Measuring Variation 2 Lecture 17 Sec. 5.3.3 Mon, Oct 3, 2005

The Five-Number Summary Five-number summary – A summary of a sample or population consisting of the five numbers Minimum First quartile Q1 Median Third quartile Q3 Maximum This does a better job of indicating the spread than any single number can do.

TI-83 – Five-Number Summary Use the TI-83 to find a five-number summary of the data in Exercise 4.29, p. 252. Min = 75.4 Q1 = 77.1 Median = 81.15 Q3 = 84.9 Max = 90.3

Excel – Five-Number Summary Use Excel to find a five-number summary of the same data. OnTimeArrivals.xls.

The Five-Number Summary From the 5-number summary of these data, can we detect skewness in the distribution? Answer: Maybe.

Boxplots Boxplot – A graphical display of a five-number summary. Draw and label a scale representing the variable. Draw a box over the scale with its left and right ends at Q1 and Q3. Draw a vertical line through the box at the median. Draw a left tail (whisker) from the box to the minimum. Draw a right tail from the box to the maximum.

Example Draw a boxplot of the data in the previous example.

Boxplots and Shape What would a boxplot for a uniform distribution look like? What would a boxplot for a symmetric distribution look like? What would a boxplot for a left-skewed distribution look like?

TI-83 – Boxplots Press STAT PLOT. Press WINDOW. Press GRAPH. Select Plot1 Turn Plot 1 On. Select the Boxplot Type. Specify list L1. Press WINDOW. Set minX and maxX appropriately. Press GRAPH. See the instructions on p. 316.

TI-83 – Boxplots Press TRACE. Use the arrow keys to see the values of the minimum, Q1, the median, Q3, and the maximum.

Modified Boxplots Modified boxplot – A boxplot in which the outliers are indicated.

Modified Boxplots Draw the box part of the boxplot as usual. Compute STEP = 1.5  IQR. The inner fences are at Q1 – STEP and Q3 + STEP. Extend the whiskers from the box to the smallest and largest values that are within the inner fences. Draw as individual dots any values that are outside the inner fences. These dots represent outliers.

Example: DePaul University For an example of modified boxplots, see DePaul University’s web page on retention.

Example Draw a modified boxplot of the data from the earlier example. Are there any outliers?

TI-83 – Modified Boxplots Follow the same steps as for a regular boxplot, but for the Type, select the modified-boxplot icon, the first icon in the second row. It looks like a boxplot with a couple of extra dots. Use the TI-83 to find a modified boxplot of the data from the previous example.

Let’s Do It! Let’s do it! 5.9, p. 320 – Five-number Summary and Outliers. Let’s do it! 5.10, p. 320 – Cost of Running Shoes. Let’s do it! 5.11, p. 321 – Comparing Ages– Antibiotic Study.