Slide 1 Interquartile Range (IQR) IQR= Upper quarile lower quartile But what are quartiles? Quartiles are points that divide a data set into quarters (4 equal parts) Slide 2 The Lower Quartile (Q 1 ) Is the point in the data where at least 25% of the data is below it and at least 75% of the data is above it Denoted by Q 1 Alternative names First Quartile 25 th percentile Slide 3 How to find the lower quartile (Q1) Order the the observations from smallest to largest Determine the sample size n Then th n + 1 Q1 = observation 4 If n + 1 4 is a fraction then the lower quartile is the average of the two observations with order closest to n + 1 4
Slide 4 Example Find the lower quartile for the data below x= 183, 163, 152, 157, 157 Slide 5 The Upper Quartile (Q 3 ) Is the point in the data where at least 75% of the data is below it And at least 25% of the data is above it Denoted by Q 3 Alternative names Third Quartile 75 th percentile Slide 6 How to find the Upper quartile (Q3) Order the the observations from smallest to largest Determine the sample size n Then n + 1 Q3 = 3 4 th observation If + 1 3 n 4 is a fraction then the lower quartile is the average of the two observations with order closest to + 1 3 n 4
Slide 7 Example Find the upper quartile for the data below x= 183, 163, 152, 157, 157 Slide 8 What are the three points that we collectively call Quartiles? 1---------------------------------------------- 2------------------------------------------------ 3-------------------------------------------------- Slide 9 Interquartile Range (IQR) Measures the range (spread) of the middle half of the data IQR=Q3-Q1 What is the percentage of observations between Q1 and Q3 IQR= 0 means ---------------------------------------- Can IQR be negative?-----------------------------------------
Slide 10 Range =0 means ----------------------------------------- S = 0 means ----------------------------- x Can range be negative?------------------------------- Slide 11 IQR vs Range IQR is a better measure of variation than range. Why? Consider the data below: x= 90,90,90,99,100,100,90,25,100,100 Find IQR Find Range Slide 12 IQR vs standard deviation For skewed distribution IQR is a better measure of variation than standard deviation. Why? Consider the data below: x=90,90,90,99,100,100,90,25,100,100 Calculate with and with out the observation 25 S x Calculate IQR with and without the observation 25
Slide 13 If a distribution is approximately symmetric, then SD is a better and most widely used measure of variation than than IQR. Slide 14 The Five Number Summary Give you a quick summary of both the central tendency and variation in your data. These are 1. The Minimum 2. The Lower quartile 3. The 2 nd Quartile (also called -----------------------------) 4. The Upper quartile 5. The maximum Slide 15 If you are given the five number summary of a particular data What measure of central tendency can you get? What measure(s) of variation can you get?
Slide 16 Box Plot The five number summary can be represented graphically using a box plot. Box plot is used for comparing the center and variability of two or more data sets. Slide 17 Measure of relative position (Z-score) Measures the position of a data point relative to the mean and in units of standard deviation. Z-score of an observation y is y mean Z score = s tan dard deviation Slide 18 Example A data set has a mean of 42.5 and SD of 13.6. Find the z-score for the value 50.1 A data set has a mean of 42.5 and SD of 13.6. Find the z-score for the value 28.4
Slide 19 Suppose the Z-score of an observation y is zero. What is the value of y? Z-score <0 means-------------------------------- Z-score >0 means ---------------------------- Slide 20 More example on Z-score The following is the score of a particular student Class His Score Class average SD STAT 72 78 10 MATH 65 60 20 In which class did the student performed well? Slide 21 Quantiles (Percentiles) Recall: given data on a variable The point that divides the ordered data into halves is called the --------------------------------- The points that divide the data into quarters are called ----------------------------- Points that divide the data into more general fractions are called Percentiles (Quantiles).
Slide 22 Definition: The sample 100p th percentile is a value such that after the data is ordered from smallest to largest, at least 100p% of the observations are below this value and at least 100 (1-p)% of the observations are above this value 0<p<1 Example: P=0.15 P=0.8 refers to the 15 th percentile or quantile of order 0.15 refers to the 80 th percentile or quantile of order 0.8 P=0.5 refers to --------------------------------- Slide 23 Home Work on Chapter 4 Read the chapter summary on page 43 Work Questions 1-11 from Worksheet 4, on page 44 Work on the practice exam (all questions) from my web page