Histograms, Frequency Polygons and Ogives These are constructions that will allow us to visually represent data, and in the first two cases to see the “shape” of a set of data.

Histograms Histograms are bar graphs in which The bars have the same width and always touch (the edges of the bars are on class boundaries which are described below). The width of a bar represents a quantitative variable x, such as age rather than a category. The height of each bar indicates frequency.

Before making a histogram, organize the data into a frequency table which shows the distribution of data into classes (intervals). The classes are constructed so that each data values falls into exactly one class, and the class frequency is the number of data in the class.

To find the class width, First compute: Largest value - smallest Value Desired number of classes Increase the value computed to the next highest whole, number even if the first value was a whole number. This will ensure the classes cover the data. The lower class limit of a class is the lowest data that can fit into the class, the upper class limit is the highest data value that can fit into the class. The class width is the difference between lower class limits of adjacent classes.

Class Boundaries Class boundaries cannot belong to any class. Class boundaries between adjacent classes are the midpoint between the upper limit of the first class, and the lower limit of the higher class. Differences between upper and lower boundaries of a given class is the class width. The midpoint of a class (class mark) is the average of its upper and lower boundaries, which is also the average of its upper and lower limits.

How long does the 1161 mile Iditarod take? (p. 67, problem 1). 261 271 236 244 279 296 284 299 288 247 256 338 360 341 333 266 287 313 311 307 303 277 283 304 305 290 289 297 332 330 309 328 285 291 295 298 306 315 310 318 320 321 323 324 327 Note the minimum time is 236 hrs and the maximum time 360 hours. We are asked to divide the data into 5 classes. Find class limits, class boundaries and frequencies, then construct a frequency distribution and frequency histogram.

It is easier to make the histogram if the data is sorted: 236 244 247 256 261 266 271 277 279 283 284 285 287 288 289 290 291 295 296 297 298 299 303 304 305 306 307 309 310 311 313 315 318 320 321 323 324 327 328 330 332 333 338 341 360

The class width is computed as (360-236)/5 which is 24. 8 The class width is computed as (360-236)/5 which is 24.8. Hence the class width is 25. Lower Limit Upper Boundary Mark Frequency 236 260 235.5 260.5 248 4 261 285 285.5 273 9 286 310 310.5 298 25 311 335 335.5 323 16 336 360 360.5 348 3

Histogram for Iditarod Data

Relative Frequencies The relative frequency of a class is f/n where f is the frequency of the class, and n is the total of all frequencies. Relative frequency tables are like frequency tables except the relative frequency is given. Relative frequency histograms are like frequency histograms except the height of the bars represent relative frequencies.

Systolic blood pressures of 50 subjects Make a histogram with 8 classes 100 102 104 108 110 112 115 116 118 120 126 128 130 132 134 136 138 140 146 148 152 156 160 190 200 208

Systolic blood pressures of 50 subjects Class Width = (208-100)/8 = 13 Systolic blood pressures of 50 subjects Class Width = (208-100)/8 = 13.5, thus use 14 L. Bndy U. Bndy L. Limit U. Limit Mark Freq. R. Freq. C. Freq 99.5 113.5 100 113 106.5 10 0.20 127.5 114 127 120.5 12 0.24 22 141.5 128 141 134.5 17 0.34 39 155.5 142 155 148.5 5 0.10 44 169.5 156 169 162.5 2 0.04 46 183.5 170 183 176.5 0.00 197.5 184 197 190.5 1 0.02 47 211.5 198 211 204.5 3 0.06 50

Frequency Histogram for Blood Pressure Data

Relative Frequency Histogram for Blood Pressure Data

Constructing Frequency Polygons Make a frequency table that includes class midpoints and frequencies. For each class place dots above class midpoint at the height of the class frequency. Put dots on horizontal axis one class width to left of first class midpoint, and one class width to right of of last midpoint. Connect dots with straight lines.

Frequency Polygon for Blood Pressure Data

Cumulative Frequencies & Ogives The cumulative frequency of a class is the frequency of the class plus the frequencies for all previous classes. An ogive is a cumulative frequency polygon.

Constructing Ogives Make a frequency table showing class boundaries and cumulative frequencies. For each class, put a dot over the upper class boundary at the height of the cumulative class frequency. Place dot on horizontal axis at the lower class boundary of the first class. Connect the dots.

Ogive for Blood Pressure Data

2.2#11(a) What number, and percentage, of winning times are under 2:07.15? (b) Estimate number, and percentage, of winning times between 2:05.15 and 2:11.15.

Distribution Shapes Symmetrical Uniform (it has a rectangular histogram) Skewed left – the longer tail is on the left side. Skewed right – the longer tail is on the right side. Bimodal (the two classes with the largest frequencies are separated by at least one class)