Statistics & Research Methods Exam 2
The proportion of scores that occur between standard scores of 0 and -1 is the same as the proportion of scores that occur between standard scores of ____ and ______. This directly follows the fact that all normal distributions are symmetrical about the mean
0 and +1
What are the values of the mean and standard deviation for any set of standard scores?
0, 1.00
A set of standard scores will always have a mean of __ and a _____ _______ of 1.00 regardless of what the original mean and standard deviation were
0, standard deviation
In fact, the standard deviation (and the variance) of a set of standard scores is always equal to ______
1.00
The percentile rank that corresponds to Chesters score on a class test is 30. This means that....
30% of the obtained scores were less than or equal to Chesters
Recall that the median is the score that 50% of the scores in a distribution are greater than and 50% of the scores are less than. The median thus corresponds to the _______ percentile. Accordingly, the procedure for calculating the ______ percentile is the same as that for calculating the median.
50th, 50th
For instance, the value that corresponds to a percentile rank of 60 is referred to as the ______ percentile
60th
What does a positive standard score indicate about the original scores position relative to the mean? What does a negative standard score indicate about the original scores position relative to the mean?
A positive standard score indicates that the original score is greater than the mean, a negative standard score indicates that the original score is less than the mean
Percentile ranks must _________ be interpreted relative to the group upon which the scores are based
ALWAYS
The mean of a normal distribution is?
Also its median
This area also defines the Z scorees that occur in that interval ?
Area under the normal curve
An important use of standard scores is to compare scores on distributions that have?
Different meand and different standard deviations
If so many potential pitfalls, why compute correlations?
Does tell us if there is a linear relationship between two variables Can tell us quantitatively the strength and directionality That's a lot of information that can help us design experiments to test things like causality
T or F: A set of scores that are normally distributed will not be normally distributed after having been converted to standard scores
FALSE, they will remain the same
Simple Probabilities Definition
The likelihood of an event occurring Given set of outcomes are mutually exclusive and exhaustive
In practice, all possible raw score values and the corresponding standard scores should be included in what?
The listing of raw, standard, and t score
The resulting value of T is usually rounded to?
The nearest whole number
Note that the only difference between the formula for population standard scores versus sample standard scores is?
The only difference between these two formulas is the notion that it is used for the mean and the standard deviation of the distribution
A standard score of zero indicates that?
The original score is equal to the mean
A posiitve Standard score indicated that?
The original score is greater than the mean
A negative standard score indicates that?
The original score is less than the mean
In the Miller Analogies Test the task of the test taker is what?
The task of the test taker is to select the answer that makes two comparisons analogous to one another
Percentile (Xp) Definition
The value corresponding to a given percentile rank Position known, score unknown
Which of the following is a problem with percentile ranks as an index of relative standing?
They do not tell us anything about the MAGNITUDE of the score of interest relative to the other scores in the distribution
Cell Definition
a unique combination of variables in a contingency table
Formula for slope (b)
b = ΣXY - (ΣX)(ΣY)/ N / ΣX^2 - (ΣX)^2 /N
Linear relationship Definition
between two variables is one in which the relationship can be most accurately represented by a straight line.
Consequently, the percentile rank and the __________ ___________ for a given score value will differ (1)
cumulative percentage (1)
Formula for the normal curve
d(X) = e ^k / square root of 2 pi sigma squared
John recieved a standard score of -2.37 on an exam. This means that John...?
did worse than the class average
The sign of the correlation coefficient indicates the __________ of the linear approximation
direction
T scores are _______ _______ to standard scores, but instead of having a mean of 0 and standard deviation of 1.00, they have a mean of 50.00 and a standard deviation of 10.00
directly analogous
The process of standardizing scores _____ ______ change the fundamental shape of a distribution
does not
Restriction of Range Definiton
effect on the correlation coefficient of examining only a portion of the range of a variable
Correlation coefficient (r) Definition
expresses quantitatively the magnitude and direction of the relationship
Marginal Frequencies Definition
frequencies summed across rows or down columns
Standard error of the estimate:
index of predictive error Syx
Some texts refers to any standard score as a Z score. However, in traditional statistics, a Z score is used only to represent a standard score in a ______ _________
normal distribution
Spurius Relationship
not being what it purports to be; false or fake
A percentile rank is one index of the relative position of a score in a set of scores, However percentile ranks reflect only an _______ measure of relative standing
ordinal
Percentile Rank (PRx) Defintion
percent of scores in a distribution that are at or below a given value Score know, position unknown
Normal distributions can be conceptualized as ______ ________
probability density functions
Thus, knowing that a set of scores approximates a normal distribution allows us to make ______________ ____________ with respect to those scores.
probability statements
Correlation Formula
r = {SQUARE ROOT OF} SCP/(SSx)(SSy)
Non linear Relationships
r not sensitive to curvilinear relationships
Whereas percentile ranks reflect only an ordinal measure of relative standing, a standard score converts a score from its original or ____ form to a form that takes into consideration its standing relative to the mean and standard deviation of the entire distribution of scores
raw
Because the mean and standard deviation of a set of standard scores always take on the same vales (0 to 1.00 respectively), it is possible to compare ______ ________ from different distributions by converting them to ______ _______
raw scores, standard scores
A set of scores that are positively skewed will ______ positively skewed after having been converted to standard scores
remain
Standard Score (z) Definition
represents the number of standard deviation units a score falls above or below the mean.
Standard scores provide information about a scores location relative to the mean in _______ units of the distribution
standard deviation
Linear Equation
y = a + bX
Definition Formula of SCP
Σ(x-XBAR)( y-YBAR) How a set of X and Y scores covary from each other (covary)
Sensitive to Outliers Definition
Both magnitude and sign can be influenced-Most sensitive when sample size is small
Some researchers round percentile calculations to units of the original scale, trading off technical precision for ease of presentation. We believe that both positions havemerit , in either case it is important to realize that both approaches yield only an approximation to the true percentile of interest. We follow the strategy of?
Carrying percentile calculations to two additional decimal places
A percentile rank is different from a ___________ __________ (2)
Cumulative Percentage (2)
We can also reverse these procedures and determine raw scores that?
Define specified areas of an approximately normal distribution
The magnitude of the correlation is indexed by its absolute value-
Degree to which linear relationship is approximated
A standard score does this:
It converts a score from its original, or raw form to a form that takes into consideration its standing relative to the mean and standard deviation of the entire distribution of scores
A standard score yields infromation about?
It yields considerable information about the relative position of a score in a normal distribution. Such scores are even more meaningful when they occur in a normal distribution
The median of absolute values of the difference between each score in a data set and the median is referred to as the?
MAD
The formula for calculating the median for a continuous variable is? Where mdn represents? L represents? N represents? nL represents? nW represents? and i represents?
Mdn = L + [ (N) (.50) - nL / nW ] i MDN= median L = lower real limit of the category that contains the median N = the total number of scores in the distribution nL = the number of scores that are less than L nW = the number of scores that are within the category that contains the median i = the size of the interval of the category that contains the median
The ______ ______ _________ is designed to measure the general aprtitude of applicants for graduate and professional study
Miller Analogies Test
Correlation & Causation Issues
The fact that two variables are correlated does not imply that one variable causes the other to vary-Very possible for two variables to be correlated and have no causal relationship X could cause Y Y could cause X Or a third variable could be responsible for their relationship Correlation coefficient does not tell us which is the case
Standard scores are particularly meaningful when they occur in ______ ________
Normal Distributions
The _______ of the standard score formula reflexts the number of units that a score is above or below the mean of its distrubution. When the numerator is divided by the standard deviation, the result expresses the number of standard deviations the score is above or below its mean.
Numerator
Percentile Rank Forumla? What does nW stand for? nL? and N?
PRx = [(.5)(nw) + nL / N] (100) nW= number of scores in the category containing the score nL= number of scores below the category containing the target score N = total number of scores
The value that corresponds to a given percentile rank is called a __________
Percentile
The percentages of scores in a distribution that occur at or below a given value, X, is the _________ _________ of that value. (1)
Percentile Rank (1)
Thus, one way to convey the relative position of a score, is in terms of its ___________ ___________ (2)
Percentile Rank (2)
Properties of Standard Scores
Positive if original score is greater than the mean Negative if original score is less than the mean A z score of zero = the mean The sum of z scores = 0 Σz = 0
Regression Equation
Predicted Y = a + bX
Computational Formula for SCP
SCP = ∑ XY - (∑X)(∑Y)/ N
Least Squares Criterion
SIGMA = ( y - y PREDICTED) ^2 = minimum
Computational Formula for the Sum of Squares (SSx & SSy)
SSx = ∑X2 - (∑X)2 / N SSy = ∑Y2 - (∑Y)2 / N
It is possible to reverse the above procedures and determine the raw scores that define specified areas of an approximately normal distribution. This is accomplished be re-arranging two equations, substiuting the values of the distribution mean and standard deviation along with the relevant value of Z into one of the revised equations solving for X
Sample data formula : X = X(bar) + (z)(s) Population data formula: X = υ +z (δ)
What is a contingency table?
Separates subjects into 2 or more categories with 2 variables. Shows frequencies within categories
This is because the standard score reflects ?
Signed deviation scores
A _______- ________ represents the number of standard deviation units that a score falls above or below its mean. In this way a ______ _______ summarizes an individuals relative standing, taking into consideration the mean and standard deviation of the distribution
Standard Score
An index of relative standing that reflects the difference is?
Standard scores
All normal distributions are?
Symmetrical and characterized by a "bell shape" ; the mode, the median, and the mean of a normal distribution all have the same value, and this value defines the point that the distribution is symmetrical about; and normal distributions are theoretical in nature
Transformation of T scores Formula
T = 50 +10 (Standard score)
The negative signs indicate?
That in both cases, the raw score is below the mean of its distribution
When do you use appendix B?
To find the normal curve
T or F: A set of scores that are platykurtic will still be platykurtic after having been converted to standard scores
True
T or F: As an aid to interpreting test scores, most educational and psychological test mannuals report the standard score equivalents of raw scores in addition to percentile tables.
True
T or F: It follows that if the sum of a set of standard scores is always equal to zero the mean of a set of standard scores is always equal to zero.
True
T or F: The Pearson Correlation coefficient (r) can range from -1.0 to + 1.0
True
T or F: The value that corresponds to a given percentile rank is called a percentile
True
T or F: We learned that the mode, median, and mean are of a normal distribution all have the same value, In fact this value defines the point that a normal distribution is symmetrical about. Another important feature of normal distributions is that they are theoretical in nature
True
Regression Definition
When two variables are used to identify a line that represents the best fit for the data points
Formula for determining the value that defines a given percentile? What does the L, N, nL, nW, and i represent?
Xp = L + [(N)(P) - nL / nW] i L = lower real limit of category containing the score N = total number of scores nL= number of scores below the category containing the target score nW= number of scores in the category containing the score i = the size of the category intervals
Regression Formula Explained
Y' = a + bX a = y intercept, b = slope b = SCP/SSx a = mean of Y - slope (mean of x)
Where Z is the value of _____ that cuts off the specified portion of the distribution
Z
Standard Score Sample Formula
Z = X - _____ / s
Standard Score Population Formula
Z = X - μ / δ
A standard score in a normal distribution is refferred to as a ___ _______
Z score
One such property is that the proportion of scores that occur at or below or at or above a given standard score is the same in all normal distributions as is the proportion of scores that occur between ?
two specified standard scores
