Organizing and Displaying Categorical Data Guide Part 2

Class 2 Measures of Spread Example 1: {25,17,18,19,25,31,34,10,14} 1. Range= Maximum - minimum = 34-10=24 2. Interquartile range Percentile/ Cumulative Probability Function (Cumulative Distribution Function) Outcome/ Cumulative Probability Bar Graph What is .111 percentile? If you ranked .11, then what is your score? 10 If you ranked .22, then what is your score? 14 If you ranked .33 then 17 is your score If you scored 18, then what is your cumulative probability/ rank? 4/9 {3,4,8,9,12,15,25,27,30,1004} Measure of Spread 1. Range = Max-min = 1004-3 = 1001 *measure is skewed because of the outlier, 1004 2. Interquartile-Range= 75th Percentile- 25th Percentile = Middle 50 percent of the data is in this range = a. 75th percentile = 26 b. 25th percentile = 6 c. IQR= 26-6 = 20 ● ● ● Stem and leaf plot Histogram table/ range RF table (histogram) Box- Plot 25 percent of the data is above 26; 25 percent of the data is below 6; 25 percent of the data is between 6 and 12; 25 percent of the data is between 12 and 26 - The data is not symmetric Example 3: {11,15,17, 8, 22, 34, 12, 100, 70 ,80} a. Draw a Score/ Rank aka Outcome/ Cumulative Probability Bar graph b. Calculate the 25th percentile and 75th percentile c. Draw a box-plot 25th percentile = (11+12)/2 = 11.5 75th percentile = (34+70)/2 = 52 Is the data symmetric? No Note: There is more than 1 way to calculate the median, aka 50th percentile. 50th percentile = 17 Last week, we said the 50th percentile (median) is the average of 17 and 22, which is 19.5 How do you find the median in your test/ HW? Excel: ● 1st Quartile: =Quartile (A:A, 1) ● 2nd Quartile: =Quartile (A:A, 2) 3rd Measure of Spread: Variance and Standard Deviation (26,32) 1. Range 2. IQR 3. Variance =outcome/ d^2 from the mean bar graph Average (outcome) =29 Average (d^2 from the mean)= (9+9)/2 Example: 11,15,19 Calculate Variance … Outcome/ d^2 from the mean bar graph ● ● Average (outcome) = 15 Average (d^2 from mean)= (16+0+16)/3 = 10.6 Variance is the average d^2 from the mean; it is a measure of how spread out the data is. The higher the variance, the more spread out the data is Note: The variance can never be negative. Example: (8,10,12,14) Calculate the Variance ● Step 1: Make an outcome/ d^2 from mean bar graph. Variance = Avg = (9+1+1+9)/4 = 5 SD = Square root of the variance = SQRT(5) Step1: Draw the outcome/ d^2 from mean bar graph ExampleA: Population Variance = (9+1+1+9)/4 = 5 Population SD = SQRT (5) Example B: Sample Variance = (9+1+1+9)/(4-1)= 20/3 Sample SD = SQRT (20/3) *Point 1: If the question says to calculate the variance, you should ask yourself is the data a sample of the population or entire population. Usually it’s a sample. Point 2: If the sample variance is the average d^2 from the mean, why do we divide 3(4-1)? ANSWER 1: That’s the formula ANSWER 2: The variance is the average distance^2 from the population mean. In B, we don’t know the population mean and therefore to correct this issue we subtract 1 from the denominator. ANSWER 3: See Bessel’s correction on BB. [8,20] A. Assuming this data represents the totality of data of the population, calculate the variance and standard deviation. B. Assuming this data is a sample of the population, calculate the variance and standard deviation. A. Calculate the variance: Population variance= (36+36)/2 = 36 a. Pop SD = SQRT (36)= 6 B. Sample variance = (36+36)/(2-1) = 72 a. Sample SD= SQRT (72) NOTE 1: Excel On your exams and HW, use excel to calculate the variance and SD. Variance ● Population ○ VAR.P ● Sample (n-1) ○ VAR.S Standard Deviation ● Population ○ STDEV.P ● Sample ○ STDEV.S NOTE 2: Difference between Sample and Population Variance The population variance is the average d^2 the data is from the mean. When we collect a sample we can only approximate the population variance. To guestimate the population variance, we calculate the sample variance. NOTE 3: Greek versus Latin Letters Pop. Variance = 36 Pop. S.D. = 6 Sample Variance = 72 Sample S.D. = SQRT(72) NOTE 4: Difference between the variance and S.D. E.G. (8,20) Population Variance = 36 Population S.D = 6 Interpretation - Average d^2 from the mean is 36 - SQRT (Avg d^2 from the mean) = 6 The S.D is the average distance from the mean but not precisely true. Z-SCORE We collect the following sample: (107.109.111) The sample mean = 109 The sample variance = 4 The sample S.D = 2 How far is 120 away from the mean? 11 How far is 100 away from the mean? -9 How many S.D. away is 120 from the mean (each sd is 2 units)? 1 sd = 2 units 11 units= +5.5 SD from the mean How many S.D away from the mean is 100? -4.5 SD Z-Score of 120? Find the standard value of 120; how many SD above or below the mean is 120? Z-score of 120= Standardize value of 120= (120-109)/ 2=5.5 Z- score of 100 = (100-109)/2= -4.5 Example: 109, 112, 115 Sample mean: 112 Sample variance: 9 Sample S.D: 3 Calculate Z111, Z107,Z106 Z106= (106-112)/3= -2 Z111= (111-112)/3 = -1/3 Z107= (107-112)/3= -5/3 Example: 108, 113, 118 Sample mean: 113 Sample variance: 25 Sample S.D: 5 Z133= (133-113)/5= 4 Z103= (103-113)/5= -2 Z-score of 133 is 4. Standard value of 133 is 4. 113 is 4 standards above the mean. Z-score of 103 is -2. Standardized value of 103 is -2. 103 is -2 S.D. below the mean. Z-SCORE/STANDARDIZED VALUE MEASURES DISTANCE IN S.D. UNITS Z-Score in EXCEL 108 113 118 Z-score of 133 where the mean is 113 and the S.D. is 5 =Standardize (133,113,5) =Standardize (number,mean,s.d.) Box & Whisker Plot 25% of data is above 30 (q3) 25% of data is below 10 (q1) It would be nicer if we could say 25% of the data is between 30 and another # It would be nicer if we could say 25% of the data is between 10 and another # With the whiskers: 25% of the data is between 30 and 35. 25% of the data is between 10 and -2 Another thing the whiskers add is the ability to find outliers (-12 is an outlier)