Answer:
To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the
90
th percentile of an exam does not mean, necessarily, that you received
90
% on a test. It means that
90
% of test scores are the same or less than your score and
10
% of the test scores are the same or greater than your test score.
Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the
75
th percentile. That translates into a score of at least
1220
.
Percentiles are mostly used with very large populations. Therefore, if you were to say that
90
% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant.
The median is a number that measures the “center” of the data. You can think of the median as the “middle value,” but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data.
1
;
11.5
;
6
;
7.2
;
4
;
8
;
9
;
10
;
6.8
;
8.3
;
2
;
2
;
10
;
1
Ordered from smallest to largest:
1
;
1
;
2
;
2
;
4
;
6
;
6.8
;
7.2
;
8
;
8.3
;
9
;
10
;
10
;
11.5
Since there are
14
observations, the median is between the seventh value,
6.8
, and the eighth value,
7.2
. To find the median, add the two values together and divide by two.
6.8
+
7.2
2
=
7
The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.
Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median or second quartile. The first quartile,
Q
1
, is the middle value of the lower half of the data, and the third quartile,
Q
3
, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:
1
;
1
;
2
;
2
;
4
;
6
;
6.8
;
7.2
;
8
;
8.3
;
9
;
10
;
10
;
11.5
The median or second quartile is seven. The lower half of the data are
1
,
1
,
2
,
2
,
4
,
6
,
6.8
. The middle value of the lower half is two.
1
;
1
;
2
;
2
;
4
;
6
;
6.8
The number two, which is part of the data, is the first quartile. One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two.
The upper half of the data is
7.2
,
8
,
8.3
,
9
,
10
,
10
,
11.5
. The middle value of the upper half is nine.
The third quartile,
Q
3
, is nine. Three-fourths (
75
%) of the ordered data set are less than nine. One-fourth (
25
%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.
The interquartile range is a number that indicates the spread of the middle half or the middle
50
% of the data. It is the difference between the third quartile (
Q
3
) and the first quartile (
Q
1
).
I
Q
R
=
Q
3
–
Q
1
The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile. Potential outliers always require further investigation.
Step-by-step explanation:
