interquartile range from frequency table

Question

I have attached a word document here with the frequency table along with my answer at the answer the textbook gives , only reason I ask is because i'm getting conflicting information on this example my answer is apparently wrong but on another question my method was used so what is going on?

q)Find the interquartile range for this data.
Ans)
Exemplar answer :

Order the data -0,2,2,2,2,3,3,4,4

Q1=(9+1)/4=2.5 which then =2

Q3=3(9+1)/4=7.5 which then =3.5

So 3.5-2 =1.5 is the iqr

My answer:

Firstly find, cumulative frequency

Cf=22

So Q1=(22+1)/4=5.75 which then= game 2

And Q3=3(22+1)/4=17.25= game 8

8-2=6

So interquartile range =6 games

What answer is right here?

Answer 1

You seem to have ordered only part of the data. For example, you seem to have ignored the two $1$'s. If I read your link correctly, the data are $X =$

(1,1, 2,2,2,2, 4,4,4, 5,5,5,5, 6,6, 7,7, 8,8,8, 9,9)

In that case, R statistical software gives:

x = c(1,1, 2,2,2,2, 4,4,4, 5,5,5,5, 6,6, 7,7, 8,8,8, 9,9)
quantile(x)
  0%  25%  50%  75% 100% 
 1.0  2.5  5.0  7.0  9.0 
IQR(x)
## 4.5

Minitab statistical software gives:

Variable   N   Mean  StDev  Minimum     Q1  Median     Q3  Maximum    IQR
x         22  5.000  2.582    1.000  2.000   5.000  7.250    9.000  5.250

As you can see, results are slightly different. Several different methods of finding quantiles are used in various textbooks and software programs. At least eight slightly different methods are in regular use. The practical differences among them disappear when sample sizes are large. If your textbook has a clearly stated definition, that's the one you should use.

One argument for saying that lower quartile of 2 is as follows: Less than 25% of the $n = 22$ observations are below 2 and less than 25% are above 2. The definition of the median (second or middle quartile) is fairly standard: Here it is the average of the 11th and 12th observations, both of which happen to be 5.

R gives the interquartile range (IQR) as $7 - 2.5 = 4.5,$ but Minitab's method gets 5.25, and I think you would get 5 using yet other methods.