3.3

Identifying Outliers for Modified Boxplots

1.Find the quartiles Q1, Q2, and Q3. 2.Find the interquartile range (IQR), where IQR = Q3 − Q1. 3.Evaluate 1.5 × IQR. 4.In a modified boxplot, a data value is an outlier if it is above Q3, by an amount greater than 1.5 × IQR or below Q1, by an amount greater than 1.5 × IQR.

5 Number Summary

5-Number Summary For a set of data, the 5-number summary consists of these five values: 1.Minimum 2.First quartile, Q1 3.Second quartile, Q2 (same as the median) 4.Third quartile, Q3 5.Maximum

Boxplot (or Box-and-Whisker Diagram)

A boxplot (or box-and-whisker diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile Q1, the median, and the third quartile Q3.

Skewness

A boxplot can often be used to identify skewness. A distribution of data is skewed if it is not symmetric and extends more to one side than to the other.

Modified Boxplots

A modified boxplot is a regular boxplot constructed with these modifications: (1)A special symbol (such as an asterisk or point) is used to identify outliers as defined above, and (2)the solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier.

Percentiles

Percentiles are measures of location, denoted P1, P2, . . . , P99, which divide a set of data into 100 groups with about 1% of the values in each group.

Quartiles

Quartiles are measures of location, denoted Q1, Q2, and Q3, which divide a set of data into four groups with about 25% of the values in each group

Finding the Percentile of a Data Value

The process of finding the percentile that corresponds to a particular data value x is given by the following (round the result to the nearest whole number):

Statistics defined using quartiles and percentiles

look at the picture

Notation

n total number of values in the data set k percentile being used (Example: For the 25th percentile, k = 25.) L locator that gives the position of a value (Example: For the 12th value in the sorted list, L = 12.) Pk kth percentile (Example: P25 is the 25th percentile.)

z score formula

z=(x-mean)/standard deviation Z =(x-μ)/σ ( Population)

Descriptions of Quartiles

•Q1 (First quartile): Same value as P25. It separates the bottom 25% of the sorted values from the top 75%. Q2 (Second quartile): Same as P50 and same as the median. It separates the bottom 50% of the sorted values from the top 50%. •Q3 (Third quartile): Same as P75. It separates the bottom 75% of the sorted values from the top 25%. Caution Just as there is not universal agreement on a procedure for finding percentiles, there is not universal agreement on a single procedure for calculating quartiles, and different technologies often yield different results.