Stat 217 – Day 10 Measures of spread, position (Topic 9)

Thought Question Three landmarks of baseball achievement are Ty Cobb’s batting average of.420, Ted William’s.406 in 1941, and George Brett’s.390 in Which performance was the most impressive?

Last Time – Measures of Spread Variability is an important property of the distribution of a quantitative variable. You have seen several ways to measure this property:  Range = max – min  Interquartile range = Upper quartile – lower quartile Width of middle 50%  Standard deviation = square root of sum of squared deviations divided by sample size minus 1 “Typical” deviation of observations from the mean Not resistant (best with symmetric data, no outliers)

Activity 9-3 (p. 166) Class F (n = 24) IQR = = 2.5 Class G (n = 25) IQR = 6.5 – 3.5 = 3 Standard deviations: F = 1.77, G = 2.04

Class FClass GClass HClass IClass J Range68888 IQR Std Dev Activity 9-3 H vs. I vs. J n=24 n=25 n=18 Because there are two clumps on both ends of the graph and very few in the middle.

Activity 9-3 “Spread” looks at “horizontal width”/”distance from center”  Not bumpiness  Not variety (i) 10 ratings between 1 and 9 inclusive with standard deviation as small as possible (j) 10 ratings with standard deviation as large as possible

Try this (Example 1) Open Dotplot Summaries applet  BB: Course Materials > Applets Click Draw Samples until you get a fairly symmetric, mound-shaped distribution Then check both Guess Std Dev and Show actual (SD) boxes. Drag out the Guess SD edges to match the actual SD Click Show Percentages

Activity 9-4 Roughly symmetric and mound-shaped Within one standard deviation of the mean (6.36, 14.08) or really (7-14): 146/213 =.685

Activity 9-4 Roughly symmetric and mound-shaped Within one standard deviation of the mean (6.36, 14.08) or really (7-14): 146/213 =.685 Within two standard deviations of the mean (2.50, 17.9): 202/213 =.948 All within 3 standard deviations

Turns out (p. 170) These percentages apply in many, many situations  Mound-shaped, symmetric distributions

Another interpretation of SD With mound-shaped symmetric distributions, standard deviation tells you the width of the middle 68% of the observations  IQR = width of middle 50%

Activity 9-5: SATs and ACTs College admissions… (a) Bobby: 240 points (b) Kathy: 9 points (d) Bobby: ( )/240 = 1 (e) Kathy: (30-21)/6 = 1.50

Z-score z = (observation – mean) standard deviation  Unitless  Positive when observation is above mean, negative when observation is below mean The standard deviation also provides a convenient “yard stick” for measuring the position of observations within a distribution Interpretation: number of standard deviations away from the mean of the distribution

Best Hitter Ty Cobb.420 Williams.406 Brett.390 Williams’ performance was further above his peers  All pretty impressive, better than 99.85% of the populations MeanSD MeanSDz

Boxplot

Example 2 From the Stat 217 Data Files page (Course Materials), choose bumpus.xls  Select the 3 rd column and copy From Stat 217 Applets page click on the Creating Boxplots link  Scroll to the bottom to Enter your data below, one per line and replace the data there by pasting in the bumpus data.  Update boxplot  Check Graph by Category radio button

Surviving Sparrows Min Q L med Q u max

Surviving Sparrows Min Q L med Q u max

Surviving Sparrows

Activity 10-1 Underneath “identify outliers,” use the scroll bar to make “modified boxplots.”

To Turn In, with partner To Turn In with partner  Activity 10-2 For Wednesday  Lab 3  See review sheet online  Submit review question in discussion board

To Turn In, with partner To Turn In with partner  Activity 10-1 parts (j) and (k) For Wednesday  Activity 10-2  See review sheet online  Submit review question in discussion board  Bring HW 2, Lab 2 to class