# Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. 8408 1374 1872 8879 2459 11413 608 14138 6452 1850 2818 1356 10498 7478 4019 4341 739 2127 3653 5794 8305 a. Provide a five-number summary. b. Compute the lower and upper limits. c. Do the data contain any outliers?

Annual sales, in millions of dollars, for 21 pharmaceutical companies follow.
8408 1374 1872 8879 2459 11413

608 14138 6452 1850 2818 1356
10498 7478 4019 4341 739 2127
3653 5794 8305
a. Provide a five-number summary.
b. Compute the lower and upper limits.
c. Do the data contain any outliers?
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a) Concept Used:
The five-number summary is the lowest value, Quartile-1, median, Quartile-3 and highest value in any data set.
The median is the middle value of the data distribution when arranged in ascending order. The Quartile-1 is the middle value of lowest value and median of data distribution. The Quartile-3 is the middle value of highest value and median of data distribution.
Calculation:
The lowest value of the data distribution is 608 and highest value is 14138.
The median is the middle value of the data when arranged in ascending order which is ${11}^{th}$ value
Median=4019
First quartile is average of ${5}^{th}$ and ${6}^{th}$ values
Quartile−1=1850+18722=1861
Third Quartile is average of ${16}^{th}$ and ${17}^{th}$ values
Quartile−3=8305+84042=8354.5
Conclusion:
The five number summary for annual sales of 21 pharmaceutical companies is 608, 1861, 4019, 8354.5, 14138.
(b)Concept Used:
The formula for boundaries of outliers are,
Lower Boundary=Q1−1.5*IQR
Upper Boundary=Q3+1.5*IQR
The difference between third quartile and first quartile is Inter Quartile range.
Calculation:
Substituting the values in Inter quartile range formula,
Inter Quartile​​​ Range=Q3−Q1=8354.5−1861=6493.5
Substituting in the boundaries of outliers formula,
Lower Boundary=Q1−1.5*IQR=1861−1.5*6493.5=−7880.25
Upper Boundary=Q3+1.5*IQR
=8354.5+1.5*6493.5=18094.75
Conclusion:
The lower and upper limits for outliers of annual sales for 21 pharmaceutical companies are -7880.25 and 18094.75, respectively. (c) The lower and upper limits for outliers of annual sales for 21 pharmaceutical companies is -7880.25 and 18094.75. All data points in the given data are within the boundaries of outliers. Hence, we can say there are no outliers in the data.
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a)

Minimum: The minimum value is 608.

First Quartile: The first quartile is middle value of the data below the median, in the other words , value at the 25% of the sorted data array.

${Q}_{1}=0.25\cdot 21=5.25\approx 6⇒{Q}_{1}=1872$

The percentile is not integer, in that case round up and read the value at the given place in the sorted array.

Median:Median is middle value of the sorted data array. In this case, there is odd number of data given, so the median is the middle value.

$Median={Q}_{2}=4019$

Third Quartile: The third quartile is middle value of the data above the median, in the other words , value at the 75% of the sorted data array.

${Q}_{3}=0.75\cdot 21=15.75\approx 16⇒{Q}_{3}=8305$

The percentile is not integer, in that case round up and read the value at the given place in the sorted array.

Maximum: The maximum value is 14138.

b)

To determine lower and upper limits, first find the interquartile range. IQR is the difference between third and first quartile.

$IQR={Q}_{3}-{Q}_{1}=8305-1872=6433$

Lower limit is the first quartile value decreased by 1.5 times the IQR.

Upper limit is the third quartile value increased by 1.5 times the IQR.

c)

The data does not cointain any outliers because there is no value that is below the lower limit nor above the upper limit.