Review: Central Measures

Transcription

1 Review: Central Measures Mean, Median and Mode When do we use mean or median? If there is (are) outliers, use Median If there is no outlier, use Mean. Example: For a data 1, 1.2, 1.5, 1.7, 1.8, 1.9, 2.3, 2.5, 2.8, 3. Which one is more appropriate? For a data 1, 1.2, 1.5, 1.7, 1.8, 1.9, 2.3, 2.5, 2.8, 3, 10, 40, which one is more appropriate? Relationship between the central measures Agresti/Franklin Statistics, 1of 25

2 A B C mean C median B mode A A A A A B C A B C A B C Agresti/Franklin Statistics, 2of 25

3 Section 2.4 How Can We Describe the Spread of Quantitative Data? Agresti/Franklin Statistics, 3of 25

4 Measuring Spread Range: difference between the largest and smallest observations. IQR =Interquartile Range =3 rd quartile-1 st quartile IQR is robust to outliers since it is the difference of two medians. Standard deviation Agresti/Franklin Statistics, 4of 25

5 Identify the minimum and maximum sugar values: a. 2 and 14 b. 1 and 3 c. 1 and 15 d. 0 and 16 mode dot plot Mean=8.8, median=10, Q 1 =3, Q 3 =12 Min=1 Max=15 Range=15-1=14 Agresti/Franklin Statistics, 5of 25

6 Standard Deviation Creates a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations s = ( x x) n 1 2 Agresti/Franklin Statistics, 6of 25

7 Sample Standard Deviation (Shortcut Formula) s = n (Σx 2 ) - (Σx) 2 n (n - 1) Formula 2-5 Agresti/Franklin Statistics, 7of 25

8 Example: Publix check-out waiting times in minutes Data: 1, 4, 10. Find the sample mean and sample standard deviation. 2 x x x ( x x) 1 1 5= x ( x x) 2 n=3 15 x = = min s ( x x) = n 1 2 = = 21 = 4.6min Agresti/Franklin Statistics, 8of 25

9 Example: Publix check-out waiting times in minutes Data: 1, 4, 10. Find the sample mean and sample standard deviation Using the shortcut formula: 2 2 n x ( x ) s = = = = ( = ( 6 n ) 3 4 ( ( 15 ) min Agresti/Franklin Statistics, 9of 25 n 1 ) 1 = )

10 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 Agresti/Franklin Statistics, 10 of 25

11 Empirical Rule For bell-shaped data sets: Approximately 68% of the observations fall within 1 standard deviation of the mean Approximately 95% of the observations fall within 2 standard deviations of the mean Approximately 99% of the observations fall within 3 standard deviations of the mean Agresti/Franklin Statistics, 11 of 25

12 Parameter and Statistic A parameter is a numerical summary of the population A statistic is a numerical summary of a sample taken from a population Agresti/Franklin Statistics, 12 of 25

13 Five summary statistics Minimum =1 1 st quartile = 3 Median =10 3 rd quartile=12 Maximum =15 Agresti/Franklin Statistics, 13 of 25

14 Boxplot Agresti/Franklin Statistics, 14 of 25

15 Boxplot of SUGARg 16 max Q 3 10 Q 2 =median SUGARg 8 6 mean min Q 1 Agresti/Franklin Statistics, 15 of 25

16 Boxplot A box is constructed from Q 1 to Q 3 A line is drawn inside the box at the median A line extends outward from the lower end of the box to the smallest observation that is not a potential outlier A line extends outward from the upper end of the box to the largest observation that is not a potential outlier Agresti/Franklin Statistics, 16 of 25

17 Boxplot A box is constructed from Q 1 to Q 3 A line is drawn inside the box at the median A line extends outward from the lower end of the box to the smallest observation that is not a potential outlier A line extends outward from the upper end of the box to the largest observation that is not a potential outlier Agresti/Franklin Statistics, 17 of 25

18 Comparison using boxplots Example: Your company makes plastic pipes, and you are concerned about the consistency of their diameters. You measure ten pipes a week for three weeks. Create a boxplot to examine the distributions. 1 Open the worksheet PIPE.MTW. 2 Choose Graph > Boxplot or Stat > EDA > Boxplot. 3 Under Multiple Y's, choose Simple. Click OK. 4 In Graph Variables, enter 'Week 1' 'Week 2' 'Week 3'. Click OK. Agresti/Franklin Statistics, 18 of 25

19 Graph window output 9 Boxplot of Week 1, Week 2, Week Data Week 1 Week 2 Week 3 Agresti/Franklin Statistics, 19 of 25

20 Skewed to the right Symmetric Skewed to the left Agresti/Franklin Statistics, 20 of 25

21 Interpreting the results Tip To see precise information for Q1, median, Q3, interquartile range, whiskers, and N, hover your cursor over any part of the boxplot. The boxplot shows: Week 1 median is 4.985, and the interquartile range is to Week 2 median is 5.275, and the interquartile range is 5.08 to An outlier appears at 7.0. Week 3 median is 5.43, and the interquartile range is 4.99 to The data are positively skewed. Conclusion: The medians for the three weeks are similar. However, during Week 2, an abnormally wide pipe was created, and during Week 3, several abnormally wide pipes were created. Agresti/Franklin Statistics, 21 of 25

22 Z-Score The z-score for an observation measures how far an observation is from the mean in standard deviation units z = observatio n - mean standard deviation An observation in a bell-shaped distribution is a potential outlier if its z-score < -3 or > +3 Agresti/Franklin Statistics, 22 of 25

23 Inverse problem If Bob s score is 1.5 standard deviation higher than the mean, what is Bob s score for the previous problem. Denote Bob s score=x, then 1.5=(x-75)/10 so x=1.5(10)+75=90. Agresti/Franklin Statistics, 23 of 25

24 Example: Converting to z-score Scores on a test have a mean of 75 and a standard deviation of 10. Bob has a score of 90. Convert Bob score to a z- score. Round to the nearest hundredth. Bob s z-score=(90-75)/10=1.50 which means that Bob s score is 1.5 standard deviation higher than the mean. Agresti/Franklin Statistics, 24 of 25

25 2.6 How are descriptive summaries misused? (read) Figure 2.18, page 75 HW4: read section 3.2 problems 2.57, 2.62, 2.63, 2.65, 2.67, 2.68, 2.69, 2.71, 2.72 Agresti/Franklin Statistics, 25 of 25