Mean, Median, and Mode

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MEAN, MEDIAN, and MODE

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Statistics is the discipline devoted to organizing, summarizing, and drawing conclusions from data.

Given a collection of data, it is often convenient to come up with a single number that somehow describes its center;
a number that in some way is representative of the entire collection.
Such a number is called a measure of central tendency.

The two most popular measures of central tendency are the mean and the median.
Another measure sometimes used to describe a "typical" data value is the mode.

The mean (or average) is already familiar to you: add up the numbers, and divide by how many there are:

DEFINITION: mean
The mean of the n data values x1, x2, x3, &ldots; , xn is denoted by x¯ (read as "x bar") and is given by the formula
x¯ = ∑ i=1 n xi n

Similarly, the mean of the n data values   y1, y2, y3, &ldots; , yn    would be denoted by y¯ (read as "y bar").
The sum    ∑ i=1 n xi n    can also be denoted by    1n ∑ i=1 n xi .

EXAMPLE:  Find the mean of these data values:  2, -1, 2, 3, 0, 25, -1, 2
There are 8 data values.
The mean is   2+ (-1)+ 2+ 3+ 0+ 25+ (-1)+2 8= 328=4 .

The mean gives the balancing point for the distribution, in the following sense:
if eight pebbles of equal weight are placed on a "number line see-saw":
two pebbles at -1 , one pebble at 0 , three pebbles at 2 , one pebble at 3 , and one pebble at 25 ;
then the support would have to be placed at 4 for the see-saw to balance perfectly!

Notice that in the previous example the number 25 seems to be unusually large, compared to the other numbers.
An outlier is an unusually large or small observation in a data set.
A drawback of the mean is that its value can be greatly affected by the presence of even a single outlier.
If the outlier 25 is changed to 250, then the new mean would be 32.125 ,
which does not seem at all representative of a "typical" number in the data set!

The median, on the other hand, is quite insensitive to outliers.
Just as the median strip of a highway goes right down the middle,
the median of a set of numbers goes right through the middle of the ordered list.
Of course, only lists with an odd number of values have a true middle:
the middle number in the ordered list   5, 7, 20   is 7 .
See how the definition below solves the problem when there are an even number of data values:

DEFINITION: median
To find the median of a set of n data values, first order the observations from least to greatest.

If n is odd, then the median is the number in the exact middle of the list.
That is, the median is the data value in position n+12 of the ordered list.

If n is even, then the median is the average of the two middle members of the ordered list.
That is, the median is the average of the data values in positions n2 and n2+1 of the ordered list.

EXAMPLE:  Find the median of these data values:  2, -1, 2, 3, 0, 25, -1, 2
Begin by ordering the eight data values from least to greatest:   -1, -1, 0, 2, 2, 2, 3, 25
There are an even number of values, so we average the values in positions four and five:
the median is 2+22=2 .
Note that, for this data set, the median seems to do a better job than the mean in representing a "typical" member.
Note also that if the outlier 25 is changed to 250 , it does not affect the median at all!

Finally, a mode is a value that occurs "most often" in a data set.
Whereas a data set has exactly one mean and median, it can have one or more modes.

For example, the mode of the data values  2, -1, 2, 3, 0, 25, -1, 2
is 2 , since this data value occurs three times, and this is the most occurrences of any data value.

Every member of the data set   3, 7, 9   is a mode, since each value occurs only once.
The data set   3, 3, 7, 7, 9   has two modes: 3 and 7 .

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