Summarizing Data With Frequency Tables Section 2 2

2 -2 Summarizing Data With Frequency Tables v Frequency Table lists categories (or classes) of scores, along with counts (or frequencies) of the number of scores that fall into each category Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 2

Table 2 -1 Axial Loads of 0. 0109 in. Cans 270 278 250 278 290 274 242 269 257 272 265 263 234 270 273 270 277 294 279 268 230 268 278 262 273 201 275 260 286 272 284 282 278 263 273 282 285 289 268 208 292 275 279 276 242 285 273 268 258 264 281 262 278 265 241 267 295 283 281 209 276 273 263 218 271 289 223 217 225 283 292 270 262 204 265 271 273 283 275 276 282 270 256 268 259 272 269 270 251 208 290 220 259 282 277 282 256 293 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 254 223 263 274 262 263 200 272 268 206 280 287 257 284 279 252 280 215 281 291 276 285 287 290 228 274 277 286 277 251 278 277 286 277 289 267 276 206 284 269 284 268 291 289 293 277 280 274 282 230 275 236 295 289 283 261 262 252 283 277 204 286 270 278 270 283 272 281 288 248 266 256 292 3

Table 2 -2 Frequency Table of Axial Loads of Aluminum Cans Axial Load Frequency 200 - 209 9 210 - 219 3 220 - 229 5 230 - 239 4 240 - 249 4 250 - 259 14 260 - 269 32 270 - 279 52 280 - 289 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 4

Frequency Table Definitions Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 5

Lower Class Limits are the smallest numbers that can actually belong to different classes Table 2 -2 Score Frequency 200 - 209 9 210 - 219 3 220 - 229 5 230 - 239 4 240 - 249 4 250 - 259 14 260 - 269 32 270 - 279 52 280 - 289 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 7

Lower Class Limits are the smallest numbers that can actually belong to different classes Table 2 -2 Lower Class Limits Score Frequency 200 - 209 9 210 - 219 3 220 - 229 5 230 - 239 4 240 - 249 4 250 - 259 14 260 - 269 32 270 - 279 52 280 - 289 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 8

Upper Class Limits are the largest numbers that can actually belong to different classes Table 2 -2 Score Frequency 200 - 209 9 210 - 219 3 220 - 229 5 230 - 239 4 240 - 249 4 250 - 259 14 260 - 269 32 270 - 279 52 280 - 289 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 10

Upper Class Limits are the largest numbers that can actually belong to different classes Table 2 -2 Upper Class Limits Score Frequency 200 - 209 9 210 - 219 3 220 - 229 5 230 - 239 4 240 - 249 4 250 - 259 14 260 - 269 32 270 - 279 52 280 - 289 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 11

Class Boundaries are obtained by increasing the upper class limits and decreasing the lower class limits by the same amount so that there are no gaps between consecutive numbers that is the “midpoint between two classes” Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 12

Class Boundaries midpoint between two classes Copyright © 1998, Triola, Elementary Statistics Addison Wesley

Class Boundaries midpoint between two classes Table 2 -2 199. 5 Score Frequency 200 - 209. 5 9 210 - 219. 5 3 220 - 229. 5 5 230 - 239. 5 4 240 - 249. 5 4 250 - 259. 5 14 260 - 269. 5 32 270 - 279. 5 52 280 - 289. 5 299. 5 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 14

Class Boundaries midpoint between two classes Table 2 -2 Class Boundaries 199. 5 Score Frequency 200 - 209. 5 9 210 - 219. 5 3 220 - 229. 5 5 230 - 239. 5 4 240 - 249. 5 4 250 - 259. 5 14 260 - 269. 5 32 270 - 279. 5 52 280 - 289. 5 38 290 - 299 14 299. 5 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 15

Class Marks midpoints within the classes Copyright © 1998, Triola, Elementary Statistics Addison Wesley

Class Marks midpoints within the classes Table 2 -2 Score Frequency 200 - 204. 5 - 209 9 210 - 214. 5 - 219 3 220 - 224. 5 - 229 5 230 - 234. 5 - 239 4 240 - 244. 5 - 249 4 250 - 254. 5 - 259 14 260 - 264. 5 - 269 32 270 - 274. 5 - 279 52 280 - 284. 5 - 289 38 290 - 294. 5 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 17

Class Marks midpoints within the classes Table 2 -2 Class Marks Score Frequency 200 - 204. 5 - 209 9 210 - 214. 5 - 219 3 220 - 224. 5 - 229 5 230 - 234. 5 - 239 4 240 - 244. 5 - 249 4 250 - 254. 5 - 259 14 260 - 264. 5 - 269 32 270 - 274. 5 - 279 52 280 - 284. 5 - 289 38 290 - 294. 5 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 18

Class Width is the difference between two consecutive lower class limits, upper class limits, class boundaries, or class marks Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 19

Class Width is the difference between two consecutive lower class limits, upper class limits, class boundaries, or class marks Table 2 -2 Score Frequency 200 - 10 - 209 9 210 - 219 3 220 - 10 - 229 5 230 - 10 - 239 4 240 - 10 - 249 4 250 - 10 - 259 14 260 - 10 - 269 32 270 - 10 - 279 52 280 - 10 - 289 38 290 - 10 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 20

Class Width is the difference between two consecutive lower class limits, upper class limits, class boundaries, or class marks Table 2 -2 Class Width Score Frequency 200 - 10 - 209 9 210 - 219 3 220 - 10 - 229 5 230 - 10 - 239 4 240 - 10 - 249 4 250 - 10 - 259 14 260 - 10 - 269 32 270 - 10 - 279 52 280 - 10 - 289 38 290 - 10 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 21

Determine the Definition Values for this Frequency Table Quiz Scores v Classes Frequency 0 -4 2 5 -9 5 10 - 14 8 15 - 19 11 20 - 24 7 v Lower Class Limits v Upper Class Limits v Class Boundaries v Class Marks v Class Width Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 22

Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 23

Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up. range class width = round up of number of classes Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 24

Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up. range class width = round up of number of classes 3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 25

Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up. range class width = round up of number of classes 3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score. 4. Add the class width to the starting point to get the second lower class limit. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 26

Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up. range class width = round up of number of classes 3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score. 4. Add the class width to the starting point to get the second lower class limit. 5. List the lower class limits in a vertical column and enter the upper class limits. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 27

Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score – lowest score) and round up. range class width = round up of number of classes 3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score. 4. Add the class width to the starting point to get the second lower class limit. 5. List the lower class limits in a vertical column and enter the upper class limits. 6. Represent each score by a tally mark in the appropriate class. Total tally marks to find the total frequency for each class. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 28

Guidelines For Frequency Tables 1. Classes should be mutually exclusive. 2. Include all classes, even if the frequency is zero. 3. Try to use the same width for all classes. 4. Select convenient numbers for class limits. 5. Use between 5 and 20 classes. 6. The sum of the class frequencies must equal the number of original data values. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 29

Relative Frequency Table Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 30

Relative Frequency Table relative frequency = class frequency sum of all frequencies Copyright ©

Relative Frequency Table 2 -2 Score Table 2 -3 Frequency Axial Load Relative Frequency 200 - 209 9 200 - 209 0. 051 210 - 219 3 210 - 219 0. 017 220 - 229 5 220 - 229 0. 029 230 - 239 4 230 - 239 0. 023 240 - 249 4 240 - 249 0. 023 250 - 259 14 250 - 259 0. 080 260 - 269 32 260 - 269 0. 183 270 - 279 52 270 - 279 0. 297 280 - 289 38 280 - 289 0. 217 290 - 299 14 290 - 299 0. 080 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 32

Relative Frequency Table 2 -2 Score Table 2 -3 Frequency Axial Load Relative Frequency 200 - 209 9 200 - 209 0. 051 210 - 219 3 210 - 219 0. 017 220 - 229 5 220 - 229 0. 029 230 - 239 4 230 - 239 0. 023 240 - 249 4 240 - 249 0. 023 250 - 259 14 250 - 259 0. 080 260 - 269 32 260 - 269 0. 183 270 - 279 52 270 - 279 0. 297 280 - 289 38 280 - 289 0. 217 290 - 299 14 290 - 299 0. 080 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. - 9 =. 051 175 3 =. 017 175 5 =. 029 175 33

Cumulative Frequency Table 2 -2 Score Table 2 -4 Frequency Axial Load Cumulative Frequency 200 - 209 9 Less than 210 9 210 - 219 3 Less than 220 12 220 - 229 5 Less than 230 17 230 - 239 4 Less than 240 21 240 - 249 4 Less than 250 25 250 - 259 14 Less than 260 39 260 - 269 32 Less than 270 71 270 - 279 52 Less than 280 123 280 - 289 38 Less than 290 161 290 - 299 14 Less than 300 175 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 34

Cumulative Frequency Table 2 -2 Score Table 2 -4 Frequency Axial Load Cumulative Frequency 200 - 209 9 Less than 210 9 210 - 219 3 Less than 220 12 220 - 229 5 Less than 230 17 230 - 239 4 Less than 240 21 240 - 249 4 Less than 250 25 250 - 259 14 Less than 260 39 260 - 269 32 Less than 270 71 270 - 279 52 Less than 280 123 280 - 289 38 Less than 290 161 290 - 299 14 Less than 300 175 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. Cumulative Frequencies 35

Frequency Tables Table 2 -3 Table 2 -2 Score Frequency Axial Load Relative Frequency 200 - 209 9 200 - 209 0. 051 210 - 219 3 210 - 219 0. 017 220 - 229 5 220 - 229 0. 029 230 - 239 4 230 - 239 0. 023 240 - 249 4 240 - 249 0. 023 250 - 259 14 250 - 259 0. 080 260 - 269 32 260 - 269 0. 183 270 - 279 52 270 - 279 0. 297 280 - 289 38 280 - 289 0. 217 290 - 299 14 290 - 299 0. 08 - Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. Table 2 -4 Axial Load Cumulative Frequency Less than 210 9 Less than 220 12 Less than 230 17 Less than 240 21 Less than 250 25 Less than 260 39 Less than 270 71 Less than 280 123 Less than 290 161 Less than 300 175 36

Pictures of Data Section 2 -3 M A R I O F. T R

Pictures of Data To depict the nature or shape of the data distribution Copyright

Table 2 -1 Axial Loads of 0. 0109 in. Cans 270 278 250 278 290 274 242 269 257 272 265 263 234 270 273 270 277 294 279 268 230 268 278 262 273 201 275 260 286 272 284 282 278 263 273 282 285 289 268 208 292 275 279 276 242 285 273 268 258 264 281 262 278 265 241 267 295 283 281 209 276 273 263 218 271 289 223 217 225 283 292 270 262 204 265 271 273 283 275 276 282 270 256 268 259 272 269 270 251 208 290 220 259 282 277 282 256 293 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 254 223 263 274 262 263 200 272 268 206 280 287 257 284 279 252 280 215 281 291 276 285 287 290 228 274 277 286 277 251 278 277 286 277 289 267 276 206 284 269 284 268 291 289 293 277 280 274 282 230 275 236 295 289 283 261 262 252 283 277 204 286 270 278 270 283 272 281 288 248 266 256 292 39

Frequency Table of Axial Loads Table 2 -2 Axial Load Frequency 200 - 209 9 210 - 219 3 220 - 229 5 230 - 239 4 240 - 249 4 250 - 259 14 260 - 269 32 270 - 279 52 280 - 289 38 290 - 299 14 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 40

Histogram of Axial Load Data 60 Frequency 50 40 30 20 10 Figure 2 -1 299. 5 289. 5 279. 5 269. 5 259. 5 249. 5 239. 5 229. 5 219. 5 209. 5 199. 5 0 Axial Load (pounds) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 41

Relative Frequency Histogram Table 2 -3 Relative Frequency 0. 080 260 - 269 0. 183 270 - 279 0. 297 280 - 289 0. 217 290 - 299 0. 080 0. 05 0 299. 5 250 - 259 0. 10 289. 5 0. 023 279. 5 240 - 249 0. 15 269. 5 0. 023 259. 5 230 - 239 249. 5 0. 029 239. 5 220 - 229 0. 20 229. 5 0. 017 219. 5 210 - 219 0. 25 209. 5 0. 051 199. 5 200 - 209 0. 30 Relative Frequency Axial Load (pounds) Figure 2 -2 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 42

Nature (Shape) of Distribution Bell Shaped or Normal 60 Relative Frequency 0. 30 40 30 20 10 0. 20 0. 15 0. 10 Axial Load (pounds) 299. 5 289. 5 279. 5 269. 5 249. 5 259. 5 239. 5 229. 5 219. 5 209. 5 199. 5 289. 5 279. 5 269. 5 249. 5 259. 5 239. 5 229. 5 219. 5 0 209. 5 0 0. 25 0. 05 199. 5 Frequency 50 Axial Load (pounds) Figure 2 -2 Figure 2 -1 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 43

Important Distributions Normal Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 44

Important Distributions Normal Uniform Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 45

Important Distributions Normal Uniform Skewed Right Copyright © 1998, Triola, Elementary Statistics Addison Wesley

Other Types of Pictures of Data v. Stem-and-Leaf Plot * v. Pareto Chart (Bar Chart) v. Pie Chart* v. Dotplot v. Scatter Diagram* vand more Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 48

Stem-and-Leaf Plot Stem 20 21 22 23 24 25 26 27 28 29 Leaves 014466889 578 03358 0046 1228 01122466677899 012222233333455567788888999 0000112222233333344445555666667777999 00011112222223333334444555666677899999 00011222334557 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 49

Pareto Chart 45, 000 40, 000 Frequency 35, 000 30, 000 25, 000 20, 000 15, 000 10, 000 5, 000 Firearms Fire Drowning Poison Falls Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. Ingestion of food or object Figure 2 -5 Motor Vehicle 0 50

Pie Chart Firearms (1400. 1. 9%) Ingestion of food or object (2900. 3. 9% Fire (4200. 5. 6%) Motor vehicle (43, 500. 57. 8%) Drowning (4600. 6. 1%) Poison (6400. 8. 5%) Falls (12, 200. 16. 2%) Figure 2 -6 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 51

Dot Plot. . . . . c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 220 240 260 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman. 280 LOAD 300 52