
 Overview
Consider the four distributions
obtained for variables in the survey done on the first day
of class. The distributions are colored with red for males
and blue for females:
We can summarize these distributions by discussing their
 Central Tendency  Mean, Median, Mode
 Central tendency measures identify a single score as
representative of an entire distribution of scores. The
goal of central tendency is to find the single score that
is most typical or most representative of the entire distribution.
 Variability  Standard Deviation, Variance, Range,
IQ Range
 Variability measures provide a quantitative indication
of the degree to which scores in a distribution are spread
out or clustered together.
 Skewness  A measure of shape
 Skewness measures provide a quantitative indication
of the degree to which scores in a distribution are located
at one end of the distribution. Positive scores mean the
distribution is positively skewed, and conversely.
 Kurtosis  Another measure of shape
 Kurtosis measures provide a quantitative indication
of the degree to which scores in a distribution are peaked
or flat. Positive values mean the distribution is peaked.
Negative values mean the distribution is flat.
The ViSta summary report presents measures of each
of these characteristics of a variable's distribution. For
these four variables, the report is:
 Three measures of variability
 The Range
The range is the distance between the largest and smallest
scores in a distribution. To determine the range, you must
use the real limits of the maximum and minimum. This
makes the formula for the range:
range = URL(Max)  LRL(Min)
where URL(Max) stands for "Upper Real Limit of
the maximum score" and LRL(min)means the "Lower
Real Limit of the minimum score".
If the distribution consists of whole numbers (integers),
the range is
range (of integers) = Max  Min + 1
Problem: Because the range doesn't consider all
of the scores in the distribution, only the extremes,
it often does not give an accurate description of the
variability.
ViSta's Range: ViSta calculates the range as
simply
range = Max  Min
ViSta leaves off the term dealing with real limits. This
is another way the range is commonly defined.
Example:For the variables in our class data (see
table above), the ranges are:
Age: (28  18 + 1) = 11
GPA: (3.93  2.20 + 1) = 2.73
MathSAT: (750  340 + 1) = 411
Satisfaction: (9  3 + 1) = 7
Questions:
 How well does the range do as a way of representing
the variation in each of these variables?
 Does it do better for some of the variables than for
others?
 If so, why?
 The Interquartile Range and SemiInterquartile Range
We begin defining the interquartile range by first defining
quartiles:
 Quartiles
 Quartiles are the percentiles at the locations which
divide a distribution into quarters. There are three
quartiles, denoted the first (Q1), second (Q2) and third
(Q3):
 Q1 has 25% of the scores below it.
 Q2 has 50% of the scores below it (and, therefore,
is the median).
 Q3 has 75% of the scores below it.
 Interquartile Range
 The interquartile range is the distance between the
first and third quarters of the scores in a distribution.
Thus, the formula is:
IQR = Q3  Q1
 SemiInterquartile Range
 The semiinterquartile range is simply onehalf of
the interquartile range.
SIQR = (Q3  Q1) / 2
Evaluation: The IQR and SIQR are more stable
than the Range because they focus on the middle half of
the scores and, therefore, can't be influenced by extreme
scores. However, the actual value of the scores aren't
used, which would be an improvement.
Example:For the variables in our class data (see
table above), the IQR values are:
Age: (21  19) = 2
GPA: (3.4  2.8) = 0.6
MathSAT: (670  540 + 1) = 130
Satisfaction: (7  7) = 0
Questions:
 How well does the IQR do as a way of representing
the variation in each of these variables?
 Does it do better for some of the variables than for
others?
 If so, why?
 Do we get a different idea of variability in the data
by looking at the IQRs than we did by looking at the
ranges?
 The Standard Deviation and Variance
 Definition of Standard Deviation
 In simple terms, the standard deviation is the average
distance of scores in a distribution from their mean.
 Definition of Variance
 The Variance is the square of the standard deviation.
Examples
Consider, once again, the four variables in our class
survey data. Here are their means and standard deviations:
 Age: Mean = 20.24, StDv = 1.77
One interprets the mean as showing that the typical
age is about 20 1/4 years (20 years and 3 months).
The standard deviation shows that the average person
is within 1 3/4 years (1 year and 9 months) of that
age. That is, most people in class are between 20.241.77=
18.47 (18 1/2 years old) and 20.24+1.77=22.01 years
old (between 18 1/2 and 22).
 GPA: Mean = 3.06, StDv = .44
The mean tells us that the average GPA is 3.06,
corresponding to just above a B average. The standard
deviation tells us that, on the average person has
a GPA that is between 2.62 and 3.50. In other words,
the typical GPA range is between a B and B+/A. Not
bad!
 Math SAT: Mean = 589.39, StDv = 94.35
The mean tells us that the average SAT score on
the Math section is about 590 (which is certainly
better than the average score in the whole population).
The standard deviation tells us that, the typical
person has a SAT score on the Math section that is
between 495 and 685, or, roughly, between 500 and
700. This seems like a fairly big variation, suggesting
that some students did quite a lot better than others
on SAT Math.
 Satisfaction: Mean = 6.88, StDv = 1.14
Typically, you rated your satisfaction with your
experience at UNC averages about 7, which is above
the middle of the scale (which was 5). Furthermore,
most of you rated your experience in the 5 to 9 range,
which is from the middle of the scale to the top of
it. Its hard to know what this means, exactly, since
we don't have a well defined reference for the scale,
as we do for the other variables.
 Most Useful Measure of Variation
 The standard deviation and variance are the most useful
measures of variation.
Don't you feel as though you've learned more about
the variability of the scores on the four variables
discussed above than you did from the range or IQR?
This is because it uses every score in the
distribution to come up with a value for the variation
in the scores, not just two scores (as for the range)
or some of the scores (as for the IQR).
Also, the standard deviation and variance are very
much involved in inferential statistics, whereas the
other measures are not involved.
For these reasons, we will see these measures repeatedly
throughout the book.
The down side is that the formula is much more complicated
than those for the range or IQR, as we will see next.
 Formula for the standard deviation
 The standard deviation is, roughly, the average difference
of scores from their mean. More precisely, it is the
square root of the "mean" of the squared
difference of scores from their mean. Here is the
formula:
We see that:
 Square root: The standard deviation is
the square root of everything inside, since the
superscript of 1/2 means square root.
 "Mean" is modified slightly (and therefore
I've put it in quotes) to involve dividing by n1
rather than n, for reasons described in the
"degrees of freedom" section of the chapter.
This is true for the sample standard deviation
(which is what we will almost always be computing),
but not the population standard deviation (which
we'll seldom see) where we divide by N, getting
the true mean.
 Squared differences are used because the
sum of the unsquared differences from the mean
is always zero, which isn't useful for an index
of variability! Always zero? Why is this?
Note that here the differences are also distances.
So we are figuring out how far the typical score
is from the mean of the scores, in distances.
 Formula for the variance
 This is simply the square of the standard deviation:
We will be using this formula in later portions of the
book.
 Population Formulas
 The population formulas for the standard deviation
and variance are the same, except they use the population
mean (denoted by "mu" rather than "xbar"), and they
use the population size N, rather than 1 less than the
sample size n. Here they are:
Check out the HyperStat
site. Pay particular attention to the first two chapters,
especially the one on Describing
Univariate Data.
Use the Histogram
Explorer to get a better understanding of histograms
and distributions. Follow the Basic Instructions given
there. Use the Practice Guessing.
Also try this interactive
demonstration of how to calculate the standard deviation
and variance.
