Data Summaries 3

The p^th percentile of an ordered data set is the value that has at most p% of the data below it and at most (100-p)% above it.

Steps on how to find the p^th percentile:
1) Order the data set
2) Find the location of the p^th percentile: np/100, round up
3) Find p^th percentile at that location.

Example: Find the 67^th percentile of Babe Ruth's Homerun data
Solution:
np/100 = 15*67/100 = 10.05 round up to 11
Find 11^th observation: 49

So the 67^th percentile of Babe Ruth's Homerun data is 49

Warning A very popular mistake is to think that the number found in step 2 is the percentile already. It is the location of the percentile in the ordered dataset. Again, if you make that mistake most of the time your answer will make no sense! In the example above we found 11, but how could that be the 67^th percentile? It is smaller than any of the observations.
Also note that the way it is defined here percentiles are always actual observations.

Example Find the 80^th percentile of th following dataset:

MINITAB Ver. 14 or lower does not have a command to find percentiles, but at least you can use the command Data > Sort to do the ordering.

1^st quartile Q₁ = 25^th percentile
3^rd quartile Q₃ = 75^th percentile

Using these we can also find the Interquartile Range IQR = Q₃ - Q₁
and the 5-number summary:

Minimum

Q1

Median

Q3

Maximum

Example: Babe Ruth again:

Minimum	Q1	Median	Q3	Maximum
22	35	46	54	60

IQR = 54-35 = 19

The Stat > Basic Statistics > Display Descriptive Statistics command finds Q₁ and Q₃ as well. Note that MINITAb uses slightly different formulas from those above so the answers can be bit different.

Example Find the 5-number-summary and the IQR of the incomes of WRInc

Minimum	Q1	Median	Q3	Maximum
8000	26600	32400	39200	88400

IQR = 39200-26600 = 12600

What is the meaning of these percentiles?
• Q₁ = P₂₅ = $26600, so 25% (or 1 in 4) of the employees make less than $26600.
• Median = 32400, so half of the employees make less than 32400, half make more.
•Q₃ = P₇₅ = $39200, so 25% (or 1 in 4) of the employees make more than $39200.

What is the meaning of IQR? Actually it is a 3^rd way to calculate a measure of "spread-out-ness", after the range and the standard deviation. To make them comparable, though, we need to devide IQR by 1.35:

Example The standard deviation of the incomes is s=9424. Now IQR/1.35 = 12600/1.35 = 9333.

Now we have several formulas (methods) for finding an "average" (mean, median) and a "spread-out-ness" (range/4, s, IQR/1.35). How do you decide which to use?

• Use the range only if you can't find either of the other two, for example if you only know the smallest and the largest observation, or if you have to do a quick calculation in your head.

Example: Babe Ruth's homeruns:

Minimum	Q1	Median	Q3	Maximum
22	35	46	54	60

and so IQR = Q₃-Q₁ = 54-35 = 19
therefore we find
LF = Q₁ - 1.5*IQR = 35-1.5·19 = 6.5 < Min = 22
UF = Q₃ + 1.5*IQR = 54+1.5·19 = 74.5 > Max = 60