1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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

8. The class width is computed as (360-236)/5 which is 24. 8
The class width is computed as ( )/5 which is 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

9. Histogram for Iditarod Data

10. 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.

11. 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

12. Systolic blood pressures of 50 subjects Class Width = (208-100)/8 = 13
Systolic blood pressures of 50 subjects Class Width = ( )/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

13. Frequency Histogram for Blood Pressure Data

14. Relative Frequency Histogram for Blood Pressure Data

15. 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.

16. Frequency Polygon for Blood Pressure Data

17. 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.

18. 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.

19. Ogive for Blood Pressure Data

20. 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.

21. 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)