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Chapter 7

PSYC 300A Chapter Notes - Chapter 7: Unimodality, Linear Map, Xm Satellite Radio

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David Medler

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mathematical properties — remember:
-adding/subtracting a constant to each value in the data set increases/
decreases the value of the SD by the same amount
-multiplying/dividing by a constant to each value in the data set
multiple/divides the value of the SD by same amount
empirical distributions: only occur when we know everything about the
distributions (i.e. we have all the data)
-allows us to use the standard distribution
-frequency distribution: an empirical distribution of recoded data
represents the observed variables in a research study
allows to create other graphic figures, such as bar graphs (if the IV
represents an ordinal or nominal scale), line graph, or histogram
graphic figures allow us to see our distribution
-shape of distribution of data varies:
(a) kurtotic: unimodal & symmetrical
(i) mesokurtic: normally shaped distribution
variability in the data is what we would consider normal
(ii) platykurtic: distribution with a flattened middle & “high” tails
large space between the axis & the tails
variability in the data is greater than normal
(iii) leptokurtic: distribution with a tall, narrow, high peaked
middle, & low, skinny tails
little space between the axis & the tails
variability in the data is less than normal
(b) skewed: either positively or negatively skewed
(c) bimodal: 2 modes
(d) rectangular: no mode
normal distribution: a specific form of a mesokurtic distribution that is
characterized by a bell-shaped curve that is mathematically defined
-may be a distribution of real values (real data), or be a theoretical
distribution of hypothetical data
-a true normal distribution has fixed properties used when applying
probability theory to inferential analysis
empirical data — can be described using:
(a) visual descriptions: graph & tables
(b) numerical descriptions: measures of CT & variability
(c) location: use descriptive information to compare one value’s placement
within a set of values
(d) univariate data: percentage, percentile ranks, standard scores
(i) percentages: compares your performance to the test
(ii) percentile ranks: percentage of values at or below a particular value,
relative to all the values in the data set
-an ordinal measure — the ordinal position
-ranks one value relative to other values (ex. 8th out of 10 students)
-doesn’t tell you what actual difference exists between values
-a measure of physical location
-unaffected by the quantitative value of a score !
(ex. both a 80% & 85% — quantitive — are an A — qualitative)
(iii) standard scores: transforms raw scores into a standard (universal)
unit of measure
-the location of a single score relative to the mean, in units of
standard deviations
-locates a raw score as the “number of standard deviations that the
score is above or below the mean of the data”
-preserves the raw score’s distance from other scores & the shape
of the distribution
-(+) computes a precise score
-(+) locates a score relative to other scores in distribution
-(+) compares scores within the distribution
-(+) compares scores from different distributions (as long as they
both have the shape, means, & SDs)
-(-) cannot compare scores across distributions unless both
distributions have the same shape
-(-) it is unusual to find 2 identically shaped distributions
-ex. z-scores, t-scores, F-scores, SkewP, Pearson r, T-scores
(e) bivariate data: correlation coefficient
z-scores: a linear transformation with a sign & number — z = (X - MX) ÷ SD
-can use a z-score to find a raw score — X = MX + z(SD)
-transforms a raw score to a standard score for which the new mean of the
distribution is always µ = 0.00 & the standard deviation is σ = 1.00
-reflects how much each score deviates from the mean & adjusts it by the
variability in the data
-sign — the score’s location above (+) or below (-) the mean
-number — a measure of the distance between a score & the mean in
number of standard deviation units
-total area under the curve always = 1.00
-all scores in the data set are included in the distribution - is applied to
every score in the data set
-(+) one of the easiest types of standard scores to compute
-(+) allows you to compare relative scores across distribution
-(+) retains the shape of the raw score distribution
-(+) it changes the numerical scale of the distribution, but not the
relationship among scores
-(-) only applied to interval & ratio data
standard normal distribution: a set of normally distributed raw scores
that have been converted into z-scores
-an essential component for inferential analysis
-retains all of the qualities of a normal distribution
-original data is measured on an interval or ratio scale
-is a linear transformation of scores
-data are normal distributed
-µ = 0.00 & σ = 1.00
-under the curve = 100% of scores (proportion will be 1.00)
-each z-score is associated with a fixed proportion under the curve !
(ex. the z-score between the values of 0.0 & +1.0 SDs is always = 0.3413
of the total area)
-because of standardization, shape of the distribution can reflect different
forms of kurtosis & retain fixed properties for area under the curve
z-score table: represents area under the curve (p) to the left of the z-score
-1st column = first 2 digits of z-score
-2nd row = 3rd digit of z-score
-positive z-score + area to left read p-value from table
-positive z-score + area to right subtract p-value from 1.0000 (1 - p)
-negative z-score + area to left subtract p-value from 1.0000 (1 - p)
-negative z-score + area to right read p-value from table
-area between negative z-score + mean subtract 0.5 from p-value
-area between positive z-score + mean subtract p-value from 1.0000
-find the z-score given a proportion by locating the closest proportion on
the table (using normal rounding rules)
if the proportion >0.5 read z-score directly from table
if proportion <0.5 subtract it from 1.00 & find that z-score
PSYC 300A - Chapter 7: Empirical Distributions
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