where the score is ____________ relative to all the other scores in the distribution
raw score, z-score, located
th z-score combines the ____, the _____, and the ________ into a single number that describes the location of a particular score relative to the other scores in the distribution
score, mean, standard deviation
in a sample with s=8, a score of z=-0.50, with m=60 what is the raw score?
x=m+zs x=60+(-0.50)(8) x=60+(-4.0) x=56
standardized distribution
z-score distribution
z-score for a sample
z=(X-m)/s
determining raw score from z-score for a sample
z=(x-m)/s SO x=m+zs
z-score for a population
z=(x-u)/o
determining raw score from z-score for a population
z=(x-u)/o SO X=u+zo
for a distribution of scores, which of the following z-score values represents the location closest to the mean?
z=+0.50
for a distribution of scores, which of the following z-score values represents the most extreme location on the left-hand side of the distribution?
z=-2.00
What is the z score for x=46 from a sample where m=40 and s=12?
z=X-m/s z=(46-40)/12 z=6/12=0.50