Q 87. - The Practice of Statistics for AP

Question

Domain names When it comes to Internet domain names, is shorter better? According to one ranking of Web sites in $2008$, the top $8$ sites (by number of

“hits”) were yahoo.com, google.com, youtube.com, live.com, msn.com, myspace.com, wikipedia.org, and facebook.com. These familiar sites certainly have short domain names. The histogram below shows the domain name lengths (in a number of letters in the name, not including the extensions .com and .org) for the $500$ most popular Web sites. 

(a) Estimate the mean and median of the distribution. Explain your method clearly.

(b) If you wanted to argue that shorter domain names were more popular, which measure of the center would you choose—the mean or the median? Justify your answer.

Step-by-Step Solution

Answer

Part (a) Median is $6$ and mean is $7.01$

Part (b) Here, median is chosen.

Part (a) Step 1: Given information

Given graph : 

Part (a) Step 2: Concept

The mean of a data set is determined by adding all of the numbers in the set and dividing by the number of values in the set. The median is the value in the middle of a data set when it is arranged from smallest to largest. The mode is the number that appears the most frequently in a data set. 

Part (a) Step 3: Calculation

For the $500$ most popular websites, a histogram of domain name length (in number of letters in the name, excluding the extensions.com and.org) was created. The following is how we calculated the mean: 

Domain name length	Frequency (f)	fx
2	10	20
3	40	120
4	42	168
5	58	290
6	105	630
7	60	420
8	58	464
9	38	342
10	27	270
11	18	198
12	20	240
13	10	130
14	5	70
15	5	75
16	1	16
17	3	51

We find the Mean $$\begin{gathered} \overline{X}= \frac{\Sigma fX}{\Sigma f} \\ \overline{X}=\frac{3504}{500} \\ \overline{X}=7.04 \end{gathered}$$

Median is calculated as follows: 

Domain name length (X)	Frequency (f)	Commutative frequency
2	10	10
3	40	50
4	42	92
5	58	150
6	105	255
7	60	315
8	59	374
9	38	412
10	30	442
11	28	470
12	20	490
13	10	500
14	5	505
15	5	510
16	2	512
17	4	516

$$\begin{gathered} M = \frac{N+1}{2} \\ N=Cumilative frequency \\ =\frac{500+1}{2} \\ M = 250.5 \end{gathered}$$

The median is $6$

Therefore, the median is $6$ and mean is $7.01$

Part (b) Step 1: Explanation

To justify that shorter domain names were more popular than the median, the data was skewed to the right, so the means were bigger than the median, and the median was shorter.