Section 1.2 Describing Distributions with Numbers

Types of Measures Measures of Center: –Mean, Median, Mode Measures of Spread: –Range (Max-Min), Standard Deviation, Quartiles, IQR

Means and Medians Consider the following sample of test scores from one of Dr. X’s recent classes (max score = 100): 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 What is the Average (or Mean) Test Score? What is the Median Test Score?

Consider the following sample of test scores from one of Dr. X.’s recent classes (max score = 100): 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 Draw a Stem and Leaf Plot (Shape, Center, Spread?) Find the Mean and the Median Let’s Use our TI-83 Calculators! –Enter data into a list via Stat|Edit –Stat|Calc|1-Var Stats What happens to the Mean and Median if the lowest score was 20 instead of 65? What happens to the Mean and Median if a low score of 20 is added to the data set (so we would now have 11 data points?) What can we say about the Mean versus the Median?

Quartiles: Measures of Position

A Graphical Representation of Position of Data (It really gives us an indication of how the data is spread among its values!)

Using Measures of Position to Get Measures of Spread IQR is a measure of how the data deviates from the median

5 Number Summary, IQR, Box Plot, and where Outliers would be for Test Score Data: 65, 65, 70, 75, 78, 80, 83, 87, 91, 94

5 Number Summary, IQR, Box Plot, and where Outliers would be for Test Score Data: 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 5-number summary: Min = 65, Q1 = 70, Median = 79, Q3 = 87, Max = 94 IQR = Q3 - Q1 = 17 Suspected Outliers: anything less than Q *IQR = 44.5 anything greater than Q *IQR = Note: MegaStat uses a slightly different definition for the quartiles, so the box plot here is not perfectly consistent with the definitions we’re using. (By MegaStat)

Histograms of Flower Lengths Problem 1.58 Generated via Minitab

Box Plot and 5-Number Summary for Flower Length Data Generated via Box Plot Macro for Excel BihaiRedYellow Median Q Min or In Fence Max or In Fence Q Outliers?

Remember this histogram from the Service Call Length Data on page 9? How do you expect the Mean and Median to compare for this data? Mean 188.6, Median 115

Box Plot for Call Length Data

More on Measures of Spread Data Range (Max – Min) IQR (75% Quartile minus 25% Quartile = range of middle 50% of data) Standard Deviation –Measures how the data deviates from the mean Recall the Sample Test Score Data: 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 Recall the Sample Mean (X bar) was 78.8…

Deviations from the Mean Recall the Sample Test Score Data: 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 Recall the Sample Mean (X bar) was What does the number 4.2 measure? How about 13.8?

Consider (again!) the following sample of test scores from one of Dr. X.’s recent classes (max score = 100): 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 What happens to the standard deviation and the location of the 1 st and 3 rd quartiles if the lowest score was 20 instead of 65? What happens to the standard deviation and the location of the 1st and 3rd quartiles if a low score of 20 is added to the data set (so we would now have 11 data points?) What can we say about the effect of outliers on the standard deviation and the quartiles of a data set? Effects of Outliers on the Standard Deviation

Example 1.18: Stemplots of Annual Returns for Stocks (a) and Treasury bills (b) On page 53 of text. What are the stem and leaf units????

Consider (again!) the following sample of test scores from one of Dr. X’s recent classes (max score = 100): 65, 65, 70, 75, 78, 80, 83, 87, 91, 94 Xbar = 78.8 s = 10.2 (rounded) Suppose we “curve” the grades by adding 5 points to every test score (i.e. Xnew=Xold+5). What will be new mean and standard deviation? Suppose we “curve” the grades by multiplying every test score times 1.5 (i.e. Xnew=1.5*Xold). What will be the new mean and standard deviation? Suppose we “curve” the grades by multiplying every test score times 1.5 and adding 5 points (i.e. Xnew=1.5*Xold+5). What will be the new mean and standard deviation? Effects of Linear Transformations on the Mean And Standard Deviation

Box Plots for Problems