1. Slide 1 Lecture 4: Measures of Variation Given a stem –and-leaf plot Be able to find »Mean (40+42+3*50+51+2*52+64+67)/10=46.7 »Median (50+51)/2=50.5 »mode 50 Stem (tens)Leaves (units) 40 2 50 0 0 1 2 2 64 7 # of phones ( x ) f fx Cum Freq 428 2 3412 6 2510 11 116 27 0130 40=n Review of Lecture 3: Measures of Center Given a regular frequency distribution Be able to find » Sample size 2+4+5+16+13=40 » Mean (8+12+10+16+0)/40=1.15 » Median: average of the two middle values=1 Median group 5 th 6 th

2. Slide 2 2.5 Measures of Variation Statistics handles variation. Thus this section one of the most important sections in the entire book Measure of Variation (Measure of Dispersion): A measure helps us to know the spread of a data set. Candidates: Range Standard Deviation, Variance Coefficient of Variation

3. Slide 3 Definition The range of a set of data is the difference between the highest value and the lowest value Range=(Highest value) – (Lowest value) Example: Range of {1, 3, 14} is 14-1=13.

4. Slide 4 Standard Deviation The standard deviation of a set of values is a measure of variation of values about the mean We introduce two standard deviation: Sample standard deviation Population standard deviation

5. Slide 5 Sample Standard Deviation Formula Formula 2-4  ( x - x ) 2 n - 1 S =S = Sample size Data value

6. Slide 6 Sample Standard Deviation (Shortcut Formula) Formula 2-5 n ( n - 1) s = n (  x 2 ) - (  x ) 2

7. Slide 7 Example: Publix check-out waiting times in minutes Data: 1, 4, 10. Find the sample mean and sample standard deviation. x 1-416 41 10525 1542  1 16 100 117 Using the shortcut formula: n=3

8. Slide 8 Standard Deviation - Key Points  The standard deviation is a measure of variation of all values from the mean  The value of the standard deviation s is usually positive and always non-negative.  The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others)  The units of the standard deviation s are the same as the units of the original data values

9. Slide 9 Population Standard Deviation 2  ( x - µ ) N  = This formula is similar to Formula 2-4, but instead the population mean and population size are used

10. Slide 10  Population variance: Square of the population standard deviation Variance  The variance of a set of values is a measure of variation equal to the square of the standard deviation.  Sample variance s 2 : Square of the sample standard deviation s

11. Slide 11 Variance - Notation standard deviation squared s  2 2 } Notation Sample variance Population variance

12. Slide 12 Round-off Rule for Measures of Variation Carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation.

13. Slide 13 Definition The coefficient of variation (or CV) for a set of sample or population data, expressed as a percent, describes the standard deviation relative to the mean CV = SamplePopulation A measure good at comparing variation between populations No unit makes comparing apple and pear possible.

14. Slide 14 Example: How to compare the variability in heights and weights of men? Sample: 40 males were randomly selected. The summarized statistics are given below. Sample meanSample standard deviation Height68.34 in3.02 in Weight172.55 lb26.33 lb Solution: Use CV to compare the variability Heights: Weights: Conclusion: Heights (with CV=4.42%) have considerably less variation than weights (with CV=15.26%)

15. Slide 15 Standard Deviation from a Frequency Distribution Use the class midpoints as the x values Formula 2-6 n ( n - 1) S =S = n [  ( f x 2 )] - [  ( f x )] 2

16. Slide 16 Example: Number of TV sets Owned by households A random sample of 80 households was selected Number of TV owned is collected given below. TV sets ( x) # of Households ( f ) fxfx 2 0400 133 22856112 3103090 452080 Total80139315 Compute: (a)the sample mean (b) the sample standard deviation

17. Slide 17 Estimation of Standard Deviation Range Rule of Thumb For estimating a value of the standard deviation s, Use Where range = (highest value) – (lowest value) Range 4 s 

18. Slide 18 Estimation of Standard Deviation Range Rule of Thumb For interpreting a known value of the standard deviation s, find rough estimates of the minimum and maximum “usual” values by using: Minimum “usual” value (mean) – 2 X (standard deviation)  Maximum “usual” value (mean) + 2 X (standard deviation) 

19. Slide 19 Definition Empirical (68-95-99.7) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply:  About 68% of all values fall within 1 standard deviation of the mean  About 95% of all values fall within 2 standard deviations of the mean  About 99.7% of all values fall within 3 standard deviations of the mean

20. Slide 20 The Empirical Rule FIGURE 2-13

21. Slide 21 The Empirical Rule FIGURE 2-13

22. Slide 22 The Empirical Rule FIGURE 2-13

23. Slide 23 Recap In this section we have looked at:  Range  Standard deviation of a sample and population  Variance of a sample and population  Coefficient of Variation (CV)  Standard deviation using a frequency distribution  Range Rule of Thumb  Empirical Distribution

24. Slide 24 Homework Assignment 4 problems 2.5: 1, 3, 7, 9, 11, 17, 23, 25, 27, 31 Read: section 2.6: Measures of relative standing.