Psychological Testing: Chapter 3

coefficient of correlation

(or correlation coefficient ) is a number that provides us with an index of the strength of the relationship between two things.

Linear transformation

A standard score obtained by a linear transformation is one that retains a direct numerical relationship to the original raw score.

Outlier

An extremely atypical point located at a relatively long distance—an outlying distance—from the rest of the coordinate points in a scatterplot

the mode

The most frequently occurring score in a distribution of scores

kurtosis

The term testing professionals use to refer to the steepness of a distribution in its centre

Normal curve

a bell-shaped, smooth, mathematically defined curve that is highest at its center. From the center it tapers on both sides approaching the X -axis asymptotically (meaning that it approaches, but never touches, the axis). In theory, the distribution of the normal curve ranges from negative infinity to positive infinity. The curve is perfectly symmetrical, with no skewness. If you folded it in half at the mean, one side would lie exactly on top of the other. Because it is symmetrical, the mean, the median, and the mode all have the same exact value.

Meta analysis

a family of techniques used to statistically combine information across studies to produce single estimates of the data under study. The estimates derived, referred to as effect size , may take several different forms. In most meta-analytic studies, effect size is typically expressed as a correlation coefficient.

Moments

a moment describes a deviation about a mean of a distribution. Individual deviations about the mean of a distribution are referred to as deviates. Deviates are referred to as the first moments of the distribution. The second moments of the distribution are the moments squared. The third moments of the distribution are the moments cubed, and so forth.

frequency distribution

all scores are listed alongside the number of times each score occurred.

Variability

an indication of how scores in a distribution are scattered or dispersed.

Nominal scales

are the simplest form 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.

T scores

can be called a fifty plus or minus ten scale; that is, a scale with a mean set at 50 and a standard deviation set at 10. this standard score system is composed of a scale that ranges from 5 standard deviations below the mean to 5 standard deviations above the mean. a raw score that fell exactly at 5 standard deviations below the mean would be equal to a T score of 0, a raw score that fell at the mean would be equal to a T of 50, and a raw score 5 standard deviations above the mean would be equal to a T of 100. One advantage in using T scores is that none of the scores is negative.

interval scales

contain equal intervals between numbers. Each unit on the scale is exactly equal to any other unit on the scale. But like ordinal scales, interval scales contain no absolute zero point. With interval scales, we have reached a level of measurement at which it is possible to average a set of measurements and obtain a meaningful result.

standard deviation

equal to the square root of the average squared deviations about the mean. More succinctly, it is equal to the square root of the variance. The variance is equal to the arithmetic mean of the squares of the differences between the scores in a distribution and their mean.

histogram

graph with vertical lines drawn at the true limits of each test score (or class interval), forming a series of contiguous rectangles. It is customary for the test scores (either the single scores or the midpoints of the class intervals) to be placed along the graph's horizontal axis (also referred to as the abscissa or X -axis) and for numbers indicative of the frequency of occurrence to be placed along the graph's vertical axis (also referred to as the ordinate or Y -axis).

Scatterplots

graphic representation of correlation referred to by many names, including a bivariate distribution , a scatter diagram , a scattergram , or—our favorite—a scatterplot . A simple graphing of the coordinate points for values of the X -variable (placed along the graph's horizontal axis) and the Y -variable (placed along the graph's vertical axis). Scatterplots are useful because they provide a quick indication of the direction and magnitude of the relationship, if any, between the two variables.

normalizing a distribution

involves "stretching" the skewed curve into the shape of a normal curve and creating a corresponding scale of standard scores, a scale that is technically referred to as a normalized standard score scale

The interquartile range

is a measure of variability equal to the difference between Q3 and Q1 . Like the median, it is an ordinal statistic.

range of a distribution

is equal to the difference between the highest and the lowest scores.

coefficient of determination

or r 2 . The coefficient of determination is an indication of how much variance is shared by the X - and the Y -variables.

distribution

may be defined as a set of test scores arrayed for recording or study. The scores in this distribution are referred to as 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. In a nonlinear transformation, the resulting standard score does not necessarily have a direct numerical relationship to the original, raw score. As the result of a nonlinear transformation, the original distribution is said to have been normalized.

bar graph

numbers indicative of frequency also appear on the Y -axis, and reference to some categorization (e.g., yes/no/maybe, male/female) appears on the X -axis. Here the rectangular bars typically are not contiguous.

bimodal distribution

occurs when there are two scores that occur with the highest frequency

ordinal scales

permit classification. However, in addition to classification, rank ordering on some characteristic is also permissible with ordinal scales. Ordinal scales imply nothing about how much greater one ranking is than another. Even though ordinal scales may employ numbers or "scores" to represent the rank ordering, the numbers do not indicate units of measurement.

Pearson correlation coefficient

r can be the statistical tool of choice when the relationship between the variables is linear and when the two variables being correlated are continuous (that is, they can theoretically take any value).

curvilinearity

refers to an "eyeball gauge" of how curved a graph is.

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. a z score is equal to the difference between a particular raw score and the mean divided by the standard deviation.

grouped frequency distribution

test-score intervals, also called class intervals, replace the actual test scores.

measurement

the act of assigning numbers or symbols to characteristics of things (people, events, whatever) according to rules. The rules used in assigning numbers are guidelines for representing the magnitude (or some other characteristic) of the object being measured.

arithmetic mean

the most commonly used measure of central tendency, which is referred to in everyday language as the "average." The mean takes into account the actual numerical value of every score. Denoted by the symbol X (and pronounced "X bar"), is equal to the sum of the observations (or test scores, in this case) divided by the number of observations.

skewness

the nature and extent to which symmetry is absent. Skewness is an indication of how the measurements in a distribution are distributed.

Negative skew

A distribution has a negative skew when relatively few of the scores fall at the low end of the distribution. Negatively skewed examination results may indicate that the test was too easy.

Positive skew

A distribution has a positive skew when relatively few of the scores fall at the high end of the distribution. Positively skewed examination results may indicate that the test was too difficult.

Quartile scores

A distribution of test scores (or any other data, for that matter) can be divided into four parts such that 25% of the test scores occur in each quarter. the dividing points between the four quarters in the distribution are the quartiles . There are three of them, respectively labeled Q1, Q2, and Q3. quartile refers to a specific point whereas quarter refers to an interval. An individual score may, for example, fall at the third quartile or in the third quarter (but not "in" the third quartile or "at" the third quarter).

Tails

A normal curve has two tails. The area on the normal curve between 2 and 3 standard deviations above the mean is referred to as a tail . The area between 2 2 and 2 3 standard deviations below the mean is also referred to as a tail.

standard score

A raw score that has been converted from one scale to another scale, where the latter scale has some arbitrarily set mean and standard deviation. Raw scores may be converted to standard scores because 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.

Frequency polygon

Data illustrated in a frequency polygon are 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).

Kurtosis descriptions

Distributions are generally described as platykurtic (relatively flat), leptokurtic (relatively peaked), or— somewhere in the middle— mesokurtic .

ratio scale

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.

Error

Measurement always involves error. In the language of assessment, error refers to the collective influence of all of the factors on a test score or measurement beyond those specifically measured by the test or measurement.

Spearman's rho

One commonly used alternative statistic is variously called a rank-order correlation coefficient , a rank-difference correlation coefficient , or simply Spearman's rho . Developed by Charles Spearman, a British psychologist ( Figure 3-11 ), this coefficient of correlation is frequently used when the sample size is small (fewer than 30 pairs of measurements) and especially when both sets of measurements are in ordinal (or rank-order) form. Special tables are used to determine whether an obtained rho coefficient is or is not significant

Stanines

Researchers during World War II developed a standard score with a mean of 5 and a standard deviation of approximately 2. Divided into nine units, the scale was christened a stanine , a term that was a contraction of the words standard and nine. Stanines are different from other standard scores in that they take on whole values from 1 to 9, which represent a range of performance that is half of a standard deviation in width

Advantages to meta analyses

Some advantages to meta-analyses are: (1) meta-analyses can be replicated; (2) the conclusions of meta-analyses tend to be more reliable and precise than the conclusions from single studies; (3) there is more focus on effect size rather than statistical significance alone; and (4) meta-analysis promotes evidence-based practice, which may be defined as professional practice that is based on clinical and research findings

measures of variability

Statistics that describe the amount of variation in a distribution. Some measures of variability include the range, the interquartile range, the semi-interquartile range, the average deviation, the standard deviation, and the variance.

scale

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

measure of central tendency

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

raw score

a straightforward, unmodified accounting of performance that is usually numerical. A raw score may reflect a simple tally, as in number of items responded to correctly on an achievement test.

median

defined as the middle score in a distribution, is another commonly used measure of central tendency. We determine the median of a distribution of scores by ordering the scores 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 remaining scores lying below it. When the total number of scores ordered is an even number, then the median can be calculated by determining the arithmetic mean of the two middle scores.