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Basic statistics in
assessment
Mean, variability, correlation
Scales of Measurement
 Measurement= A set of rules for assigning numbers to
represent objects, traits, attributes, or behaviors
 Scale of measurement= A system or scheme for
assigning values or scores to the characteristic being
measured
 The four scales of measurement are nominal, ordinal,
interval, and ratio
 Each scale of measurement has its own limitations
Nominal Scales
 The simplest of the four scales
 Provide a qualitative system for categorizing people or
objects into categories, classes, or sets
 Categories are typically mutually exclusive (do not overlap)
 Numbers may be assigned to each category for ease of
interpretation, but the numbers are arbitrary values and not
ordered, and thus should not be manipulated or ranked
(e.g., Male = 1, Female = 2 in an excel spreadsheet)
 As a result, not many statistical procedures can be used with
this type of scale
Ordinal Scales
 Allows one to rank people or objects according to the amount or
quantity of a characteristic they possess
 Provide more information than nominal scales
 Traditionally, ranking is ordered from “most” to “least”
 Intervals between the ranks may not be consistent (for example the
1st rank student may have a score of 95, the 2nd rank may have 90,
and the 3rd rank may have 89)
 Somewhat limited in the measurement information they can supply
(we cannot report an “average” rank for a student based on various
subjects.
 Other examples include percentile ranks, age equivalents, and
grade equivalents
Interval Scale
 Provides more information that either nominal or ordinal scales
 Allows you to rank people or objects like an ordinal scale, but on a scale with
equal units (so 81, 82, and 83 are equidistant from each other)
 Many educational and psychological tests are designed to produce interval
level scores
 Interval level data can be manipulated using math (addition, subtraction,
multiplication, and division) and most statistical procedures – typical of
teacher-made tests
 However, these scales do not have a true zero point (a score of zero on a test
does not mean an attribute, like understanding, is completely absent
 Seen in schools in the form of standard scores (e.g, scaled scores)
Ratio Scales
 Have the properties of interval scales along with a true
zero point
 Examples include miles per hour, length, and weight
 Can be used to interpret ratios between scores (e.g, 24
is twice of 12, and 4 times of 6).
 Relatively few ratio scales used in schools
Description of Test Scores
 On its own, an individual’s test score provides little
information
 To meaningfully interpret test scores, you need a frame
of reference
 Often times, the frame of reference is how other
students performed on the test (e.g., a score of 75 is
much more meaningful if you know that 99% of the
students scored below 75).
Distributions – basic concepts
 Distribution= A set of scores
 Can be represented in a table or graph
 Symmetrical= If a distribution is divided into two halves,
they will mirror each other
 Not all distributions are symmetrical (but a common one is
the Bell curve)
 Skewed= A distribution that is not symmetrical
 Negatively skewed distribution= Very few scores at the low
end (typical of achievement tests)
 Positively skewed distribution= Very few scores at the high
end
Measures of Central
Tendency
 Central tendency= When scores tend to concentrate
around a center
 Mean= The average, or the sum of scores divided by
the number of scores
 Median= The score or potential score that divides a
distribution in half; an equal amount of scores fall
above and below it
 Mode= The most frequently occurring score
Choosing Between Mean,
Median, and Mode
 The mean is essential when calculating certain (traditional)
statistics
 For descriptive purposes, the median is often the most
versatile and useful measure of central tendency, especially
when a distribution is skewed
 If you are dealing with nominal level data, the mode is the
only useful measure of central tendency
 For small groups (less than 10), it is better to use median
than the mean (because an extreme score can drag the
mean in its direction)
Measures of Variability
 Two distributions can have the same mean, median,
and mode, but differ in the way scores are distributed
around the measure of central tendency
 Therefore, measures of variability are used to
describe distributions
 The three measures are the range, standard deviation,
and variance
Measures of Variability
 Range= The distance between the smallest and largest score in a
distribution (or, highest score - lowest score) – provides limited info
 Standard deviation (SD)= A measure of the average distance that
scores vary from the mean of the distribution
 To find the standard deviation:
1. Calculate the mean
2. Subtract each score from the mean, then square the
difference
3. Sum all of the squared difference scores
4. Divide the sum by the number of scores (this is the variance)
5. The standard deviation is the positive square root of the
variance

Variance= A measure of variability that has special meaning as a
theoretical concept in measurement theory and statistics
Choosing Between the Range, Standard
Deviation, and Variance
 The range tells us the limits of a distribution, but not
how scores are dispersed within these limits
 The standard deviation indicates the average distances
that scores vary from the mean of the distribution
 The larger the standard deviation, the more variability
there is in the distribution
 Variance represents an important theoretical concept
and is based on standard deviation. You can conduct
further analysis of numbers using variance.
Correlation Coefficients
 Correlation= The relationship between two variables
 Correlation coefficient (r)= A quantitative measure of the
relationship between two variables
 Range from -1.0 to +1.0
 A positive r means that when one variable increases, the other
increases
 A negative r means that when one variable increases, the other
decreases
 The closer r is to 0, the weaker the relationship between the
variables (1.0 indicates a perfect relationship, which is a rarity)
 Coefficient of determination ()= The amount of variance
shared by the two variables
Scatterplots
 Scatterplot= A graph that visually depicts the relationship
between two variables
 To create a scatterplot, you need two different scores for
each individual
 Each plot represents two scores, one graphed on the X axis
and the other on the Y axis
 The stronger the correlation between the two variables, the
more the scatterplot will resemble a straight line
 A positive correlation will slope up, while a negative
correlation will slope down
Correlation and Prediction
 Linear regression= A mathematical procedure that allows
you to predict the values of one variable based on the
information you have about the other
 This is usually based on correlation. So, if a correlation
between two scores is high (e.g., math and physics scores),
then you can predict a student’s physics score based on
his/her math score. This is one method to use to set learning
targets (e.g., if last years students scores in math and
physics are correlated, we can predict physics scores for
this year’s students based on their math scores; this is
helpful because we usually have scores on the core subjects
like math). Another example is ELA and Social Studies
scores (which tend to be correlated)
Types of Correlation
Coefficients
 Specific correlation coefficients are appropriate for specific
situations
 Perhaps the most common one is the Pearson productmoment correlation
 Appropriate when the variables being measured are on an
interval or ratio scale
 Spearman’s rank correlation coefficient is used when
variables are measured on an ordinal scale
 The point-biserial correlation coefficient is used when one
variable only has two possible scores and the other is
measured on an interval or ratio scale
Correlation and Causality
 Just because two variables are related does not mean that
one causes the other!
 It is possible that the variables are causally related, but also
that a third variable influences the relationship (e.g.,
absences and academic achievement scores may be
correlated but caused by parents’ socio-economic status)
 Or, all the variables may mutually cause each other (e.g.,
self-esteem, academic achievement, good instructional
practices, and student interest in the subject)
 The use of inferential statistics can allow one to infer
causality in multivariate studies. This is outside the scope of
the current discussion.