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Chapter 3 Descriptive Statistics: Numerical Methods

Chapter 3 Descriptive Statistics: Numerical Methods. Measures of Location The Mean (A.M, G.M and H. M) The Median The Mode Percentiles Quartiles. Summary Measures. Describing Data Numerically. Center and Location. Variation. Mean. Range. Weighted Mean. Variance. Median. Mode.

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Chapter 3 Descriptive Statistics: Numerical Methods

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  1. Chapter 3Descriptive Statistics: Numerical Methods

  2. Measures of Location • The Mean (A.M, G.M and H. M) • The Median • The Mode • Percentiles • Quartiles

  3. Summary Measures Describing Data Numerically Center and Location Variation Mean Range Weighted Mean Variance Median Mode Standard Deviation Percentiles Coefficient of Variation Quartiles

  4. Mean • The Mean is the average of data values • The most common measure of central tendency • Mean = sum of values divided by the number of values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 Mean = 4

  5. Mean The mean (or average) is the basic measure of location or “central tendency” of the data. • The sample meanis a sample statistic. • The population mean  is a population statistic.

  6. Mean • Sample mean • Population mean n = Sample Size N = Population Size

  7. Example: College Class Size We have the following sample of data for 5 college classes: 46 54 42 46 32 We use the notation x1, x2, x3, x4, and x5 to represent the number of students in each of the 5 classes: X1 = 46 x2 = 54x3 = 42 x4 = 46 x5 = 32 Thus we have: The average class size is 44 students

  8. Median The median is the value in the middle when the data are arranged in ascending order (from smallest value to largest value). • For an odd number of observations the median is the middle value. • For an even number of observations the median is the average of the two middle values.

  9. The College Class Size example First, arrange the data in ascending order: 32 42 46 46 54 Notice than n = 5, an odd number. Thus the median is given by the middle value. 32 42 46 46 54 The median class size is 46

  10. Median Starting Salary For a Sample of 12 Business School Graduates A college placement office has obtained the following data for 12 recent graduates:

  11. First we arrange the data in ascending order 2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325 Notice that n = 12, an even number. Thus we take an average of the middle 2 observations: 2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325 Middle two values Thus

  12. Mode The mode is the value that occurs with greatest frequency • A measure of central tendency • Value that occurs most often • There may be no mode • There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 Mode = 5 No Mode

  13. The Mode MODE The value of the observation that appears most frequently.

  14. Characteristics of the Mode • Mode: the value of the observation that appears most frequently. • Advantage: Not affected by extremely high or low values. • Disadvantages: • For many sets of data, there is no mode because no value appears more than once. • For some data sets there is more than one mode. Characteristics of the Mean • The most widely used measure of location. • Major characteristics: • All values are used. • It is unique. • It is calculated by summing the values and dividing by the number of values. • Weakness: Its value can be unclear when extremely large or extremely small data compared to the majority of data are present. Properties and Uses of the Median • There is a unique median for each data set. • Not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

  15. Weighted Mean • When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. • In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. • When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

  16. Weighted Mean x =  wi xi  wi where: xi= value of observation i wi = weight for observation i

  17. Mean for Grouped Data • Sample Data • Population Data where: fi = frequency of class i Mi = midpoint of class i

  18. Weighted Mean • Used when values are grouped by frequency or relative importance Example: Sample of 26 Repair Projects Weighted Mean Days to Complete:

  19. Example: Apartment Rents Given below is the previous sample of monthly rents for one-bedroom apartments presented here as grouped data in the form of a frequency distribution.

  20. Example: Apartment Rents • Mean for Grouped Data This approximation differs by $2.41 from the actual sample mean of $490.80.

  21. Five houses on a hill by the beach Review Example House Prices: $2,000,000 500,000 300,000 100,000 100,000

  22. Mean: ($3,000,000/5) = $600,000 Median: middle value of ranked data = $300,000 Mode: most frequent value = $100,000 Summary Statistics House Prices: $2,000,000 500,000 300,000 100,000 100,000 Sum 3,000,000

  23. Percentiles The pth percentile is a value such that at least p percent of the observations are less than or equal to this value and at least (100 – p) percent of the observations are greater than or equal to this value. I scored in the 70th percentile on the Graduate Record Exam (GRE)—meaning I scored higher than 70 percent of those who took the exam

  24. Calculating the pth Percentile • Step 1: Arrange the data in ascendingorder (smallest value to largest value). • Step 2: Compute an index i where p is the percentile of interest and n in the number of observations. • Step 3: (a) If i is not an integer, round up. The next integer greater than i denotes the position of the pth percentile. (b) If i is an integer, the pth percentile is the average of values in i and i + 1

  25. Example: Starting Salaries of Business Grads Let’s compute the 85th percentile using the starting salary data. First arrange the data in ascending order. Step 1: 2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325 Step 2: Step 3: Since 10.2 in not an integer, round up to 11.The 85thpercentile is the 11th position (3130)

  26. Quartiles Quartiles are just specific percentiles Let: Q1 = first quartile, or 25th percentile Q2 = second quartile, or 50th percentile (also the median) Q3= third quartile, or 75th percentile Let’s compute the 1st and 3rd quartiles using the starting salary data. Note we already computed the median for this sample—so we know the 2nd quartile

  27. 2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325 Now find the 25th percentile: Note that 3 is an integer, so to find the 25th percentile we must average together the 3rd and 4th values: Q1 = (2850 + 2880)/2 = 2865 Now find the 75th percentile: Note that 9 is an integer, so to find the 75th percentile we must average together the 9th and 10th values: Q1 = (2950 + 3050)/2 = 3000

  28. Quartiles for the Starting Salary Data 2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325 Q1 = 2865 Q3 = 3000 Q1 = 2905 (Median)

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