Psychological Testing: Chapter 3: A Statistics Refresher

Used to describe, make inferences from, and draw conclusions about numbers

Act of assigning numbers or symbols to characteristics of things (people, events, etc) according to rules

Guidelines for representing the magnitude (or some other characteristic) of the object being measured

Set of numbers (or other symbols) whose properties model empirical properties of the objects to which the numbers are assigned

Continuous Scale

Discrete Scale

A scale used to measure a continuous variable; exists when it is theoretically possible to divide any of the values of the scale

A scale used to measure a discrete variable; example, male or female

Depends on factors such as the purpose of the measurement and practicality

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

A distracting thunderstorm

Particular selection of test items the instructor chose to use for the test

Continuous if used for psychological and educational assessment and therefore can be expected to contain this sort of error

Nominal, Ordinal, Interval, and Ratio Scales; Within these, different levels or scales of measurement, assigned numbers convey different kinds of information;

May or may not be appropriate, depending upon the leel or scale of measurement

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

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

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

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.

Uses ordinal form of measurement

Without meaning in such a test because the number of units that separate one testtaker’s score from another’s is simply not known

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

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

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

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

Defined as a set of test scores arrayed for recording or study

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

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

What a frequency distribution is referred to, to indicate that individual scores have been used and the data have not been grouped

Test-score intervals replace the actual test scores

Test-score intervals

A diagram or chart composed of lines, points, bars, or other symbols that describe and illustrate data

The place of a single score in relation to a distribution of test scores can be understood easily

Histogram
Bar Graph
Frequency Polygon

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

X axis, the graph's horizontal axis; where test scores should be placed

Y axis, the graph's vertical axis; where numbers indicative of the frequency of occurrence should be placed

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

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)

Normal graphic representation of data

Statistic that indicates the average or midmost score between the extreme scores in a distribution

Arithmetic Mean
Median
Mode

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

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

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

Most frequenctly occurring score in a distribution of scores;

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;

Indication of how scores in a distribution are scattered or dispersed

Include the range, the interquartile range, the semi-interquartile range, the average deviation, the standard deviation and the variance

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

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

Dividing test scores into four parts such that 25% of the test scores occur in each quarter; refers to a specific point

Refers to an interval

Measure of variability equal to the difference between Q3 and Q1; ordinal statistic

Equal to the interquartle range divided by 2;

Provided by the relative distances from Q1 and Q3 from Q2 (the median)

Q1 and Q3 will be esactly the same distance from the median

Lack of symmetry

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 deviatio

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;

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

1 Standard Deviation Unit is approximately equal to 14 units of measurement or to 14 test-score points

only makes a difference if n is 10 or more

The nature and extent to which symmetry is absent; indication of how the measurements in a distribution are distributed

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

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

Distances from Q1 and Q3 to the median are the sme

Used to refer to the steepness of a distribution in its center;

Platykurtic
Leptokurtic
Mesokrtic

Relatively flat

Relatively peaked

Somewhere in the middle

First ones to develop the concept of normal curve

Made some substantial contributions to the concept of normal curve; LaPlace-Gaussian Curve

Credited with being the first to refer to the curve as the normal curve; Gaussian 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

It approaches but never touches the axis

Ranges from negative infinity to positive infinity

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

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

Raw score that has been converted from one scale to another scale, where the latter scale has some arbitrarily set mean and standard deviation

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

z Scores
T Scores
Stanines
Other standard scores

Type of standard score scale that may be thought of as the zero plus or minus one scale

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

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;

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

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

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

Involves stretching the skewed curve into the shape of a normal curve and creating a corresponding scale of standard scores