1. Statistics Lecture Notes Dr. Halil İbrahim CEBECİ Chapter 03 Numerical Descriptive Techniques

2.  Measures of Central Location  Mean, Median, Mode  Measures of Variability  Range, Standard Deviation, Variance, Coefficient of Variation  Measures of Relative Standing  Percentiles, Quartiles  Measures of Linear Relationship  Covariance, Correlation, Least Squares Line Numerical Descriptive Techniques Statistics Lecture Notes – Chapter 03

3. Arithmetic Mean It is computed by simply adding up all the observations and dividing by the total number of observations Measures of Central Location Statistics Lecture Notes – Chapter 03 is seriously affected by extreme values called “outliers”. E.g. as soon as a billionaire moves into a neighborhood, the average household income increases beyond what it was previously

4. Ex3.1 – Weights of the students of a classroom in elementary school are given below. Calculate the arithmetic mean. If a new student with weight of 140lbs came in, what the new mean of the classroom would be. Discuss the validity of results. Arithmetic Mean Statistics Lecture Notes – Chapter 03 89779010166 76596475 88657572706466688280

5. Median The value that falls in the middle of the pre-ordered (ascending or descending) observation list Where there is an even number of observations, the median is determined by averaging the two observation in the middle. Measures of Central Location Statistics Lecture Notes – Chapter 03

6. Measures of Central Location Statistics Lecture Notes – Chapter 03 5964 6566 687072 75 76778082888990101140

7. Mode Observation that occurs with the greatest frequency. There are several problems with using the mode as a measure of central location  In a small sample it may not be a very good measure  It may not unique Measures of Central Location Statistics Lecture Notes – Chapter 03

8. Mode Statistics Lecture Notes – Chapter 03 89779010166 76596475 88657572706466688280

9. Geometric Mean Statistics Lecture Notes – Chapter 03 4537403035455095

10. Measures of central location fail to tell the whole story about the distribution; that is, how much are the observations spread out around the mean value? Measures of Variability Statistics Lecture Notes – Chapter 03 For example, two sets of class grades are shown. The mean (=50) is the same in each case… But, the red class has greater variability than the blue class.

11. Measures of Variability Statistics Lecture Notes – Chapter 03

12. Variance: A measure of the average distance between each of a set of data points and their mean value. Variance is the difference between what is expected (Budget) and the actuals (Expenditure). it is the difference between "should take" and "did take". Measures of Variability Statistics Lecture Notes – Chapter 03

13. Ex3.6 – The following are the number of summer jobs a sample of six student applied for. Find the mean and variance of these data. Variance Statistics Lecture Notes – Chapter 03 1715237913

14. Standart Deviation: shows how much variation or "dispersion" exists from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values Measures of Variability Statistics Lecture Notes – Chapter 03

15. Ex3.7 – Find the standard deviation of the values of Ex3.6 Standard Deviation Statistics Lecture Notes – Chapter 03 1715237913

16. Interpreting the Standard Deviation Statistics Lecture Notes – Chapter 03 Approximately 68% of all observations fall within one standard deviation of the mean. Approximately 95% of all observations fall within two standard deviations of the mean. Approximately 99.7% of all observations fall within three standard deviations of the mean If the histogram is bell-shaped

17. Chebysheff’s Theorem Statistics Lecture Notes – Chapter 03

18. Chebysheff’s Theorem Statistics Lecture Notes – Chapter 03

19. Chebysheff’s Theorem Statistics Lecture Notes – Chapter 03

20. Ex3.9 – if we know that the distribution of the values in Ex3.8 is bell-shaped, new answers are a.66 and 78?  Approximately 68% of the grades fall between this interval (1 standard deviation) b.60 and 84?  Approximately 95% of the grades fall between this interval (2 standard deviation) c.54 and 90?  Approximately 99.7% of the grades fall between this interval (3 standard deviation) Chebysheff’s Theorem Statistics Lecture Notes – Chapter 03

21. Measures of Relative Standing Statistics Lecture Notes – Chapter 03

22. Ex3.10 – The weights in pounds of a group of workers are as follows: Measures of Relative Standing Statistics Lecture Notes – Chapter 03 173165171175188 183177160151169 162179145171175 168158186182162 154180164166157 a.Find the 25th percentile of the weights b.Find the 50th percentile of the weights. c.Find the 75th percentile of the weights.

23. A3.10a Measures of Relative Standing Statistics Lecture Notes – Chapter 03 A3.10b A3.10c Range Weights Range Weights 1 145 14 171 2 151 15 171 3 154 16 173 4 157 17 175 5 158 18 175 6 160 19 177 7 162 20 179 8 162 21 180 9 164 22 182 10 165 23 183 11 166 24 186 12 168 25 188 13 169

24. Box Plots Statistics Lecture Notes – Chapter 03

25. Box Plots Statistics Lecture Notes – Chapter 03

26. Ex3.11 – Use the weights in pounds of a group of workers in Ex3.10 and,Construct a box plot for these weights. Compute the interquartile range and identify any outliers. A3.11 – The first step in constructing a box plot is to rank the data, in order to determine the numerical values of the five descriptive statistics to be plotted. Box Plots Statistics Lecture Notes – Chapter 03 Descriptive StatisticsNumerical Value 145 161 169 178 188 17 135.5 and 203,5 * There is no outlier

27.  Wendy’s service time is shortest and least variable  Hardee’s has the greatest variability,  Jack-in-the-Box has the longest service times. Box Plots Statistics Lecture Notes – Chapter 03

28. Covariance: a measure of how much two variables change together Measures of Linear Relationship Statistics Lecture Notes – Chapter 03

29. Coefficient of Correlation: Correlation is a scaled version of covariance that takes on values in [−1,1] with a correlation of ±1 indicating perfect linear association and 0 indicating no linear relationship. Measures of Linear Relationship Statistics Lecture Notes – Chapter 03

30. Ex3.12 - Based on the sample of data shown below. Measure how these two variables are related by computing their covariance and coefficient of correlation. Measures of Linear Relationship Statistics Lecture Notes – Chapter 03 1125 1233 1122 1541 818 1028 1132 1224 1753 1126

31. Measures of Linear Relationship Statistics Lecture Notes – Chapter 03 1125- 0.80.64-5,227.044.16 12330.20.042,87.840.56 1122-0.80.64-8,267.246.56 15413.210.2410,8116.6434.56 818-3.814.44-12,2148.8446.36 1028-1.83.24-2,24.843.96 1132-0.80.641,83.24-1.44 12240.20.04-6,238.44-1.24 17535.227.0422,8519.84118.56 1126-0.80.64-4,217.643.36

32. Measures of Linear Relationship Statistics Lecture Notes – Chapter 03

33. Least Square Method: Produces a straight line drawn through the points so that the sum of squared deviations between the points and the line is minimized. Measures of Linear Relationship Statistics Lecture Notes – Chapter 03

34. Ex3.13 - Find the least squares (regression) line for the data in Ex3.12 using the shortcut formulas for the coefficients. Measures of Linear Relationship Statistics Lecture Notes – Chapter 03

35. Q3.1 - Consider the following sample of measurements Compute each of the following: a.the mean b.the median c.the mode Exercises Statistics Lecture Notes – Chapter 03 37323028303235283229

36. Q3.2 - Consider the following sample of data Compute each of the following for this sample: a.the mean b.the range c.the variance d.the standard deviation Exercises Statistics Lecture Notes – Chapter 03 172518142821

37. Q3.3 - Consider the following sample of data, a.Construct a box plot for these data b.Compute the interquartile range and identify any outliers Exercises Statistics Lecture Notes – Chapter 03 208160175334228211179354 265215191239298226220260 173163226165252422284232 225348290180300200245204 256281230275158224315217

38. Q3.4 – based on the sample of data shown below, a.Calculate Covariance b.Calculate the coefficient of Correlation c.Find the equation of least square line Exercises Statistics Lecture Notes – Chapter 03 5528 12876 8653 6571 4556 151058 10963 7719