Chapter 3: A Statistics Refresher

measurement the act of assigning numbers or symbols to characteristics of things (people, events,
whatever).

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

Discrete scale has a sample space that can be counted. A categorical variable like year in high school
has four members in its sample space: {freshman, sophomore, junior, senior}
numbers between the sample space members are not allowed.

Continuous Scale the values can be any real number in the scale’s sample space. Continuous scales
therefore can have fractions or numbers with as many decimals as needed.

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.

Levels of measurement.

Nominal Scales the simplest form of measurement.

involve classification or categorization based on one or more distinguishing
characteristics, where all things measured must be placed into mutually exclusive and
exhaustive categories.

E.g. True or False

Ordinal Scales assign people to categories. Unlike nominal scales, ordinal scales have categories with
a clear and uncontroversial order.

Interval Scales Each unit on the scale is exactly equal to any other unit on the scale.
it is possible to add and subtract scores, which allows for calculating means and standard
deviations.
interval scales contain no absolute zero point.

Ratio Scales has a true zero point

Describing Data

distribution . defined as a set of test scores arrayed for recording or study

raw score a straightforward, unmodified accounting of performance that is usually numerical.

 Chapter 3: A Statistics Refresher

may reflect a simple tally, as in the number of items responded to correctly on an

achievement test.

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

simple frequency indicate that individual scores have been used and the data have not been grouped.
distribution

grouped frequency distribution used to summarize data

frequency
distribution,

test-score replace the actual test scores.

intervals/class The number of class intervals used and the size or width of each class interval (or, the
intervals, range of test scores contained in each class interval)

The highest class interval (95–99) and the lowest class interval (40–44) are referred to,
respectively, as the upper and lower limits of the distribution.

Graph Frequency distributions of test scores can also be illustrated graphically.

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

Histogram with vertical lines drawn at the true limits of each test score (or class interval),
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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.

Frequency are expressed by a continuous line connecting the points where test scores or class
Polygon intervals (as indicated on the X-axis) meet frequencies (as indicated on the Y-axis).

Measures of a statistic that indicates the average or midmost score between the extreme scores in a
Central Tendency distribution.
(mean, median,
mode)

The arithmetic denoted by the symbol ¯ X (and pronounced “X bar”), is equal to the sum of the
mean observations

The arithmetic mean is typically the most appropriate measure of central tendency for
interval or ratio data when the distributions are believed to be approximately normal.

X = Σ(X/n),
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Formula for frequency distribution

median the middle score in a distribution,

determine the median of a distribution of scores by ordering the scores in a list by
magnitude, in either ascending or descending order.
an appropriate measure of central tendency for ordinal, interval, and ratio data.
may be a particularly useful measure of 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.

The mode frequently occurring score in a distribution of scores

Except with nominal data, the mode tends not to be a very commonly used measure of
central tendency.

Measures of Statistics that describe the amount of variation in a distribution

Variability

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

two or more distributions of test scores can have the same mean even though
differences in the dispersion of scores around the mean can be wide.

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

standard as a measure of variability equal to the square root of the average squared deviations
deviation about the mean. More succinctly, it is equal to the square root of the variance.

variance calculated by squaring and summing all the deviation scores and then dividing by the
total number of scores.

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

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.
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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.

Kurtosis steepness of a distribution in its center.

platykurtic relatively flat

leptokurtic relatively peaked), or—somewhere in the middle

mesokurtic somewhere in the middle

high kurtosis characterized by a high peak and “fatter” tails compared to a normal distribution.

lower kurtosis indicate a distribution with a rounded peak and thinner tails

normal curve 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).

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.

As a general rule (with ample exceptions), the larger the sample size and the wider the
range of abilities measured by a particular test, the more the graph of the test scores
will approximate the normal curve.
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STANDARD SCORES
z Scores 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

In addition to providing a convenient context for comparing scores on the same test,
standard scores provide a convenient context for comparing scores on different tests.

z scores is called a 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. Devised by W. A. McCall
(1922, 1939) and named a T score in honor of his professor E. L. Thorndike, this standard
 Chapter 3: A Statistics Refresher

score system is composed of a scale that ranges from 5 standard deviations below the
mean to 5 standard deviations above the mean

One advantage in using T scores is that none of the scores is negative. By contrast, in a z
score distribution, scores can be positive and negative;

stanine, Researchers during World War II developed a standard score with a mean of 5 and a
standard deviation of approximately 2.
Standard and nine

IQ scores distribution typically has a mean set at 100 and a standard deviation set at 15.

linear one that retains a direct numerical relationship to the original raw score.
transformation

nonlinear may be required when the data under consideration are not normally distributed yet
transformation 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.

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

Correlation and Inference

Inference (deduced conclusions) about how some things (such as traits, abilities, or interests) are
related to other things (such as behavior)

coefficient of A coefficient of correlation (r) expresses a linear relationship between two (and only two)
correlation (or variables,
correlation usually continuous in nature.
coefficient) It reflects the degree of concomitant variation between variable X and variable Y.
the numerical index that expresses this relationship: It tells us the extent to which X and Y
are “co-related.”

correlation is an expression of the degree and direction of correspondence between two things.

A perfect is almost impossible to achieve.

Positive If two variables simultaneously increase or simultaneously decrease. They are directly
correlation correlated.

A negative (or occurs when one variable increases while the other variable decreases.
inverse)
correlation

The Pearson r Devised by carl pearson

Pearson Most widely used
correlation
coefficient and
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the Pearson can be the statistical tool of choice when the relationship between the variables is linear and
product-mome when the two variables being correlated are continuous (or, they can theoretically take any
nt coefficient value).
of correlation
0= suggest a weak or no correlation.
±0.75 =are often considered strong,
±0.30 and ±0.49 = are considered moderate
= below ±0.29 are weak.

Pearson r should be used only if the relationship between the variables is linear.

coefficient of The coefficient of determination is an indication of how much variance is shared by the X-
determination, and the Y-variables Simply square the correlation coefficient and multiply by 100; the result
or r2 is equal to the percentage of the variance accounted for. If, for example, you calculated r to
be .9, then r2 would be equal to .81. The number .81 tells us that 81%.

The Spearman Developed by Charles Spearman,

Rho this coefficient of correlation is frequently used when the sample size is small (fewer than 30
(rank-order pairs of measurements)
correlation especially when both sets of measurements are in ordinal (or rank-order)
coefficient/ran
k-difference
correlation
coefficient,

Graphic Representations of Correlation

bivariate a simple graphing of the coordinate points for values of the X-variable (placed along the
distribution/ graph’s horizontal axis) and the Y-variable (placed along the graph’s vertical axis).
scatter are useful because they provide a quick indication of the direction and magnitude of the
diagram, relationship, if any, between the two variables.
/scattergram,
/scatterplot Scatterplots are useful in revealing the presence of curvilinearity in a relationship. As you
may have guessed.
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curvilinearity in this context refers to an “eyeball gauge” of how curved a graph is.

Outlier An outlying distance—from the rest of the coordinate points in a scatterplot

sometimes help identify a test taker who did not understand the instructions, was not able to
follow the instructions etc.
 Chapter 3: A Statistics Refresher

unrestricted
and restricted
ranges.

People who have occasion to use or make interpretations from graphed data need to
know if the range of scores has been restricted in any way.

Meta-Analysis defined as a family of techniques used to statistically combine information across

studies to produce single estimates of the data under study

researchers raise (and strive to answer) the question: “Combined, what do all of these
studies tell us about the matter under study?”

Effective size The estimates derived,

may take several different forms. In most meta-analytic studies, effect size is typically
expressed as a correlation coefficient.