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# Measures of center un grouped data

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### Measures of center un grouped data

1. 1.  ∑-denotes addition of a set of values  X-the variable used to represent the individual data values  N-the number of values in a population  n-the number of values in a sample
2. 2.  Population --complete collection of all elements to be studied  Sample --subcollection of members selected from a population
3. 3. Population Sample Weighted 1 N i i x N     1 i i n x x n    1 1 iWi i i i k x x k w     
4. 4.  =lower case Greek mu =denotes all values of the population  =pronounced as x- bar =denotes all the data values from a sample of larger population  x
5. 5. PopulationSample  If N is odd  If N is even  If n is odd  If n is even 1 ( )2 N x   ( 1)2 2 1 [ ] 2 N N x x   ( 1)2 2 1 [ ] 2 n n x x x   1 ( )2 n x x  
6. 6.  Observations: Number of hours spent by the students in studying 2 3 3 3 6 2 3 2 4 2.5
7. 7. 10 1 1 2 3 4. .... 10 10 30.5 10 3.05 xi x n x x x x x          
8. 8. ( 1)2 2 1 [ ] 2 1 10 10 [ ( 1)] 2 2 2 1 ( 5 6) 2 1 (3 3) 2 3 n n x x x x x x x           
9. 9.  3  -- the highest frequency
10. 10. Observations=60 87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77, 73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75, 84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82, 85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68 μ = (87+85+ 79 +….+72+68)/60 = 4751/60 = 79.183
11. 11. We now find the median of the population of temperature readings 87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77, 73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75, 84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82, 85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68 Arrange these 60 measurements in ascending order 65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92 Since N/2 = 30 and both the 30th and 31st values in the list are the same, we obtain median = 78
12. 12. One further parameter of a population that may give some indication of central tendency of the data is the mode Define: mode = most frequently occurring value in the population From the previous data we see: 65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92 That the value 81 occurs 8 times mode = 81 Note! If two different values were to occur most frequently, the distribution would be bimodal. A distribution may be multi-modal.
13. 13. Grade 99 92 90 91 88 85 82 Units 3 2 6 1 4 3 3
14. 14. 1 1 iWi i i i k x x k w      99(3) 92(2) 90(6) 91(6) 88(4) 85(3) 82(3) 3 2 6 1 4 3 3 89.32GPA              