NUMERICAL DESCRIPTIVE STATISTICS Measures of Central Tendency

1. NUMERICAL DESCRIPTIVE STATISTICS Measures of Central Tendency

2. MEASURES OFCENTRAL TENDENCY • The following are typical measures of central tendency for a population • Mean -- the average • Median -- the middle observation after the data has been ordered • Mode -- the observation that occurs most often

3. Population Mean () • One measure of central tendency is the mean • The population mean is the average (or weighted average) of all observations of the population • Population mean for a population of size N:  = xi/N

4. Example • N = 2000 students took an introductory statistics last year at CSUF. Using the 4-point scale (A=4, B=3, etc.) the following were the grades 4,2,1,3,3,3,2,… 2. • The mean grade of all statistics students was:  = (4+2+1+3+3+3+2+…+2)/2000 = 2.39

5. Calculating  Using Frequency Data • Would not add up the 2000 numbers this way if we knew how many A’s, B’s, C’s, D’s, and F’s • Example A(4) = 304, B(3) = 530, C(2) = 852, D (1) = 270 F(0) = 44  = (304(4) +530(3) +852(2) +270(1) +44(0))/2000 = 2.39 • Note the relative frequencies are found by dividing by N (which is 2000): A(4) = .152, B(3) = .265, C(2) = .426, D (1) = .135 F(0) = .022 • We can also calculate  as a weighted average (weighted by relative frequencies):  = .152(4) +.265(3) +.426(2) +.135(1) +.022(0) = 2.39

6. Sample Data • The only way to know  for sure is to have access to all the data in the population • We rarely know all the members of the population • Thus we try to estimate a parameter such as  by taking a sample of n members from the N members of the population • Result: POINT ESTIMATE of the parameter

7. EXAMPLE Suppose 10 students were surveyed. Their grades were: 4,2,3,3,2,2,1,4,3,2

8. Sample Mean The sample mean is the point estimate for the population mean. DATA: 4,2,3,3,2,2,1,4,3,2

9. Frequency Calculation for the Sample Mean Grade Frequency A -- 4 2 B -- 3 3 C -- 2 4 D -- 1 1

10. 2,3 Median = Average of the middle two (2+3)/2 = 2.5 Sample Median(Even Number of Data Points) • The the average of the two middle observations • First put in either ascending or descending order • DATA: 4,2,3,3,2,2,1,4,3,2 • There is an even number of data points (10) • ASCENDING ORDER: 1,2,2,2,2,3,3,3,4,4

11. 3 Median is middle observation Median = 3 Sample Median(Odd Number of Data Points) • Suppose there were an odd number of observations -- suppose an 11 person was surveyed and she got an “A” (4) • There is an odd number of data points (11) • ASCENDING ORDER: 1,2,2,2,2,3,3,3,4,4,4

12. Highest Frequency Mode = 2 4 2 Sample Mode • Observation that occurs most often in the sample data • Create a frequency distribution Grade Frequency A -- 4 2 B -- 3 3 C -- 2 4 D -- 1 1

13. Central Tendency in Excel • Suppose data are in cells A2 to A11 • Mean -- =AVERAGE(A2:A11) • Median -- =MEDIAN(A2:A11) • Mode -- =MODE(A2:A11) • Can also use Descriptive Statistics Option from Data Analysis in the Tools Menu

14. =AVERAGE(A2:A11) =MEDIAN(A2:A11) =MODE(A2:A11)

16. Check Labels Where data values are stored Enter Name of Output Worksheet Check both: Summary Statistics Confidence Level

17. Drag to make Column A wider Sample Mean Sample Median Sample Mode

18. Review • Difference Between Population and Sample Data • Calculation of the Mean: • Definition • Relative Frequency Approach • Median • Mode • Excel • Functions • Descriptive Statistics