Module 8: Statistics Flashcards
Measurement
the act of assigning numbers or symbols to characteristics of things according to rules
Descriptive Statistics
methods used to provide concise description of a collection of quantitative information
Inferential Statistics
method used to make inferences from observations of a small group of people known as sample to a larger group of individuals known as population
Magnitude
the property of “moreness”
Equal Intervals
the difference between two points at any place on the scale has the same meaning as the difference between two other points that differ by the same number of scale units
Absolute 0
when nothing of the property being measured exists
Scale
+ a set of numbers who properties model empirical properties of the objects to which the numbers are assigned
+ the way numbers are categorized or assigned
What are the types of scales?
- Continuous Scale
- Discrete Scale
Continuous Scale
+ takes on any value within the range and the possible value within that range is infinite
+ used to measure a variable which can theoretically be divided
Discrete Scale
+ can be counted; has distinct, countable values
+ used to measure a variable which cannot be theoretically be divided
Error
+ refers to the collective influence of all the factors on a test score or measurement beyond those specifically measured by the test or measurement
+ degree to which the test score/measurement may be wrong, considering other factors like state of the testtaker, venue, test itself etc.
What kind of scale always involves error?
Measurement with continuous scale always involve with error
What are the four levels of scales of measurement?
- Nominal
- Order
- Interval
- Ratio
Nominal
+ involve classification or categorization based on one or more distinguishing characteristics
+ label and categorize observations but do not make any quantitative distinctions between observations
+ mode
Ordinal
+ rank ordering on some characteristics is also permissible
+ median
Interval
+ contains equal intervals, has no absolute zero point (even negative values have interpretation to it)
+ zero value does not mean it represents none
Ratio
+ has true zero point (if the score is zero, it means none/null)
+ easiest to manipulate
Distribution
defined as a set of test scores arrayed for recording or study
Raw Scores
straightforward, unmodified accounting of performance that is usually numerical
Frequency Distribution
all scores are listed alongside the number of times each score occurred
Independent Variable
being manipulated in the study
Quasi-Independent Variable
+ nonmanipulated variable to designate groups
+ Factor: for ANOVA
Post-Hoc Tests
used in ANOVA to determine which mean differences are significantly different
Turkey’s HSD Test
allows the compute a single value that determines the minimum difference between treatment means that is necessary for significance
Measures of Central Tendency
statistics that indicates the average or midmost score between the extreme scores in a distribution
What is the goal of measures of central tendency?
Identify the most typical or representative of entire group
Mean
+ the average of all the raw scores
+ Equal to the sum of the observations divided by the number of observations
+ Interval and ratio data (when normal distribution)
+ Point of least squares
+ Balance point for the distribution
Median
+ the middle score of the distribution
+ Ordinal, Interval, Ratio
+ Useful in cases where relatively few scores fall at the high end of the distribution or relatively few scores fall at the low end of the distribution
+ In other words, for extreme scores, use median (skewed)
+ Identical for sample and population
+ Also used when there has an unknown or undetermined score
+ Used in “open-ended” categories (e.g., 5 or more, more than 8, at least 10)
+ For ordinal data
Mode
+ Most frequently occurring score in the distribution
+ Not commonly used
+ Useful in analyses of qualitative or verbal nature
+ For nominal scales, discrete variables
+ Value of the mode gives an indication of the shape of the distribution as well as a measure of central tendency
Bimodal Distribution
if there are two scores that occur with highest frequency
Variability
an indication how scores in a distribution are scattered or dispersed
Measures of Variability
statistics that describe the amount of variation in a distribution
Range
+ Equal to the difference between highest and the lowest score
+ Provides a quick but gross description of the spread of scores
+ When its value is based on extreme scores of the distribution, the resulting description of variation may be understated or overstated
Quartile
+ dividing points between the four quarters in the distribution
+ specific point
Quarter
refers to an interval
Interquartile Range
measure of variability equal to the difference between Q3 and Q1
Semi-interquartile Range
equal to the interquartile range divided by 2
Standard Deviation
+ equal to the square root of the average squared deviations about the mean
+ Equal to the square root of the variance
+ Distance from the mean
Variance
equal to the arithmetic mean of the squares of the differences between the scores in a distribution and their mean
Normal Curve
+ also known as Gaussian Curve
+ Bell-shaped, smooth, mathematically defined curve that is highest at its center
Asymptotic
approaches but never touches the axis
Tail
2 – 3 standard deviations above and below the mean
Symmetrical Distribution
+ right side of the graph is mirror image of the left side
+ has only one mode and it is in the center of the distribution
+ mean = median = mode
Skewness
nature and extent to which symmetry is absent
Positive Skewed
+ few scores fall the high end of the distribution
+ Mean > Median > Mode
+ More items that was easier would have been desirable in order to better discriminate at the lower end of the distribution of test scores
What does it mean when the test scores are positively skewed?
The exam is difficult
Negative Skewed
+ when relatively few of the scores fall at the low end of the distribution
+ More items of a higher level of difficulty would make it possible to better discriminate between scores at the upper end of the distribution
+ Mean < Median < Mode
What does it mean when the test scores are negatively skewed?
The exam is easy
What are skewed graphs associated with?
Skewed is associated with abnormal, perhaps because the skewed distribution deviates from the symmetrical or so-called normal distribution
Kurtosis
steepness if a distribution in its center
Leptokurtic
relatively peaked
Platykurtic
relatively flat
Mesokurtic
somewhere in the middle
High Kurtosis
high peak and fatter tails
Lower Kurtosis
rounded peak and thinner tails
Standard Score
raw score that has been converted from one scale to another scale
Z-scores
+ results from the conversion of a raw score into a number indicating how many SD units the raw score is below or above the mean of the distribution
+ Identify and describe the exact location of each score in a distribution
+ Standardize an entire distribution
+ Zero plus or minus one scale
+ Have negative values
+ Requires that we know the value of the variance
T-scores
+ a scale with a mean set at 50 and a standard deviation set at 10
+ Fifty plus or minus 10 scale
+ 5 standard deviations below the mean would be equal to a t-score of 0
+ Raw score that fell in the mean has T of 50
+ Raw score 5 standard deviations about the mean would be equal to a T of 100
+ No negative values
+ Used when the population or variance is unknown
Stanine
a method of scaling test scores on a nine- point standard scale with a mean of five (5) and a standard deviation of two (2)
Linear Transformation
one that retains a direct numerical relationship to the original raw score
Nonlinear Transformation
required when the data under consideration are not normally distributed
Normalized Standard Score Scale
Normalizing the distribution involves stretching the skewed curve into the shape of a normal curve and creating a corresponding scale of standard scores
What is generally preferrable to do for a skewed curve?
Generally preferrable to fine-tune the test according to difficulty or other relevant variables so that the resulting distribution will approximate the normal curve
STEN
standard to ten; divides a scale into 10 units
Hypothesis Testing
statistical method that uses a sample data to evaluate a hypothesis about a population
Alternative Hypothesis
states there is a change, difference, or relationships
Null Hypothesis
no change, no difference, or no relationship
Alpha Level or Level of Significance
used to define concept of “very unlikely” in a hypothesis test
Critical Region
composed of extreme values that are very unlikely to be obtained if the null hypothesis is true
What is the hypothesis if the sample data falls in the critical region?
If sample data fall in the critical region, the null hypothesis is rejected
What is the alpha level for a hypothesis test?
The alpha level for a hypothesis test is the probability that the test will lead to a Type I error
Directional Hypothesis Test or One-Tailed Test
statistical hypotheses specify either an increase or a decrease in the population mean
T-Test
+ used to test hypotheses about an unknown population mean and variance
+ Can be used in “before and after” type of research
+ Sample must consist of independent observationsꟷthat is, if there is not consistent, predictable relationship between the first observation and the second
+ The population that is sampled must be normal
+ If not normal distribution, use a large sample
Correlation Coefficient
number that provides us with an index of the strength of the relationship between two things
Correlation
an expression of the degree and direction of correspondence between two things
What are the two possible directions for a correlation?
- Positive
- Negative
What is the magnitude of a correlation?
number anywhere to -1 to 1
Positive Correlation
same direction, either both going up or both going down
Negative Correlation
Inverse Direction, either DV is up and IV goes down or IV goes up and DV goes down
0 in correlation
no correlation
Pearson r/Pearson Correlation
Coefficient/Pearson Product-Moment Coefficient of Correlation
+ used when two variables being correlated are continuous and linear
+ Devised by Karl Pearson
Coefficient of Determination
an indication of how much variance is shared by the X- and Y- variables
Spearman Rho/Rank-Order Correlation
Coefficient/Rank-Difference Correlation
Coefficient
+ frequently used if the sample size is small and when both sets of measurement are in ordinal
+ Developed by Charles Spearman
Outlier
extremely atypical point located at a relatively long distance from the rest of the coordinate points in a scatterplot
Regression Analysis
+ used for prediction
+ predict the values of a dependent or response variable based on values of at least one independent or explanatory variable
Residual (in regression analysis)
the difference between an observed value of the response variable and the value of the response variable predicted from the regression line
The Principle of Least Squares
The least squares method is a statistical procedure to find the best fit for a set of data points
Standard Error of Estimate
standard deviation of the residuals in regression analysis
Slope
determines how much the Y variable changes when X is increased by 1 point
T-Test (Independent)
+ comparison or determining differences
+ 2 different groups/independent samples interval/ratio scales (continuous variables)
Equal Variance
2 groups are equal
Unequal Variance
groups are unequal
T-test (Dependent)/Paired Test
one groups nominal (either matched or repeated measures) + 2 treatments
One-Way ANOVA
3 or more IV, 1 DV comparison of differences
Two-Way ANOVA
2 IV, 1 DV
Critical Value
reject the null and accept the alternative if [ obtained value > critical value ]
P-Value (Probability Value)
reject null and accept alternative if [ p-value < alpha level ]
Norms
refer to the performances by defined groups on a particular test
Percentiles
an expression of the percentage of people whose score on a tests or measure falls below a particular raw score
Age Norms
average performance of different samples of testtakers who were at various ages at the time the test was administered
Grade Norms
developed by administering the test to representative samples of children over a range of consecutive grade levels
Developmental Norms
developed on the basis of any trait, ability, skill, or other characteristics that is presumed to develop, deteriorate, or otherwise be affected by chronical age, school grade, or stage of life
National Norms
derived from a normative sample that was nationally representative of the population at the time the norming study was conducted
Subgroup Norms
normative sample can be segmented by any criteria initially used in selecting subjects for the sample
Local Norms
provide normative information with respect to the local population’s performance on some tests
Age-related Norms
certain tests have different normative groups for age groups
Tracking
tendency to stay at about the same level relative to one’s peers
Norm-Referenced Tests
compares each person with the norm
Criterion-Referenced Tests
describes specific types of skills, tasks, or knowledge that the test taker can demonstrate
