Statistics
Mean: Its also
known as the arithmetic mean, a value that helps summarize an entire set of
numbers. A set’s mean is calculated by adding the numbers in the set together
and dividing their sum by the number of members of the set. For example, the
mean of the set {3, 4, 8} is 5, calculated (3 + 4 + 8)/3 = 5. Similarly, the set
{16, 13, 9, 2} has a mean of (16 + 13 + 9 + 2)/4 = 10.
In examining large collections of numbers, such as census
data, it is helpful to be able to present a number that provides a summary of
the data. Such numbers are often called descriptive statistics. The
arithmetic mean is probably the best-known descriptive statistic. The mean is
often called the average, but it is actually only one of several kinds of
averages, such as the median and the mode.
Median:
Median, the
value of the middle member of a set of numbers when they are arranged in order.
Like the mean (or average) and mode of a set of numbers, the median can be used
to get an idea of the distribution or spread of values within a set when
examining every value individually would be overwhelming or tedious. The median
of the set {1, 3, 7, 8, 9}, for example, is 7, because 7 is the member of the
set that has an equal number of members on each side of it when the members are
arranged from lowest to highest. If a set contains an even number of values,
there is no single middle member. In such cases the median is the mean of the
two values closest to the middle. The median of the set {1, 3, 9, 10}, for
example, is (3 + 9)/2 = 6.
The mean is a more precise measure than the median, but
can be greatly affected by a few numbers that are very different from the other
members of a set. For example, the mean of the set {2, 4, 5, 7, 8,
934}—calculated by adding the members of the set together and dividing the sum
by the total number of members—is 160, which is much higher than all but one of
the values in the set. In cases such as this the median, 6, is used to give a
better overall impression of the typical values of the numbers because it
ignores outlying values.
Mode: The number in a given set of
numbers that appears most frequently. In the set {3, 4, 6, 7, 10,
10, 13}, for example, the mode of the set is 10. If two or more numbers are tied
for most frequent appearances the set has multiple modes. The modes of the set
{1, 1, 2, 2, 3, 4, 4, 5}, for example, are 1, 2, and 4. Other sets, such as {5,
7, 9, 11}, have no modes because all the numbers occur with equal frequency.
Standard Deviation: A
number representing how closely bunched a set of numbers is around its
mean, or average value. The standard deviation is an important
concept in statistics because it is a precise indicator of the
degree of variability within a set of numbers. A set with a smaller standard
deviation consists of more closely bunched numbers than a set with a larger
standard deviation. If the test scores of a class were 76, 80, 82, 85, and 91,
for example, the standard deviation would be about 5.04, reflecting the fact
that the scores are fairly close together. A class with the wildly varying test
scores 0, 53, 77, 91, and 100, however, would have a standard deviation of about
35.8.
The standard deviation of a set of numbers is calculated
using the deviation of individual numbers from the set’s mean. Each deviation is
found by subtracting the number from the mean. If there are n members in
a set and the deviations of the members from the mean are symbolized by
x1 through xn, the standard
deviation (σ) is given by the formula:
To compute the standard deviation of the set {2, 3, 5, 6,
9}, for example, first find the mean. The mean is the sum of all the members of
a set divided by the number of members of the set: (2 + 3 + 5 + 6 + 9)/5 = 25/5
= 5. The individual deviations from the mean are 5 – 2 = 3, 5 – 3 = 2, 5 – 5 =
0, 5 – 6 = -1, and 5 - 9 = -4. Square the individual deviations and add them
together: 32 + 22 + 02 + (-1)2 +
(-4)2 = 9 + 4 + 0 + 1 + 16 = 30. Divide the result by the number of
members of the set: 30/5 = 6. Take the square root and round to two
decimal places: √6 = 2.45. The standard deviation of {2, 3, 5, 6, 9}, therefore,
is 2.45.
This equation works for sets in which all members are
specified. A more complex equation is necessary to determine the standard
deviation of a set using only a sample of that set’s elements.
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