Bioststistic-Definitions

Measures of Central Tendency




What is the most common, most typical, or
most often-occurring value of a variable? The following chart shows which
measures of central tendency can be used with variables measured at the
nominal, ordinal, or interval/ratio level.

	Nominal	Ordinal	Interval/Ratio
Mode	X	X	X
Median		X	X
Mean			X

Mode




The mode, or modal value, is the most
commonly occurring value or category of a variable. To find the mode, look for
the category that contains the highest number of observations. For example, in
a survey of 88 cities, the most common form of city governance is the
council/manager form.

Type of Government	Number of Cities
Commission	4
Weak Mayor	17
Strong Mayor	22
Council/Manager	45
Total	88

Note, however, that a variable may have two modal
categories. For example, if the type of government had looked like this,

Type of Government	Number of Cities
Commission	14
Weak Mayor	18
Strong Mayor	28
Council/Manager	28
Total	88

Then the variable would have two modal categories,
"Strong Mayor" and "Council/Manager". This means that there
is no one central tendency within the data for this variable. (More will
be said about this under "Skew" below under "Measures of
Dispersion.")



Median




The median is the value of the category
or case that divides an ordered distribution into two equal parts. One half of
the values will be higher than the median value; the other half of the values
will be lower than the median value. To find the median, you must first put all
the observations in order, from lowest to highest. Then use the formula
(N+1)/2.

For example, if there are 7 categories of employee pay,
the median category will be category number 4, or (7+1)/2=4. In the example
below, the median pay category is $24,000. The value of this category can also
be interpreted as the median pay value.

Pay:

$12,000

$17,000

$18,000

$24,000

$25,000

$27,000

$30,000

However, if there are 8 categories of employee pay, the
median pay value will fall in between two categories. The median category is
category 4.5, or (8+1)/2=9/2=4.5 Add the fourth and the fifth categories
and divide by two, or ($24,000+$25,000)/2=$49,000/2=$24,500.

Pay

$12,000

$17,000

$18,000

$24,000

$25,000

$27,000

$30,000

$58,000

If you have grouped data, that is, ranges of values, as
well as the number of people found in each group or range, there is a more
precise way to calculate the median. For example, assume that the pay
categories are ranges of pay, and employees are distributed among them as
follows:

Pay Range	Number of employees	Cumulative Frequency
$20,000- $29,000	9	9
$30,000- $39,000	14	23
$40,000- $49,000	16	39
$50,000- $59,000	21	60
Total	60	-

The median pay is found by using the formula for grouped
data, N/2. In this case, there are 60 employees, so the median = 60/2=the 30th
observation.

We can see from the cumulative distribution that the 30th
observation will be found in the category of $40,000-$49,000. We can estimate
the median by calculating the mid-point of this category, by adding the lower
boundary value to the higher boundary value and dividing by two, or
($40,000+$49,000)/2=$89,000/2=$44,500.

To calculate the median more exactly, we can look at how
many observations into the $40,000-$49,000 category we must go to find the 30th
observation.

There are 16 observations in this category. We must go to
the 7th observation to find the 30th total observation of the sample. So we can
calculate the value of going 7/16th of the way through this category.

The category has 10 values
(40,41,42,43,44,45,46,47,48,49). So 7/16 x 10 = 4.375.

We add this to the lower boundary value of the category
to get the median salary value of $40,000+$4,375=$44,375.

Note that we must assume that the observations are evenly
distributed within the categories; if the sample size is large enough in
relation to the number of categories, this is usually not a problem.

Mean




The mean, or average, is the arithmetic
balance point of the distribution, but is not the same as the median or the
mode. If you subtract from the mean each observation in the sample that is
above the mean, that sum will be equal to the sum of subtracting each
observation in the sample that is below the mean.



To find the mean


1) add up the value of all the
observations in the sample, and


2) divide that sum by the total
number of observations.

The average of the following employee salaries is equal
to $21,857.14

Salaries:

$12,000

$17,000

$18,000

$24,000

$25,000

$27,000

$30,000

However, the average of the following salaries is equal
to $26,375.

Salaries

12,000

17,000

18,000

24,000

25,000

27,000

30,000

58,000

This demonstrates the fact that the value of the mean is
sensitive to very high, or very low values. In this case, it may be better to
use the median.



To find the mean from grouped data:






1) first find the midpoint for each
range or category. This is found by adding the lower boundary value to
the upper boundary value and dividing by two.


2) Then multiply the number of
employees in each range by the midpoint of the range.


3) Add up all the products of
(number of employees times range midpoint)


4) Divide that sum by the total
number of observations.

Pay Range	Number of employees	Range Midpoint	Product
$20,000- $29,000	9	$24,500	$220,500
$30,000- $39,000	14	$34,500	$438,000
$40,000- $49,000	16	$44,500	$712,000
$50,000- $59,000	21	$54,500	$1,144,500
Total	60	-	$2,560,000

In this case, $2,560,000/60=$42,666.67



Measures of Dispersion




Measures of dispersion are the opposite
of measures of central tendency. The former attempt to describe the most
typical or central value of a distribution of values of a variable. Measure of
dispersion, in contrast, attempt to give an idea of how widely dispersed the
values are and how different the observations are from one another.

The following chart shows which measures of variation and dispersion be used
with variables measured at the nominal, ordinal, or interval/ratio level.

	Nominal	Ordinal	Interval/Ratio
Range		X	X
Percentiles		X	X
Standard Deviation			X
Variance			X

Range




The range is the difference between the
highest and the lowest values in an ordered distribution of the values of a
variable. For example, if the highest paid employee makes $58,000 per year and
the lowest paid employee makes $12,000 per year, the salary range is $46,000.
(Note the average is $26,400)

Pay:

12,000

17,000

18,000

24,000

25,000

27,000

30,000

58,000

However, if the highest paid employees makes $29,000 per
year and the lowest paid employee makes $22,500 per year, the salary range is
$3,000. (Note the average is $26,000)

Pay:

22,500

24,500

25,000

26,000

27,000

28,000

28,500

29,000

Although these two organizations
have very similar averages, they have very different ranges. For which
organization would you rather be working?

Percentiles




In general, percentiles are points in the
distribution of the ordered values of a variable at which a known number of
observations fall below the point and a known number of observations remain
above the point.

For example, the 50th percentile is the same as the
median; at the 50th percentile, half of the observations have higher values and
half of the observations have lower values.

Percentiles are often used on standardized tests, such as
the SAT or GRE. If you scored at the 75th percentile, that means that 75% of
the other people scored below your score and 25% scored at or above your score.


Sometimes on tests for civil service, applicants are
advised that they must score at a certain percentile or above to be considered
for an interview, a promotion, etc.

When two organizations have very different ranges but
similar averages, you may want to use the interquartile range, or the range
between the 25th and 75th percentiles. The interquartile range contains the
middle 50% of the observations.



To arrive at the interquartile range,






1) ignore the bottom 25% of the
categories and the top 25% of the categories.


2) re-calculate the range,
subtracting the new bottom category from the new top category.

For example, in this case, there are 8 categories, so
one-quarter of 8 categories=2 categories. Ignoring the top two and the bottom
two categories, the interquartile range for this organization is
$27,000-$18,000=$9,000.

Pay:

12,000

17,000

18,000

24,000

25,000

27,000

30,000

58,000

The interquartile range for this organization is
$28,000-$25,000=$3,000.

Pay:

22,500

24,500

25,000

26,000

27,000

28,000

28,500

29,000

Standard Deviation




The standard deviation is a measure of
the average difference of each observation in a distribution from the average
(mean) of the distribution. Given that you can calculate a mean, how different
are most of the observations from that mean?

If most of the observations are near the mean in value,
the standard deviation will be small. But if most of the observations are far
from the mean in value, the standard deviation will be large.



The formula for calculating the standard deviation is






1) calculate the mean


2) subtract the value of each
observation from the value of the mean


3) square the difference obtained
in step 2 for each observation


4) add up all the squared
differences obtained in step 3


5) divide the sum of the squared
differences by the total number of observations


6) take the square root of the
result of step 5=the value of the standard deviation






Variance




The variance is an expression of the
total amount of variability of the observations for a variable. The value of
the variance is obtained by squaring the value of the standard deviation.

A variable with a large variance has a great deal of
difference in the values of the various observations, while a variable with a
small variance has less difference in the values of the various observations.



Skew




The skew refers to the "shape"
of the distribution of the values of a variable. The values of a variable can
be plotted on a chart.

If the values of the observations are distributed
symmetrically around the mean of a variable, that is called a normal
distribution. In this case, the mean, median, and mode will all coincide.

If the values of most of the observations are lower than
the value of the mean, then the distribution is called a negatively skewed or
left skewed distribution. In this case, the mode will have a lower value than
the median, and the mean will have a higher value than the median.

If the values of most of the observations are higher than
the value of the mean, then the distribution is called a positively skewed or
right skewed distribution. In this case, the mode will have a higher value than
the median, and the mean will have a lower value than the median.

An inspection of the skewness of a variable will help the
researcher to decide which of the three measures of central tendency to
use--mean, median, or mode--as the best indicator of the central tendency of
the distribution of values for that variable.



Normal Curve




The Normal Curve is a graph of the values
of a variable where those values are distributed symmetrically about the mean
of the variable. It has the following characteristics:

1) it has a bell-shaped, symmetrical curve

2) the mean, median, and mode all have the same value

3) the properties of the curve are known

4) it is useful in calculating estimates in inferential statistics

If the distribution of the values on a variable approach
a normal curve, we know that approximately 68% of the values will be within
plus or minus one standard deviation from the mean; 95% of the values will be
within plus or minus two standard deviations from the mean; and 99% of the
values will be within plus or minus three standard deviations from the mean.

This is useful because the value of any one observation
can be converted to a standardized score, or z-score. A standardized score or
z-score converts any observation to a measure of standard deviation units,
where the value of the mean equals zero and the value of a standard deviation
equals one.



The formula for the z-score is






1) calculate the mean of the
variable


2) calculate the standard deviation
of the variable


3) subtract the value of the
observation from the value of the mean


4) divide the result of step 3 by
the standard deviation of the variable=z-score

A z-score of +1.5 means that the value of the observation
lies 1.5 standard deviation units above the mean. A z-score of -2.0 means that
the value of the observation lies 2.0 standard deviation units below the mean.

If a student scores 40 out of 50 on a test of mathematics
(with a mean of 41 and a standard deviation of 5), and 60 out of 75 on a test
of language (with a mean of 55 and a standard deviation of 10), the scores are
not directly comparable. Converting each of them to their respective z-scores
allows them to be compared directly.

z-score for 45= (45-40)/5=-0.2

z-score for 53=(53-40)/15=+0.5

Although the student scored 80% of the total points
available on each test, the student did slightly better than average (+0.5
standard deviations) on the language test, and slightly worse than average
(-0.2 standard deviations) on the math test.

See this also

http://www.nd.edu/~rwilliam/stats1/x03.pdf

See this also

http://www.stat.colostate.edu/~hari/regression_book/chapter1.pdf