Measure of Location and Variability

Histogram Multimodal Multimodal

Mean, Median and Mode Mean: average value Mean: average value Sample mean: denoted by x-bar (English) Sample mean: denoted by x-bar (English) Population mean: Mu (Greek) Population mean: Mu (Greek) Median: value in the middle when the data are arranged in ascending order (smallest to largest). Median: value in the middle when the data are arranged in ascending order (smallest to largest). odd number of observations: the middle value odd number of observations: the middle value even number of observations: the average of the two middle values even number of observations: the average of the two middle values

Mean, Median and Mode II Mode: the value that occurs with greatest frequency (may be more than one mode in a set of data). Mode: the value that occurs with greatest frequency (may be more than one mode in a set of data). Example: Example: Data: 1,1,2,2,2,2,3,3,4,4,5 Data: 1,1,2,2,2,2,3,3,4,4,5 Mode=2 Mode=2

Percentile the pth percentile is the 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. the pth percentile is the 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. For example: If you scored in the 90th percentile on the verbal part of your SAT’s, this would mean that you scored above 90% of all verbal scores taken for the SAT’s at that time. For example: If you scored in the 90th percentile on the verbal part of your SAT’s, this would mean that you scored above 90% of all verbal scores taken for the SAT’s at that time.

Percentile II To calculate the pth percentile: To calculate the pth percentile: arrange the data in ascending orderarrange the data in ascending order compute an index i=(p/100)*ncompute an index i=(p/100)*n if i is not an integer, round up, the next integer greater than i denotes the position of the pth percentileif i is not an integer, round up, the next integer greater than i denotes the position of the pth percentile if i is an integer the pth percentile is the average of the values in positions i and i+1.if i is an integer the pth percentile is the average of the values in positions i and i+1.

Quartile used to divide the data into 4 parts. used to divide the data into 4 parts. Q1: first quartile, 25th percentile Q1: first quartile, 25th percentile Q2: second quartile, 50th percentile, median Q2: second quartile, 50th percentile, median Q3: third quartile, 75th percentile Q3: third quartile, 75th percentile

5 Number Summary 1. Smallest value 1. Smallest value 2. first quartile (Q1) 2. first quartile (Q1) 3. Median (Q2) 3. Median (Q2) 4. third quartile (Q3) 4. third quartile (Q3) 5. Largest value 5. Largest value

Variability We have talked about measure of location. We have talked about measure of location. Mean, median, modeMean, median, mode Why do we need to look at variability? Why do we need to look at variability? Data with same mean or median may have different variabilityData with same mean or median may have different variability Example: two sets of quiz gradesExample: two sets of quiz grades mean= mean= mean= mean= The two means are similar but apparently the second set is more spreadout  greater variability The two means are similar but apparently the second set is more spreadout  greater variability

How to quantify variability Range Range Inter-Quantile Range (IQR) Inter-Quantile Range (IQR) Variance Variance Population variancePopulation variance Sample varianceSample variance Standard Deviation (S.D., std dev) Standard Deviation (S.D., std dev) Population S.D.Population S.D. Sample S.D.Sample S.D. Coefficients of Variation Coefficients of Variation

Range Range=max — min Range=max — min In our example: In our example: Set 1: Range=20-11=9Set 1: Range=20-11=9 Set 2: Range=20-7=13Set 2: Range=20-7=13

Inter-Quantile Range Set 1: Set 1: Median=15, Q1=12, Q3=18Median=15, Q1=12, Q3=18 IQR = 18-12=6IQR = 18-12=6 Set 2: Set 2: Median: 18, Q1=11, Q2=19Median: 18, Q1=11, Q2=19 IQR=19-11=8IQR=19-11=8

Boxplot a graphical summary of data that is based on a five-number summary. a graphical summary of data that is based on a five-number summary. Example: Example: