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Wize AP Statistics Textbook > Calculating and Interpreting Statistics

Percentiles

Skewness and SymmetryPrevious Section
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Percentiles

A percentile gives you a sense where a given number is ranked, relative to other numbers, within a dataset.

Example
If your midterm grade is in the top 10% percentile in your class, that means your midterm grade is in the top 10% when being compared to all other students' midterm grades in your class.

Steps for Determining Percentiles

  1. Arrange your data set (sample) from lowest to highest numbers.
  2. Compute i=p100n\displaystyle \boxed{i=\frac{p}{100} n} where
  3. pp is the desired %
  4. nn is sample size, and
  5. ii is the location of the percentile
  6. Look at ii.
  7. If ii is not a whole number, then you round up to the (i+1)th(i+1)^{th} number to get your desired percentile.
  8. If ii is a whole number, the desired percentile is the average of iith and (i+1)(i+1)th numbers.

Wize Tip
The idea of a percentile is to “beat ii numbers in the sample (data set)”.

Quartiles
  • To solve for the first quartile, Q1, p=0.25p=0.25
  • To solve for the median, Q2, p=0.50p=0.50
  • To solve for the third quartile, Q3, p=0.75p=0.75

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Example: Percentiles

Determine the 14th and the 40th percentiles for the following list of fifteen numbers.

12, 13, 13, 14, 19, 20, 34, 45, 66, 88, 90, 92, 200, 220, 400

(a) 14th percentile

  1. If ii is not a whole number, then you round up to the (i+1)th(i+1)^{th} number to get your desired percentile.
  2. If ii is a whole number, the desired percentile is the average of iith and (i+1)(i+1)th numbers.
i=(14100)15=2.1→3rd number=13 i=\left(\frac{14}{100}\right)15=2.1\rightarrow3^{rd}\ number=13\ is the 14th percentile

Remember: If ii is not a whole number, then you round up to the (i+1)th(i+1)^{th} number to get your desired percentile.



(b) 40th percentile



i=(0.40)(15)=6 → 6th+7th2=20+342=27 is the 40th percentile.i=\left(0.40\right)\left(15\right)=6\ \rightarrow\ \frac{6th+7th}{2}=\frac{20+34}{2}=27\ is\ the\ 40^{th}\ percentile.

Remember: If ii is a whole number, the desired percentile is the average of iith and (i+1)(i+1)th numbers.


Practice: Percentiles

The amount earned (+) or lost ( -) by an investor per month over an eight-month period, in dollars, is given by −$200,−$10, $25, $80, $100, $180, $200, $490

(a) What is the 25th percentile?
(b) If the last number is changed from $490 to $6000, the 25th percentile remain unchanged. True or false?