Measures of Central Tendency Lecture 5 22 February 2006 R. Ryznar

Transcription

1 Measures of Central Tendency Lecture 5 22 February 2006 R. Ryznar

2 Today s Content Wrap-up from yesterday Frequency Distributions The Mean, Median and Mode Levels of Measurement and Measures of Central Tendency Measures of Dispersion Measures of Central Tendency and Dispersion Together

3 Definitions for Frequency Distributions Variable the trait or characteristic on which the classification is based. Class one of the grouped categories of the variable. Class Boundaries the lowest and highest values that fall within the class. Class Midpoints the point halfway between the upper and lower class boundaries. Class Interval the distance between the upper limit of one class and the upper limit of the next higher class. Class Frequency the number of observations or occurrences of the variable with a given class. Total Frequency the total number of observations or cases in the table.

4 Creating and Using Frequency Distributions Bin the data Bins must be exclusive and exhaustive Look for overall patterns Look for any striking deviations to the pattern Look for any outliers Give the center and the spread Describe the shape

5 Percentage of Adult Population who are Obese by U.S. State (2001) 10 8 Frequency Percentage Mean = Std. Dev. = N = 50

6 Stem-and-Leaf Plot Percentage of adult population who are obese (2001) Frequency Stem & Leaf Stem width: 1.0 Each leaf: 1 case(s)

7 Babe Ruth s homeruns, 1920 to 1934: Barry Bonds home runs, 1986 to 2004:

8 Back to back stemplot Bonds Ruth

9 Number of Crews Tons of Garbage Normal Moore Total Crews

10 Percentage of Work Crews Tons of Garbage Normal Moore Total Percent N =

11 Opinion on president's performance (number of responses by gender) Females (n=110) Males (n=683) Strongly approve Approve Neither approve nor disapprove Disapprove Strongly disapprove

12 Opinion on president's performance (percentage of responses by gender) 60% 50% 40% Females Males 30% 20% 10% 0% Strongly approve Approve Neither approve nor disapprove Disapprove Strongly disapprove

13 Response Time Percentage (Cumulative) of Response Times Under 5 minutes 27.6 Under 10 minutes 82.4 Under 15 minutes 98.8 Under 20 Minutes 100 Cumulative Percentage Response Time of Metro Fire Department Under 5 minutes Under 10 minutes Under 15 minutes Under 20 Minutes

14 Atlantis Fire Department, 2001 Metro Fire Department, 2001 Response Time Percentage (Cumulative) of Response Times Response Time Percentage (Cumulative) of Response Times Under 5 minutes 21.2 Under 10 minutes 63.9 Under 15 minutes 86.4 Under 20 Minutes Under 5 minutes 27.6 Under 10 minutes 82.4 Under 15 minutes 98.8 Under 20 Minutes Fire Department Response Time Under 5 minutes Metro Atlantis Under 10 minutes Under 15 minutes Under 20 Minutes

15 μ = Population mean: N X i i =1 ( ) N = X + X X 1 2 N N x = Sample mean: i n xi = = n n ( x + x + + x ) n The median is the middle number in an ordered dataset. It can be found by ordering the values from lowest to highest and then finding the observation at (n+1)/2. Mode? The most frequently occurring value in the dataset.

16 The mean is not a resistant measure of center. f Mode Median Mean x Figure by MIT OCW. Classic illustration of the relationship between skew, mean, median and mode Journal of Statistics Education Volume 13, Number 2 (2005), Copyright 2005 by Paul T. von Hippel

17 Bonds Ruth Ruth Bonds Mean 659/15= /19=37 Median Mode 46 multiple

18 Percentage of Adult Population who are Obese by U.S. State (2001) 10 8 A normal distribution has a skew of 0 since it is symmetric. Frequency Percentage Skewness = Median = 20.3 Mean = Std. Dev. = N = 50

19 PctObese N Mean Median Mode Std. Deviation Variance Skewness Std. Error of Skewness Kurtosis Std. Error of Kurtosis Percentiles Statistics Valid Missing a. Multiple modes exist. The smallest value is shown a

20 A normal distribution has kurtosis 0, since it is symmetric.

21 Percent of Population of Hispanic Origin by U.S. State (2000) Frequency Percent Hispanic Median = 4.7 Skewness = Kurtosis = 5.12 Mean = 7.73 Std. Dev. = N = 50

22 Percentage of Population without Health Insurance for One Year by U.S. State (Three Year Average from 2000 to 2002) 8 6 Frequency Percent with no Health Insurance Median = Skewness =.732 Kurtosis =.262 Mean = Std. Dev. = N = 50

23 Levels of Measurement and Measures of Central Tendency Nominal Ordinal Interval Nominal Ordinal Interval Mean Mean X Median Median X X Mode Mode X X X

24 Would you prefer to ride on this facility? Active Living Visual Preference Survey, Collingwood, Ontario - November Mean (No -3 to Yes +3) Slide Number

25 Histogram Histogram Frequency Frequency Slide1 Mean = 3.31 Std. Dev. = N = Slide40Rec Mean = Std. Dev. = N = 180

26 Slide Frequency Slide19 Mean = 3.97 Std. Dev. = N = 186

27 The Five Number Summary Minimum, Q1, Median, Q3, Maximum The third quartile Q3 is the median of the observations whose position in the ordered list is to the right of the location of the overall median. To calculate quartiles: Arrange the observations in increasing order and locate the median M in the ordered list of observations. The first quartile Q1 is the median of the observations whose position in the ordered list is to the left of the location of the overall median.

28 The ends of the box (hinges) are at the quartiles, so that the length of the box is the IQR The median is marked by a line within the box The two vertical lines (called whiskers) outside the box extend to the smallest and largest observations within 1.5 X IQR of the quartiles. Observations that fall outside of 3 X IQR are called extreme outliers and are marked, for example, with an open circle. incws Observations between 1.5 X IQR and 3 X IQR are called mild outliers and are distinguished by a different mark, e.g., a closed circle male

29 Variance and Standard Deviation Measures the average distance of the observations from their mean. Variance Standard Deviation 1 ) (... ) ( ) ( = n x x x x x x s n 1 ) ( 2 = n x x s i

30 Visitors x i x 2 ( x i x) = = 36, = = 4, = = 56, = = = = 19, = = 71, = = 25,921 Sum = 0 Sum = 214,870 s 2 = 214,870 6 = 35, s = 35, =

31 Percentage of Adult Population who are Obese by U.S. State (2001) 10 8 Frequency Percentage Mean = Std. Dev. = N = 50

32 Variable 1 Variable Histogram Histogram Frequency 2 Frequency Variable 1 Mean = Std. Dev. = N = Variable 2 Mean = Std. Dev. = N = 11