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

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 Mean = Mean = 4

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

Mean  Sample mean  Population mean n = Sample Size N = Population Size

Example: College Class Size We have the following sample of data for 5 college classes: We use the notation x 1, x 2, x 3, x 4, and x 5 to represent the number of students in each of the 5 classes: X 1 = 46 x 2 = 54 x 3 = 42 x 4 = 46 x 5 = 32 Thus we have: The average class size is 44 students

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

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

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

Notice that n = 12, an even number. Thus we take an average of the middle 2 observations: Middle two values First we arrange the data in ascending order Thus

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 Mode = No Mode

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

Characteristics of the Mean 1. The most widely used measure of location. 2. Major characteristics:  All values are used.  It is unique.  It is calculated by summing the values and dividing by the number of values. 3. 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 1. There is a unique median for each data set. 2. Not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. Characteristics of the Mode 1. Mode: the value of the observation that appears most frequently. 2. Advantage: Not affected by extremely high or low values. 3. 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.

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.

Weighted Mean x =  w i x i  w i where: x i = value of observation i w i = weight for observation i

Sample Data Population Data where: f i = frequency of class i M i = midpoint of class i Mean for Grouped Data

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

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.

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

Five houses on a hill by the beach Review Example House Prices: $2,000, , , , ,000

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

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 70 th percentile on the Graduate Record Exam (GRE)—meaning I scored higher than 70 percent of those who took the exam

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

Example: Starting Salaries of Business Grads Let’s compute the 85 th percentile using the starting salary data. First arrange the data in ascending order.starting salary data Step 1: Step 2: Step 3: Since 10.2 in not an integer, round up to 11.The 85 th percentile is the 11 th position (3130)

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

Now find the 25 th percentile: Note that 3 is an integer, so to find the 25 th percentile we must average together the 3 rd and 4 th values: Q 1 = ( )/2 = 2865 Now find the 75 th percentile: Note that 9 is an integer, so to find the 75 th percentile we must average together the 9 th and 10 th values: Q 1 = ( )/2 = 3000

Quartiles for the Starting Salary Data Q 1 = 2865 Q 1 = 2905 (Median) Q 3 = 3000