As you may have observed already, when we select a sample and collect measurements for one or more characteristics, these measurements tend to be different from one another.

**MEASURES OF DISPERSION**

As you may have observed already, when we select a
sample and collect measurements for one or more characteristics, these
measurements tend to be different from one another. To give a simple example,
height measurements taken from a sample of persons obviously will not be all
identical. In fact, if we were to take measure-ments from a single individual
at different times during the day and compare them, the measurements also would
tend to be slightly different from one another; i.e., we are shorter at the end
of the day than when we wake up!

How do we account for differences in biological and
human characteristics? While driving through Midwestern cornfields when
stationed in Michigan as a post-doctoral fellow, one of the authors (Robert
Friis) observed that fields of corn stalks generally resemble a smooth green
carpet, yet individual plants are taller or shorter than others. Similarly, in
Southern California where oranges are grown or in the al-mond orchards of
Tuscany, individual trees differ in height. To describe these dif-ferences in
height or other biological characteristics, statisticians use the term
vari-ability.

We may group the sources of variability according
to three main categories: true biological, temporal, and measurement. We will
delimit our discussion of the first of the categories, variation in biological
characteristics, to human beings. A range of factors cause variations in human
biological characteristics, including, but not limited to, age, sex, race,
genetic factors, diet and lifestyle, socioeconomic status, and past medical
history.

There are many good examples of how each of the
foregoing factors produces variability in human characteristics. However, let
us focus on one—age, which is an important control or demographic variable in
many statistical analyses. Biological characteristics tend to wax and wane with
increasing age. For example, in the U.S., Europe, and other developed areas,
systolic and diastolic blood pressures tend to in-crease with age. At the same
time, age may be associated with decline in other char-acteristics such as
immune status, bone density, and cardiac and pulmonary func-tioning. All of
these age-related changes produce differences in measurements of
characteristics of persons who differ in age. Another important example is the
im-pact of age or maturation effects on children’s performance on achievement
tests and intelligence tests. Maturation effects need to be taken into account
with respect to performance on these kinds of tests as children progress from
lower to higher levels of education.

Temporal variation refers to changes that are
time-related. Factors that are capa-ble of producing temporal variation include
current emotional state, activity level, climate and temperature, and circadian
rhythm (the body’s internal clock). To illus-trate, we are all aware of the
phenomenon of jet lag—how we feel when our normal sleep–awake rhythm is
disrupted by a long flight to a distant time zone. As a conse-quence of jet
lag, not only may our level of consciousness be impacted, but also physical
parameters such as blood pressure and stress-related hormones may fluctu-ate.
When we are forced into a cramped seat during an extended intercontinental
flight, our circulatory system may produce life-threatening clots that lead to
pulmonary embolism. Consequently, temporal factors may cause slight or
sometimes major variations in hematologic status.

Finally, another example of a factor that induces
variability in measurements is measurement error. Discrepancies between the
“true” value of a variable and its measured value are called measurement
errors. The topic of measurement error is an important aspect of statistics. We
will deal with this type of error when we cover regression (Chapter 12) and
analysis of variance (Chapter 13). Sources of measure-ment error include
observer error, differences in measuring instruments, technical errors,
variability in laboratory conditions, and even instability of chemical reagents
used in experiments. Take the example of blood pressure measurement: In a
multi-center clinical trial, should one or more centers use a faulty
sphygmomanometer, that center would contribute measures that over- or
underestimate blood pressure. Another source of error would be inaccurate
measurements caused by medical per-sonnel who have hearing loss and are unable
to detect blood pressure sounds by lis-tening with a stethoscope.

Several measures have been developed—measures of
dispersion—to describe the variability of measurements in a data set. For the
purposes of this text, these measures include the range, the mean absolute
deviation, and the standard deviation. Percentiles and quartiles are other
measures, which we will discuss in Chapter 6.

The range is defined as the difference between the
highest and lowest value in a dis-tribution of numbers. In order to compute the
range, we must first locate the highest and lowest values. With a small number
of values, one is able to inspect the set of numbers in order to identify these
values.

When the set of numbers is large, however, a simple
way to locate these values is to sort them in ascending order and then choose
the first and last values, as we did in Chapter 3. Here is an example: Let us
denote the lowest or first value with the symbol *X*_{1} and the highest value with *X _{n}*. Then the range (

*d *=* X _{n}
*–

with indices 1 and *n* defined after sorting the values.

Calculation is as follows:

Data set: 100, 95, 125, 45, 70

Sorted values: 45, 70, 95, 100, 125

Range = 125 – 45

Range = 80

A second method we use to describe variability is
called the mean absolute devi-ation. This measure involves first calculating
the mean of a set of observations or values and then determining the deviation
of each observation from the mean of those values. Then we take the absolute
value of each deviation, sum all of the deviations, and calculate their mean.
The mean absolute deviation for a sample is

where *n* =
number of observations in the data set.

The analogous formula for a finite population is

where *N* =
number of observations in the population.

Here are some additional symbols and formulae. Let

where:

*X _{i} *= a particular observation, 1

* *= sample mean

*d _{i} *= the deviation of a value from
the mean

The individual deviations (*d _{i}*) have the mathematical property such that when we sum
them

Thus, in order to calculate the mean absolute
deviation of a sample, the formula must use the absolute value of *d _{i}* (|

Suppose we have the following data set {80, 70, 95,
100, 125}. Table 4.5 demonstrates how to calculate a mean absolute deviation
for the data set.

Historically, because of computational
difficulties, the mean absolute deviation was not used very often. However,
modern computers can speed up calculations of the mean absolute deviation,
which has applications in statistical methods called robust procedures.

Common measures of dispersion, used more frequently because of their desirable mathematical properties, are the interrelated measures variance and standard deviation. Instead of using the absolute value
of the deviations about the mean, both the variance and standard deviation use
squared deviations about the mean, de-fined for the ith observation as (*X _{i}* –

where *N* =
the total number of elements in the population.

A related term is the population standard deviation
(*σ*), which
is the square root of the variance:

Table 4.6 gives an example of the calculation of *σ* for a small finite population.

The data are the same as those in Table 4.5 (*μ* = 94).

What do the variance and standard deviation tell
us? They are useful for compar-ing data sets that are measured in the same
units. For example, a data set that has a “large” variance in comparison to one
that has a “small” variance is more variable than the latter one.

**TABLE 4.6. Calculation of Population Variance**

Returning to the data set in the example (Table
4.6), the variance *σ*^{2} is 354. If the numbers differed more from one
another, e.g., if the lowest value were 60 and the highest value 180, with the
other three values also differing more from one another than in the original
data set, then the variance would increase substantially. We will provide
several specific examples.

In the first and second examples, we will double
(Table 4.6a) and triple (Table 4.6b) the individual values; we will do so for
the sake of argument, forgetting mo-mentarily that some of the blood sugar values
will become unreliable. In the third

What may we conclude from the foregoing three
examples? The individual val-ues (*X _{i}*)
differ more from one another in Table 4.6a and Table 4.6b than they did in
Table 4.6. We would expect the variance to increase in the second two data sets
be-cause the numbers are more different from one another than they were in
Table 4.6; in fact, σ

Effect of multiplying *X _{i}* by a constant

1. Adding *a*: the mean *μ* becomes *μ* *+ a*; the variance *σ*^{2} and standard deviation *σ** *remain unchanged.

2. Multiplying by *a*: the mean *μ* becomes *μ** a*, the variance *σ*^{2} becomes *σ*^{ 2}*a*^{2}, and the
standard deviation *σ* becomes *σ** a*.

The standard deviation also gives us information
about the shape of the distribu-tion of the numbers. We will return to this
point later, but for now distributions that have “smaller” standard deviations
are narrower than those that have “larger” stan-dard deviations. Thus, in the
previous example, the second hypothetical data set also would have a larger
standard deviation (obviously because the standard deviation is the square root
of the variance and the variance is larger) than the original data set. Figure
4.4 illustrates distributions that have different means (i.e., different
locations) but the same variances and standard deviations. In Figure 4.5, the
distributions have the same mean (i.e., same locations) but different variances
and standard deviations.

Calculation of sample variance requires a slight
alteration in the formula used for population variance. The symbols *S*^{2} and *S* shall be used to denote sample variance and standard deviation,
respectively, and are calculated by using Formulas 4.10a and 4.10b (deviation
score method).

*Figure 4.4. **Symmetric distributions with the
same variances and different means. (Source: Centers for Disease Control and Prevention (1992). Principles of Epidemiology,
2nd Edition, Figure 3.4)*

Note that *n*
– 1 is used in the denominator. The sample variance will be used to estimate
the population variance. However, when *n*
is used as the denominator for the estimate of variance, let us denote this
estimate as *S _{m}*

**Figure 4.5. **Distributions with the same mean
and different variances. (Source: Centers for Disease** **Control and Prevention (1992). *Principles
of Epidemiology,* 2nd Edition, Figure 3.4, p. 150.)

Before the age of computers, finding the difference between each score and the mean was a cumbersome process. Statisticians
developed a shortcut formula for the sample variance that is computationally
faster and numerically more stable than the difference score formula. With the
speed and high precision of modern computers, the shortcut formula is no longer
as important as it once was. But it is still handy for doing computations on a
pocket calculator.

This alternative calculation formula of sample variance (Formula 4.11) is algebraically equivalent to the deviation score method. The formula speeds the computation by avoiding the need to find the difference between the mean and each individual value:

Using the data from Table 4.7, we see that:

*S*^{2 }= [696651
– (10)(67444.09)] / 9

= 2467.789* *

*S *= √2467.789 = 49.677

For larger samples (e.g., above *n* = 30), the use of individual scores in
manual cal-culations becomes tedious. An alternative procedure groups the data
and estimates *σ ^{2} *from the
grouped data. The formulas for sample variance and standard deviation for
grouped data using the deviation score method (shown in Formulas 4.12a and b)
are analogous to those for individual scores.

Table 4.8 provides an example of the calculations.
In Table 4.8, is the
grouped mean [Σ*f X*/Σ*f *= 19188.50/373 ≈ 51.44 (by rounding to two
decimal places)].

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