Statistics
The mean of a set of numbers is the sum of the values divided by the total
number of values. For example, the mean of
{2, 3, 30, 14, -1} is
= 
= 9.6. There are two
other "averages" that statisticians take from a set of data -- the
median and the mode.
The median is the middle value (in numerical order) in a set of
data. If there are an even number of data values, and hence no exact
middle, then the median is the number in between the two middle values,
or the mean of those two values. In either case, there are the same
number of values above the median as there are values below the
median.
The median of
{1, 2, 3, 4, 6} is
3. The median of
{6, 2, 8, 1, 9} is
6.
The median of
{1, 2, 3, 4, 6, 10} is
= 3.5.
To find the median of a set of data, first arrange the data in order
from smallest value to largest value. Then take the middle value or
the mean of the two middle values.
The mode is the value(s) that appears the greatest number of times
in a set of data. For example, the mode of {1, 2, 3, 2, 4} is
2, and the mode of {30, 30, 33, 33, 33} is 33. A set can have
multiple modes. For example, the modes of {1, 2, 2, 3, 3, 4} are
2 and 3, and the modes of {10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13} are 10, 11, and 12. A set of data with two modes is
called bimodal.
The range of a set of data is the highest value minus the lowest
value. For example, the range of {3, 4, 1, 8, 12} is 12 - 1 = 11 and the range of {9, 15, -2} is 15 - (- 2) = 17. The range
is a measure of how wide an interval a set of data spans. A set of
data with a larger range spans a wider interval, and thus is more
"spread out." {9, 15, -2} spans a wider interval than 3, 4, 1, 8, 12.
Example 1: If the mean of
{5, 2, - 19,2x
, 23, 7} is
8,
what is the value of
x?
= 8
= 8
2x + 18 = 48
2x = 30
x = 15
Check:
= 8 ? Yes.
Thus,
x = 15
Example 2: If the median of {3,x, 15, 6, 3, 8, 9} is 6,
which of the following values could x take? { -3, 1, 3, 6, 7, 18}
Median of {3, -3, 15, 6, 3, 8, 9} = 6.
Median of {3, 1, 15, 6, 3, 8, 9} = 6.
Median of {3, 3, 15, 6, 3, 8, 9} = 6.
Median of {3, 6, 15, 6, 3, 8, 9} = 6.
Median of {3, 7, 15, 6, 3, 8, 9} = 7.
Median of {3, 18, 15, 6, 3, 8, 9} = 8.
Thus, x could take any of the following values: { -3, 1, 3, 6}.
In fact, x can be any number such that x≤6.
Example 3: Which of the following values can be added to the
set {1, 2, 2, 3, 3, 3, 4} without changing the mode? {1, 2, 3, 4}
Mode of {1, 2, 2, 3, 3, 3, 4} = 3.
Add 1: Mode of {1, 1, 2, 2, 3, 3, 3, 4} = 3.
Add 2: Modes of {1, 2, 2, 2, 3, 3, 3, 4} = {2, 3}.
Add 3: Mode of {1, 2, 2, 3, 3, 3, 3, 4} = 3.
Add 4: Mode of {1, 2, 2, 3, 3, 3, 4, 4} = 3.
Thus, we can add 1, 3, or 4, but not 2.
Example 4: If the range of {4, 6, -1, 12, x} is no greater
than 16, what are the possible values of x?
If x is the lowest value: 12 - x≤16
- x≤4
x≥ - 4
If x is the highest value: x - (- 1)≤16
x + 1≤16
x≤15
Thus, -4≤x≤15
