Slide 1. Slide 2. Slide 3. Interquartile Range (IQR)

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1 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 is a fraction then the lower quartile is the average of the two observations with order closest to n + 1 4

2 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 n 4 is a fraction then the lower quartile is the average of the two observations with order closest to n 4

3 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? 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?

4 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

5 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?

6 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 Find the z-score for the value 50.1 A data set has a mean of 42.5 and SD of Find the z-score for the value 28.4

7 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 MATH 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).

8 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