Chapter 3: A Statistics Refresher Flashcards
Statistical Tools
Used to describe, make inferences from, and draw conclusions about numbers
Measurement
Act of assigning numbers or symbols to characteristics of things (people, events, etc) according to rules
Rules Used to Assign Numbers
Guidelines for representing the magnitude (or some other characteristic) of the object being measured
Scale
Set of numbers (or other symbols) whose properties model empirical properties of the objects to which the numbers are assigned
Ways to Categorize Scales
Continuous Scale
Discrete Scale
Continuous Scale
A scale used to measure a continuous variable; exists when it is theoretically possible to divide any of the values of the scale
Discrete Scale
A scale used to measure a discrete variable; example, male or female
Units into which a continuous scale will be divided
Depends on factors such as the purpose of the measurement and practicality
Error
Refers to the collective influence of all of the factors on a test score or measurement beoynd those specifically measured by the test or measurement
Sources of Error
A distracting thunderstorm
Particular selection of test items the instructor chose to use for the test
Measuring Scale
Continuous if used for psychological and educational assessment and therefore can be expected to contain this sort of error
Four levels or Scales of Measurement
Nominal, Ordinal, Interval, and Ratio Scales; Within these, different levels or scales of measurement, assigned numbers convey different kinds of information;
Statistical Manipulation
May or may not be appropriate, depending upon the leel or scale of measurement
Nominal Scales
These scales involve classification or categorization based on one or more distinguishing characteristics, where all things measured must be placed into mutually exclusive and exhaustive categories
Arithmetic Operations
That can be performed with norminal data include counting for the purpose of determining how many cases fall into each category and a resulting determination of proportion or percentages
Ordinal Scales
Permits classification rank ordering on some characteristic is also permissible with ordinal scales; have no absolute zero point; notes how much greater one ranking is than another; limited statistical analysis
Alfred Binet
Developer of the intelligence test that bears his name, believed strongly that the data derived from an intelligence test are ordinal in nature; Emphasized that what he tried to do in the test was not to measure people but merely to classify (and rank) people on the basis of their performance on the tasks.
Rokeach Value Survey
Uses ordinal form of measurement
Zero in a Survey
Without meaning in such a test because the number of units that separate one testtaker’s score from another’s is simply not known
Interval Scales
Contain equal intervals between numbers; each unit on the scale is exactly equal to any other unit on the scale; contain no absolute zero point; it is possible to average a set of measure ments and obtain a meaningful result
Ratio Scales
Has a true zero point; all mathematical operations can meaningfully be performed because there exist equal intervals between the numbers on the scale as well as a true or absolute zero point
Ratio-Level Measurement
Employed in some types of tests and test items, especially those involving assessment of neurological functioning; Test of hand grip, timed test of perceptual-motor ability (completion of a puzzle); no testtaker can ever obtain a score of zero on an assembly task
Measurement used in Psychology
Ordinal level of measurement; intelligence, aptitude, and personality tests are basically and strictly speaking, ordinal; these tests indicate with more or less accuracy not the amount of intelligence, aptitude, and personality traits of individuals, but rather the rank-order positions of individuals
Distribution
Defined as a set of test scores arrayed for recording or study
Raw Score
Straighforward, unmodified accounting of performance that is usually numberical; may reflect a simple tally, as in the number of items responded to correctly on an achievement test
Frequency Distribution
All scores are listed alongside the number of times each score occurred; scores may be listed the frequency of occurrence of each score in one column and the score itself in the other column
Simple Frequency Distribution
What a frequency distribution is referred to, to indicate that individual scores have been used and the data have not been grouped
Grouped Frequency Distribution
Test-score intervals replace the actual test scores
Class Intervals
Test-score intervals
Graph
A diagram or chart composed of lines, points, bars, or other symbols that describe and illustrate data
Good Graph
The place of a single score in relation to a distribution of test scores can be understood easily
Types of Graphs
Histogram
Bar Graph
Frequency Polygon
Histogram
Graph with vertical lines drawn at the true limits of each test score (or class interval), forming a series of contiguous rectangles; customary for the test scores to be placed along the graph’s horizontal axis (x) and for numbers indicative of the frequency of occurrence to be placed along the graph’s vertical axis
Abcissa
X axis, the graph’s horizontal axis; where test scores should be placed
Ordinate
Y axis, the graph’s vertical axis; where numbers indicative of the frequency of occurrence should be placed
Bar Graph
Numbers indicative of frequency also appear on the Y axis, and reference to some categorization appears on the x-axis; rectangular bars are not contiguous
Frequency Polygon
Expressed by a continuous line connecting the points where test scores or class intervals (as indicated on the X-axis) meet frequencies (as indicated on the Y-axis)
Bell-Shaped Curve
Normal graphic representation of data
Measure of Central Tendency
Statistic that indicates the average or midmost score between the extreme scores in a distribution
Center of a Distribution
Arithmetic Mean
Median
Mode
Mean
The average, takes into account the actual numerical value of every score; Sum of observations divided by the number of ebservations or test scores; most appropriate measure of central tendency for interval or ratio data when distributions are believed to be approximately normal; the most stable and useful measure of a central tendency
Median
Defined as the middle score in a distribution; scores are ordered in a list by magnitude, in either ascending or descending order. If the total number of scores ordered is an odd number, then the median will be the score that is exactly in the middle, with one-half of the remaining scores lying above it and the other half of the scores lying below it; average calculated by obtaining the average of the two middle scores; appropriate measure of central tendency for ordinal, interval, and ratio data
When Median is Useful to measure Central Tendency
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
Mode
Most frequenctly occurring score in a distribution of scores;
Bimodal Distribution
When there are two scores that occur with the highest frequency; not a commonly used measure of central tendency; not calculated; one merely counts and determines which score occurs most frequently; not necessarily a unique point in a distribution; useful in conveying certain types of information; useful in analyses of a qualitative or verbal nature; useful to convey information IN ADDITION to the mean;
Variability
Indication of how scores in a distribution are scattered or dispersed
Measures of Variability
Include the range, the interquartile range, the semi-interquartile range, the average deviation, the standard deviation and the variance
Range
Equal to the difference between the highest and the lowest scores; simplest measure of variability to calculate; Because it is based on values of the lowest and highest scores, one extreme score can radically alter the value of the range; provides a quick but gross description of the spread of scores
Nature of the Range
When its value is based on extreme scores in a distribution, the resulting description of variation may be understated or overstated. Better measures include interquartile range and the semiquartile range
Quartiles
Dividing test scores into four parts such that 25% of the test scores occur in each quarter; refers to a specific point
Quarter
Refers to an interval
Interquartile Range
Measure of variability equal to the difference between Q3 and Q1; ordinal statistic
Semi-Interquartile Range
Equal to the interquartle range divided by 2;
Shape of distribution
Provided by the relative distances from Q1 and Q3 from Q2 (the median)
Perfectly symmetrical Distribution
Q1 and Q3 will be esactly the same distance from the median
Skewness
Lack of symmetry
Average Deviation (AD)
Tool that could be used to describe the amount of variability in a distribution; Deviation scored are added and divided by the total number of scores to arrive at the average deviation
Standard Deviation
Square of each score is used; a measure of variability equal to the square root of the average squared deviations about the mean; Equal to the square root of the variance; Standard deviation measures how much - on average - individual scores of a given group vary (or deviate) from the average (or mean) score for this same group. In other words, the value of standard deviation helps show how many subjects in the group score within a certain range of variation from the mean score for the entire group. Still other way to explain it is that standard deviation measures the spread of individual results around a mean of all the results;
Variance
Equal to the arithmetic mean of the squares of the differences between the scores in a distribution and their mean; squaring and summing all the deviation scores and then dividing by the total number of scores
If Standard Deviation is 14.10
1 Standard Deviation Unit is approximately equal to 14 units of measurement or to 14 test-score points
n-1
only makes a difference if n is 10 or more
Skewness
The nature and extent to which symmetry is absent; indication of how the measurements in a distribution are distributed
Positively Skewed Distribution
When relatively few of the scores fall at the high end of the distribution; for examination result, may indicate that the test was too difficult;the distance between Q3-Q2 > Q2-Q1
Negatively Skewed Distribution
When relatively few of the scores fall at the low end of the distribution; for examination results, may indicate that the test was too easy; Q3-Q2<Q2-Q1
Symmetrical Distribution
Distances from Q1 and Q3 to the median are the sme
Kurtosis
Used to refer to the steepness of a distribution in its center;
Descriptions of Distributions
Platykurtic
Leptokurtic
Mesokrtic
Platykurtic
Relatively flat
Leptokurtic
Relatively peaked
Mesokurtic
Somewhere in the middle
Abraham DeMoivre & Marquis de Laplace
First ones to develop the concept of normal curve
Karl Friedrich Gauss
Made some substantial contributions to the concept of normal curve; LaPlace-Gaussian Curve
Karl Pearson
Credited with being the first to refer to the curve as the normal curve; Gaussian curve
Normal Curve
Bell-shaped, smooth, mathematically defined curve that is at its center. From the center, it tapers on both sides apporaching the X-axis asymptotically; symettrical, the mean, median, and mode all have the same exact value; has two tails
Asymptotically
It approaches but never touches the axis
Distribution of a normal curve
Ranges from negative infinity to positive infinity
Characteristics of all Normal Distributions
50% occur above the mean and 50% of the scores occur below the mean
Approximately 34% of all scores occur between the mean and 1 standard deviation above the mean
Approximately 34% of all scores occur between the mean and 1 standard deviation below the mean
Approximately 68% of all scores occur between the mean and +_ 1 standard deviation
Approximately 95% of all scores occur between the mean and +-2 standard deviations
Tail
The area on the normal curve between 2 and 3 standard deviations above the mean; the area on the normal curve between -2 and -3 standard deviations below the mean
Standard Score
Raw score that has been converted from one scale to another scale, where the latter scale has some arbitrarily set mean and standard deviation
Why convert raw scores to standard scores
Standard scores are more easily interpretable than raw scores; with a standard score, the position of a testtaker’s performance relative to other testtakers is readily apparent
Systems for Standard Scores
z Scores
T Scores
Stanines
Other standard scores
Zero Plus or Minus One Scale
Type of standard score scale that may be thought of as the zero plus or minus one scale
z Score
Results from the conversion of a raw score into a number indicating how many standard deviation units the raw score is below or above the mean of the distribution; provides a convenient context for comparing scores on the same and different tests; zero plus or minus one scale
T Scores
Fifty plus or minus ten scale; a scale with a mean set at 50 and a standard deviation set at 10; discovered by W.A. McCall named it as such in honor of EL Thorndike; standard score system composed of a scale that ranges from 5 standard deviations below the mean to 5 standard deviations above the mean;
Stanine
Contraction of words standard and nine; test scores are often represented as stanines; they take on whole values from 1 to 9 which represent a range of performance that is half of a standard deviation width
Scores Obtained By Linear Transformation
One that retains a direct numerical relationship to the original raw score; the magnitude of difference between standard scores exactly parallels the differences between corresponding raw scores
Nonlinear Transformation
May be required when the data under consideration are not normally distributed yet comparisons with normal distributions need to be made; resulting standard score does not necessarily have a direct numerical relationship to the original, raw score
Normalizing the Distribution
Involves stretching the skewed curve into the shape of a normal curve and creating a corresponding scale of standard scores
Normalized standard Score Scale
Corresponding scale of standard scores
