Question

Hown below are the top 15 market research firms in the United States in 2014 according to the AMA Gold Top 50 Report. Compute the mean, median, P50, P40, P85, Q1, Q3, range, and the interquartile range on these data.

Company Sales ($ millions)
Nielson Holdings N.V. $3,194.3
Kantar 952.6
IMS Health Inc. 935.0
Ipsos 574.1
Westat Inc. 563.7
Information Resources, Inc. 504.0
GfK USA 334.5 comScore Inc. 202.7
The NPD Group Inc. 202.3
Symphony Health Solutions 196.5
J.D. Power and Associates 172.9
ICF International Inc. 172.0
Abt SRBI Inc. 155.7
Maritz Research 138.9
dunnhumbyUSA LLC 121.5

Answer 1

Number Values (X)
`$\sum (X_i - \bar X)^2$`

1 121.5 193494.4144

2 138.9 178489.3504

3 155.7 164576.2624

4 172 151616.7844

5 172.9 150916.7104

6 196.5 133137.4144

7 202.3 128938.4464

8 202.7 128651.3424

9 334.5 51474.5344

10 504 3292.4644

11 563.7 5.3824

12 574.1 161.7984

13 935 139591.9044

14 952.6 153053.0884

15 3194.3 6932267.726

Total 8420.7 8509667.624

Mean
`$\bar X = \sum (X_i)/(n)$`

`$=(8420.7)/(15)$`

= 561.38

Median :

`$Q_2 = \left((1)/(2)n+(1)/(2)\right)^(th)$`value

`$ = \left((1)/(2)* 15+(1)/(2)\right)^(th)$`value

= 202.7

Percentile =
`$\left((P)/(100) * n + (1)/(2)\right)^(th)$` value

=
`$\left((50)/(100) * 15 + (1)/(2)\right)^(th)$` value

= 202.7

Percentile =
`$\left((P)/(100) * n + (1)/(2)\right)^(th)$` value

=
`$\left((40)/(100) * 15 + (1)/(2)\right)^(th)$` value

= 198.82

Percentile =
`$\left((P)/(100) * n + (1)/(2)\right)^(th)$` value

=
`$\left((90)/(100) * 15 + (1)/(2)\right)^(th)$` value

= 1849.28

`$Q_1 = \left((1)/(4)n+(1)/(4)\right)^(th)$`value

`$ = \left((1)/(4)* 15+(1)/(4)\right)^(th)$`value

= 172

`$Q_3 = \left((3)/(4)n+(3)/(4)\right)^(th)$`value

`$ = \left((3)/(4)* 15+(3)/(4)\right)^(th)$`value

= 574.1

Therefore, the range = highest value - lowest value

= 3194.3 - 121.5

= 3072.8

Now the interquartile range =
`$Q_3-Q_1$`

= 574.1 - 172

= 402.1