Percentiles Def: The kth percentile is the value such that at least k% of the measurements are less than or equal to the value. I.E. k% of the measurements.

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Presentation transcript:

Percentiles Def: The kth percentile is the value such that at least k% of the measurements are less than or equal to the value. I.E. k% of the measurements are less than or equal to the kth percentile.

Calculating Percentiles A random sample yields the dataset: We wish to calculate percentiles and construct a boxplot from this data. Step 1: Order the data

Order the data from smallest to largest: The data value 90 is what percentile? Order the Data

Calculate Percentiles 2 values are =< 50 3 values are =< 74 4 values are =< 78 5 values are =< 81 6 values are =< 89 7 values are =< is therefore the 100*(7 / 20) = 35th percentile

Quartiles To find quartiles, we first calculate their position in the ordered dataset. Position of Q1 =.25*(n + 1).25* (20 + 1) = 5.25 Position of Q3 =.75* (n + 1).75* (20 + 1) = 15.75

First Quartile Since 5.25 is not an integer, calculate a value 1/4 of the way between the 5th and 6th numbers * (89 – 81) = 83 Q1 =

First Quartile Since is not an integer, calculate a value 3/4 of the way between the 15th and 16th numbers (104 – 103) = Q3 =

Boxplot Make an axis for the graph covering the range of the data Draw bar for the: min, Q1, Median, Q3, and max Connect edges of bars at quartiles, and middles of min/Q1 and Q3/max

Boxplot The boxplot is now complete The plot tells us several things: –Location: the median is at 99 –Spread: the IQR is –Shape: This distribution is left-skewed. Note that the left tail is much longer and the median is closer to Q3

Outliers We will define two more measures: –Lower Fence = Q1 – 1.5*(IQR) = 83 – 1.5*(20.75) = –Upper Fence = Q *(IQR) = – 1.5*(20.75) = Any values outside of the fences are suspected to be outliers.

Outliers We can mark the fences to the boxplot. –Lower Fence = , Upper Fence = Any data point outside the fences are marked with *, the ‘whiskers’ extend only to the last data points still inside the fences.

Comments It may be easier to calculate the fences before constructing the plot. Boxplots can be vertical. Boxplots using * for outliers are sometimes called modified boxplots. Sometimes “Outer Fences” are used to denote “extreme outliers”, calculated by: Q1 – 3*(IQR) Q3 + 3*(IQR)