Presentation on theme: "Histograms, Frequency Polygons and Ogives"— Presentation transcript:
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 ValueDesired number of classesIncrease the value computed to the next highest whole,number even if the first value was a whole number. Thiswill ensure the classes cover the data.The lower class limit of a class is the lowest data that canfit into the class, the upper class limit is the highest datavalue that can fit into the class. The class width is thedifference 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). 261271236244279296284299288247256338360341333266287313311307303277283304305290289297332330309328285291295298306315310318320321323324327Note the minimum time is 236 hrs and the maximum time360 hours. We are asked to divide the data into 5 classes.Find class limits, class boundaries and frequencies, thenconstruct a frequency distribution and frequency histogram.
7 It is easier to make the histogram if the data is sorted: 236244247256261266271277279283284285287288289290291295296297298299303304305306307309310311313315318320321323324327328330332333338341360
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.LowerLimitUpperBoundaryMarkFrequency236260235.5260.52484261285285.52739286310310.529825311335335.532316336360360.53483
10 Relative FrequenciesThe 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 100102104108110112115116118120126128130132134136138140146148152156160190200208
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 14L. BndyU. BndyL. LimitU. LimitMarkFreq.R. Freq.C. Freq99.5113.5100113106.5100.20127.5114127120.5120.2422141.5128141134.5170.3439155.5142155148.550.1044169.5156169162.520.0446183.5170183176.50.00197.5184197190.510.0247211.5198211204.530.0650
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.
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 OgivesMake 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.
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)