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| When describing a set of data, without listing all of the values, we
have seen that we can use measures of location such as the
mean and median. It is also
possible to get a sense of the data's distribution by examining a
five statistical summary
(or five number summay), |
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the
(1) minimum, (2) maximum, (3) median (or second
quartile), (4) the first quartile, and (5) the third quartile.
Such information will show the extent to which the data is located near
the median or near the extremes.
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See how to use
your
TI-83+/TI-84+ graphing calculator with
five number studies.
Click calculator. |
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We know that the median
of a set of data separates the data into two equal parts.
Data can be further separated into quartiles. Quartiles separate the
original set of data into four equal parts.
Each of these parts
contains one-fourth of the data.
Quartiles are percentiles
that divide the data into fourths.
• The first
quartile is the middle (the median) of the
lower half of the data. One-fourth of the data lies below the
first quartile and three-fourths lies above.
(the 25th percentile) |
•
The second quartile
is another name for the median of the entire set of data.
Median of data set = second quartile of data set.
(the 50th percentile) |
•
The third quartile
is the middle (the median) of the upper half of the data.
Three-fourths of the data lies below the third quartile
and one-fourth lies above.
(the 75th percentile) |
A quartile is a number, it is not a range of values.
A value can be described as "above" or "below" the first quartile, but a
value is never "in" the first quartile.
Consider: Check out
this five statistical summary for a set of tests scores.
|
minimum
|
first quartile
|
second quartile
(median)
|
third quartile
|
maximum |
|
65 |
70 |
80 |
90 |
100 |
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While we do not know every test
score, we do know that half of the scores is below 80 and half
is above 80. We also know that half of the scores is
between 70 and 90.
(The difference between the third and first quartiles is called
the interquartile range.)
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A five statistical summary
can be represented graphically as a
The first and third
quartiles are at the ends of the box, the median is
indicated with a vertical line in the interior of the box,
and the maximum and minimum are at the ends of the whiskers.
Box-and-whisker plots are helpful in interpreting the
distribution of data. |
| How to construct
a box-and-whisker plot: |
Construct a box-and-whisker plot for the following
data:
The data:
Math test scores 80, 75, 90, 95, 65, 65, 80, 85, 70, 100
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Write the data in numerical
order. Find the first quartile, the median, the third
quartile, the minimum (smallest value) and the maximum (largest value). These
are referred to as a five statistical
summary.
median
(2nd quartile) =
80
first quartile = 70
third quartile = 90
minimum = 65
maximum = 100 |
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Place a circle beneath each of these
values in relation to their location on an equally spaced number line. |
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Draw a box with ends through the points
for the first and third quartiles. Then draw a vertical line through the box at the
median point. Now, draw the whiskers (or lines) from each end of the
box to these minimum and maximum values. |
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Special
Case:
You may see a box-and-whisker plot, like the one below,
which contains an asterisk.
Sometimes there is ONE piece of data that falls well outside the range
of the other values. This single piece of data is called an outlier.
If the outlier is included in the whisker, readers may think that there are
grades dispersed throughout the whole range from the first quartile to the
outlier, which is not true. To avoid this misconception, an * is used to
mark this "out of the ordinary" value.
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See how to use
your
TI-83+/TI-84+ graphing calculator with
box and whisker plots.
Click calculator. |
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