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pogonyaev
3 years ago
5

Hown below are the top 15 market research firms in the United States in 2014 according to the AMA Gold Top 50 Report. Compute th

e 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
Mathematics
1 answer:
Maru [420]3 years ago
8 0

Solution :

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 $\bar X = \sum \frac{X_i}{n}$

              $=\frac{8420.7}{15}$

              = 561.38

Median :

$Q_2 = \left(\frac{1}{2}n+\frac{1}{2}\right)^{th}$value

     $ = \left(\frac{1}{2}\times 15+\frac{1}{2}\right)^{th}$value

    = 202.7

Percentile = $\left(\frac{P}{100} \times n + \frac{1}{2}\right)^{th}$  value

                 = $\left(\frac{50}{100} \times 15 + \frac{1}{2}\right)^{th}$ value

                 = 202.7

Percentile = $\left(\frac{P}{100} \times n + \frac{1}{2}\right)^{th}$  value

                 = $\left(\frac{40}{100} \times 15 + \frac{1}{2}\right)^{th}$ value

                 = 198.82

Percentile = $\left(\frac{P}{100} \times n + \frac{1}{2}\right)^{th}$  value

                 = $\left(\frac{90}{100} \times 15 + \frac{1}{2}\right)^{th}$ value

                 = 1849.28

$Q_1 = \left(\frac{1}{4}n+\frac{1}{4}\right)^{th}$value

     $ = \left(\frac{1}{4}\times 15+\frac{1}{4}\right)^{th}$value

    = 172

$Q_3 = \left(\frac{3}{4}n+\frac{3}{4}\right)^{th}$value

     $ = \left(\frac{3}{4}\times 15+\frac{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

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