1. 1 Probabilistic and Statistical Techniques Lecture 4 Dr. Nader Okasha

2. 2 Descriptive measures of data: 1.Measure of Center 2.Measure of Variation 3.Measure of Position

3. 3 Measure of Center The value at the center or middle of a data set. The purpose of a measure of center is to pinpoint the center of a set of values.

4. 4 Arithmetic Mean (Mean) The measure of center obtained by adding the values and dividing the total by the number of values The mean is affected by outliers

5. 5 Notation   denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of values in a sample. N represents the number of values in a population.

6. 6 Population Mean The population mean is the sum of all values in the population divided by the number of the values in the population. is pronounced ‘mu’ and denotes the mean of all values in a population

7. 7 Example There are 12 automobile manufacturing companies in the United States. Listed below is the no. of patents granted by U.S. government to each company in a recent year No. of PatentsCompanyNo. of PatentsCompany 210Mazda511General Motors 97Chrysler385Nissan 50Porsche275DaimlerChrysler 36Mitsubishi257Toyota 23Volvo249Honda 13BMW234Ford

8. 8 Example This is an example of a population mean because we are considering all the automobiles manufacturing companies obtaining patents. = 195

9. 9 Sample Mean The sample mean is the sum of all the sampled values divided by the total number of sampled values. is pronounced ‘x-bar’ and denotes the mean of a set of sample values

10. 10 Sample Mean Example SunCom is studying the number of minutes used monthly by clients in a particular cell phone rate plan. A random sample of 12 clients showed the following number of minutes used last month 90 77 94 89 119 112 91 110 100 92 113 83 What is the arithmetic mean number of minutes used? Sample mean = 97.5 minutes

11. 11 Median The middle value when the original data values are arranged in order of increasing (or decreasing) magnitude It is not affected by an extreme value If the number of values is even, the median is found by computing the mean of the two middle numbers. If the number of values is odd, the median is the number located in the exact middle of the list.

12. 12 5.40 1.10 0.42 0.73 0.48 1.10 0.42 0.48 0.73 1.10 1.10 5.40 (in order - even number of values) MEDIAN is = 0.915 0.73 + 1.10 2 Example

13. 13 Mode The value that occurs most frequently  Mode is not always unique  A data set may be: Bimodal Multimodal No Mode

14. 14 Example a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10  Mode is 1.10  Bimodal - 27 & 55  No Mode

15. 15 Midrange The value midway between the maximum and minimum values in the original data set Midrange = maximum value + minimum value 2

16. 16 Example Table 3.1 Table 3.2 Data Set I Data Set II For each data set determine: Mean Median Mode

17. 17 Example Solution

18. 18 Mean from a Histogram Use interval midpoint of classes for variable x Interval midpoint Number of counts

19. 19 Example Interval limits Interval Mid point x No. of counts ff. x 21 - 3025.528714 31 - 4035.5301065 41 - 5045.512546 51 - 6055.52111 61 - 7065.52131 71 - 8075.52151 Sum762718