Numerical Summary of Quantitative Data Chapter 2 – Class 15 1

Class Work What kind of numerical summary have you learned so far? 2

5 number summary Min Q1 Median Q3 Max 3

4 Example 2.14 Fastest Speeds for Men Ordered Data (in rows of 10 values) for the 87 males: Median = (87+1)/2 = 44 th value in the list = 110 mph Q 1 = median of the 43 values below the median = (43+1)/2 = 22 nd value from the start of the list = 95 mph Q 3 = median of the 43 values above the median = (43+1)/2 = 22 nd value from the end of the list = 120 mph

5 Numerical Summaries of Quantitative Data Notation for Raw Data: n = number of individuals in a data set x 1, x 2, x 3,…, x n represent individual raw data values Example: A data set consists of handspan values in centimeters for six females; the values are 21, 19, 20, 20, 22, and 19. Then, n = 6 x 1 = 21, x 2 = 19, x 3 = 20, x 4 = 20, x 5 = 22, and x 6 = 19

6 Notation and Finding the Quartiles Split the ordered values into the half that is below the median and the half that is above the median. Q 1 = lower quartile = median of data values that are below the median Q 3 = upper quartile = median of data values that are above the median

7 Percentiles The k th percentile is a number that has k% of the data values at or below it and (100 – k)% of the data values at or above it. Lower quartile = 25 th percentile Median = 50 th percentile Upper quartile = 75 th percentile

8 Describing the Location of a Data Set Mean: the numerical average Median: the middle value (if n odd) or the average of the middle two values (n even) Symmetric: mean = median Skewed Left: mean < median Skewed Right: mean > median

9 Determining the Mean and Median The Mean where means “add together all the values” The Median If n is odd: M = middle of ordered values. Count (n + 1)/2 down from top of ordered list. If n is even: M = average of middle two ordered values. Average values that are (n/2) and (n/2) + 1 down from top of ordered list.

10 Example 2.12 Will “Normal” Rainfall Get Rid of Those Odors? Mean = inches Median = inches Data: Average rainfall (inches) for Davis, California for 47 years In , a company with odor problem blamed it on excessive rain. That year rainfall was inches. More rain occurred in 4 other years.

Mean VS Median : Salaries of Los Angeles Lakers Find the five number salary Find the mean 11 Kobe Bryant25.2 millionDerek Fisher3.4 million Pau Gasol18.7 millionMatt Barnes1.9 million Andrew Bynum 15.2 millionTroy Murphy1.4 million Lamar Odom8.9 millionJason Kapono1.2 million Metta World Peace 6.8 millionDerrick Caracter 0.8 million Luke Walton5.7 millionDevin Ebanks0.8 million Steve Blake4.0 million

Choose a Summary Skewed Distribution –Use 5 number summary Reasonably symmetric distribution – free of outliers –Mean and standard deviation (Since they are strongly affected by outliers) 12

13 The Influence of Outliers on the Mean and Median Larger influence on mean than median. High outliers will increase the mean. Low outliers will decrease the mean. If ages at death are: 76, 78, 80, 82, and 84 then mean = median = 80 years. If ages at death are: 46, 78, 80, 82, and 84 then median = 80 but mean = 74 years.

14 Describing Spread: Range and Interquartile Range Range = high value – low value Interquartile Range (IQR) = upper quartile – lower quartile Standard Deviation (covered later )

15 Example 2.13 Fastest Speeds Ever Driven Five-Number Summary for 87 males Median = 110 mph measures the center of the data Two extremes describe spread over 100% of data Range = 150 – 55 = 95 mph Two quartiles describe spread over middle 50% of data Interquartile Range = 120 – 95 = 25 mph

16 Outlier: a data point that is not consistent with the bulk of the data. How to Handle Outliers Look for them via graphs. Can have big influence on conclusions. Can cause complications in some statistical analyses. Cannot discard without justification.

17 Possible Reasons for Outliers and Reasonable Actions Outlier is legitimate data value and represents natural variability for the group and variable(s) measured. Values may not be discarded — they provide important information about location and spread. Mistake made while taking measurement or entering it into computer. If verified, should be discarded/corrected. Individual in question belongs to a different group than bulk of individuals measured. Values may be discarded if summary is desired and reported for the majority group only.

18 Example 2.16 Tiny Boatsmen Weights (in pounds) of 18 men on crew team: Note: last weight in each list is unusually small. They are the coxswains for their teams, while others are rowers. Cambridge:188.5, 183.0, 194.5, 185.0, 214.0, 203.5, 186.0, 178.5, Oxford: 186.0, 184.5, 204.0, 184.5, 195.5, 202.5, 174.0, 183.0, 109.5

Homework Assignment: Chapter 2 – Exercise 2.43 and 2.44 Chapter 2 – Exercise 2.74 and 2.81 Reading: Chapter 2 – p