5-Number Summaries, Outliers, and Boxplots Section 2.3 Day 2 5-Number Summaries, Outliers, and Boxplots

5-Number Summary AKA 5-point summary If you include the minimum and maximum values of the data set along with the median and quartiles, you get the 5-number summary. AKA 5-point summary

5-Number Summary

Find 5-number summary.

Find 5-number summary….don’t forget key! 3I2 represents 32 mph

Find 5-number summary….n=18 3I2 represents 32 mph

5-Number Summary Graphical display of a 5-number summary is a boxplot or box-and-whiskers plot

How do we construct a boxplot?

1. Plot the 5 points

1. Plot the 5 points 2. Draw box from Q1 to Q3

1. Plot the 5 points 2. Draw box from Q1 to Q3 3. Draw vertical line at median

1. Plot the 5 points 2. Draw box from Q1 to Q3 3. Draw vertical line at median 4. Extend whiskers to min and max values

1. Plot the 5 points 2. Draw box from Q1 to Q3 3. Draw vertical line at median 4. Extend whiskers to min and max values 5. Label graph (context)

Outliers What are outliers?

Outliers Recall outliers in a set of data are any values that differ significantly from the other values.

For this data, are there any outliers?

Formula for Outliers A value is an outlier if it lies more than 1.5 times the IQR from the nearest quartile.

Formula for Outliers A value is an outlier if it lies more than 1.5 times the IQR from the nearest quartile. Thus, a value is an outlier if it is < Q1 – 1.5(IQR) or > Q3 + 1.5(IQR)

For this data, are there any outliers?

IQR = Q3 – Q1 = 42 – 30 = 12 Lower end:Q1 – 1.5(IQR) = 30 – 1.5(12) = 12 Upper end:Q3 + 1.5(IQR) = 42 + 1.5(12) = 60

Modified Boxplot Modified boxplot is like a basic boxplot except the whiskers only go as far as the largest and smallest nonoutliers (sometimes called adjacent values). Any outliers appear as individual dots or other symbols.

Modified Boxplot Modified boxplot is like a basic boxplot except the whiskers only go as far as the largest and smallest nonoutliers (sometimes called adjacent values). Any outliers appear as individual dots or other symbols.

Boxplots Useful when plotting a single quantitative variable and you want to compare shapes, centers, and spreads of two or more distributions you don’t need to see individual values, even approximately you don’t need to see more than the 5-number summary but would like outliers to be clearly indicated

Graphing Calculator You can use graphing calculator to find 5-number summary and draw boxplot. Use data from Display 2.46 on page 61

Graphing Calculator You can use graphing calculator to find 5-number summary and draw boxplot. Use data from Display 2.46 on page 61 Press “STAT” Select 1:Edit Enter the data elements in list Note: no need to reorder data first

Graphing Calculator is Your Friend! Your calculator will compute the summary statistics for a set of data. After entering data in list: Press “STAT” Arrow right to “CALC” Select “1: 1-Var Stats” 1-Var Stats L1 Enter

1-Var Stats Display 2.46 on page 61

1-Var Stats

Draw Boxplot 2nd STAT PLOT 1: Plot 1 …on Type: select modified boxplot symbol Xlist: L1 Freq: 1 Mark: Graph

Draw Boxplot If you can not see the boxplot, press “Zoom” Select 9: ZoomStat

Standard Deviation Differences from the mean, x – x, are called deviations.

∑(x – x ) = 0 Standard Deviation Differences from the mean, x – x, are called deviations. Mean is balance point of distribution so the set of deviations from the mean will always sum to zero. ∑(x – x ) = 0

Standard Deviation Formula for standard deviation, s, is:

Standard Deviation Formula for standard deviation, s, is: Dividing by n - 1 gives a slightly larger value than dividing by n. This is useful because otherwise the standard deviation of the sample would tend to be smaller than the standard deviation of the population the sample came from.

Computing Standard Deviation

Computing Standard Deviation Use 1-Var Stats. Symbol for standard deviation is sx

Summary from Frequency Table

Summary from Frequency Table Page 68

Summary from Frequency Table Enter “values” in List 1 Enter “frequency” in List 2 “STAT”, “CALC”, “1: 1-Var Stats” 1-Var Stats L1, L2 Enter

Important Note When homework says to use the formulas to compute something, you may use your calculator

Questions?