# interquartile ranges

Discussion in 'Mathematics' started by tobyboy, Nov 3, 2007.

1. ### tobyboy

Hi everyone,

Just got a really quick quesion on an easy topic!

If we have a set of values, and the number of values is evan, when calculating the lower quartile, upper quartile and interquartile values, do we work with the numbers that are at x decimal points?

eg values: 6 11 33 39 41 49

lower quartile:
(n+1)/4
(6+1)/4
7/4 = 1.75th value

The 1.75th value is 0.75 between 6 (the 1st value)11 which is 9.75. So the lower quartile is 9.75. Is this right?

thanks
Toby

2. ### tobyboy

Hi everyone,

Just got a really quick quesion on an easy topic!

If we have a set of values, and the number of values is evan, when calculating the lower quartile, upper quartile and interquartile values, do we work with the numbers that are at x decimal points?

eg values: 6 11 33 39 41 49

lower quartile:
(n+1)/4
(6+1)/4
7/4 = 1.75th value

The 1.75th value is 0.75 between 6 (the 1st value)11 which is 9.75. So the lower quartile is 9.75. Is this right?

thanks
Toby

3. ### emilyisobel

No, only when you are dealing with large sets of grouped data.
Your example is a small set of discrete data so if you draw a line cutting the set in half, your median is halfway between the 3rd and 4th values. The quartiles are the halfway points of the data sets that are left and right of the line. So they are the 2nd value and the 5th value in this set.

4. ### emilyisobel

I haven't forgotten about those sheets on graph transformations, I just can't remember which file I put them in and haven't found them yet :0S

5. ### tobyboy

Ok thanks! No probs about the graph tansformations - I fine with it now.

thanks again!

;-)

6. ### likeglue2

The problem is that there is no agreed procedure on how to find quartiles. Roger Porkess has a very good piece on the topic in his Statistics Dictionary. Apparently the Statistician who developed quartiles now thinks we should now abandon quartiles and use "Hinges" instead, due to the confusion!

7. ### tobyboy

so in light of the above post likeglue2 & emilyisobel, is my way technically correct but just not the method used? I guess I followed the formula....would it be wrong in an exam?

8. ### emilyisobel

It would be wrong in an exam for a set of numbers.
If you used it to find the quartiles from a cumulative frequency curve it would be correct.

9. ### tobyboy

Ah ok, ta very much!

10. ### emilyisobel

No problem. Keep posting :0)

11. ### Subjectknowledge

Does anyone know of a practical application where one is interested in the interquartile range of a small set of results? In most cases, we are interested in the distribution of a population but have data concerning a sample. In finding the quartiles/IQR of a sample, we are presumably attemting to estimate the quartiles/IQR of the underlying distribution.

I set up a spreadsheet to generate 100 samples of size 10 and 100 samples of size 100 from a Normal Distribution with mean 100 and standard deviation 15 (the values commonly used for standardised tests like CATs). The theoretical values of the quartiles are 89.9, 100.0 and 110.1 and the IQR is theoretically 20.2

For the samples of size 100, I calculated the LQ, median, UQ and IQR for each sample, and then found the mean of each sample statistic, the least and greatest values. I estimated a 95% confidence interval for the error each statistic by finding the 2.5% percentile and the 97.5% percentile from the 100 samples. The results are as follows:

95% conf int
Mean Minimum Maximum lower higher
LQ 89.71 84.43 95.20 -3.1 3.8
Median 99.69 95.08 106.26 -4.4 2.5
UQ 109.89 105.53 116.17 -4.4 4.1
IQR 20.18 15.81 28.48 -4.2 5.2

With samples of size 100, the estimates of quartiles and IQR are likely to be within 5 (ie 1/3 of a standard deviation) of the correct values for the distribution. The average absolute error in estimating the IQR was 1.88, which is about 10% of the true value.

The samples of size 10 tell a different story:
Now the errors are much worse
95% conf int
Mean Minimum Maximum lower higher
LQ 91.34 77.81 110.80 -8.4 12.3
Median 99.98 85.68 115.09 -10.5 9.9
UQ 109.64 92.49 126.78 -12.1 11.1
IQR 18.30 5.56 35.83 -12.2 12.1
With samples of size 10, the estimates of quartiles and IQR are likely to be within 12 (ie 4/5 of a standard deviation) of the correct values for the distribution. The average absolute error in estimating the IQR was 5.97, which is about 30% of the true value.

The moral of the story is that quartiles of samples of size 10 aren't much use in estimating the population quartiles. It isn't clear what useful inference can be made from the quartiles of a set of 10 numbers. Quartiles of very small samples seems to be one of those things that is taught 'to keep the data simple' but never used in practice. What's more, it doesn't even keep things simple.

12. ### emilyisobel

I agree, but there are a lot of things that are in the curriculum (or indeed left out of it) that I don't agree with as it is either not useful in the long run or in the fact that it leaves a gap in subject knowledge needed for university maths.

13. ### goldbug

One of my students has pointed out that the method for calculating quartiles for a small data set and for an ungrouped frequency table lead to the quartiles being in slightly different positions. She was correct!

14. ### emilyisobel

There are differences of opinion across statistics. Due to that I would recommend looking at past exam papers for your board and see which method they expect.
The (n+1)/4 is generally used for large continuuous data sets where a single data item cannot be identified.

15. ### Minkowski

When this question appears on exam papers n is always odd. So you get the right answer which ever method is used.

I had to do a marking exercise as part of an interview. We were given the markscheme to use, and in one of the questions the candidate had used a different method for the quartiles than that used in the course (and arguably a more appropriate one). An interesting dilemma!

17. ### goldbug

Thanks Emily Isobel - all very enriching things for students to think about!

18. ### weggsterNew commenter

The chief examinaer at AQA has responded that both solutions are valid. It all depends on your definitions to start with.

For exam purposes, both would be acceptable solutions but the exam boards wouldn't set a question in like the one I posed due to the different answers possible.

It's been a good discussion point for my Optional Maths group (they take GCSE Maths, GCSE Statistics and OCR Additional Maths FSMQ over 2 years, 11 lessons a fortnight).