This

**preview**shows half of the first page. to view the full**1 pages of the document.**• 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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