2 Frequency Distributions and Their Graphs § 2.1Frequency Distributions and Their Graphs
3 LEARNING OBJECTIVESConstruct a frequency distribution that includes classes, frequencies, midpoints, relative frequencies, and cumulative frequencies.Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives.
4 Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number in each class. The frequency f of a class is the number of data points in the class.ClassFrequency, f1 – 445 – 859 – 12313 – 1617 – 202Lower Class LimitsUpper Class LimitsFrequencies
5 Frequency Distributions The class width is the distance between lower (or upper) limits of consecutive classes.ClassFrequency, f1 – 445 – 859 – 12313 – 1617 – 2025 – 1 = 49 – 5 = 413 – 9 = 417 – 13 = 4The class width is 4.The range is the difference between the maximum and minimum data entries.
6 Constructing a Frequency Distribution GuidelinesDecide on the number of classes to include. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns.Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number.Find the class limits. You can use the minimum entry as the lower limit of the first class. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper class limits.Make a tally mark for each data entry in the row of the appropriate class.Count the tally marks to find the total frequency f for each class.
7 Constructing a Frequency Distribution Example:The following data represents the ages of 30 students in a statistics class. Construct a frequency distribution that has five classes.Ages of Students1820212729193032342437382239443346544951Continued.
8 Constructing a Frequency Distribution Example continued:1. The number of classes (5) is stated in the problem.2. The minimum data entry is 18 and maximum entry is 54, so the range is 36. Divide the range by the number of classes to find the class width.365Class width == 7.2Round up to 8.Continued.
9 Constructing a Frequency Distribution Example continued:3. The minimum data entry of 18 may be used for the lower limit of the first class. To find the lower class limits of the remaining classes, add the width (8) to each lower limit.The lower class limits are 18, 26, 34, 42, and 50.The upper class limits are 25, 33, 41, 49, and 57.4. Make a tally mark for each data entry in the appropriate class.5. The number of tally marks for a class is the frequency for that class.Continued.
10 Constructing a Frequency Distribution Example continued:Number of studentsAgesAges of StudentsTallyFrequency, fClass18 – 251326 – 33834 – 41442 – 493Check that the sum equals the number in the sample.50 – 572
11 MidpointThe midpoint of a class is the sum of the lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark.(Lower class limit) + (Upper class limit)2Midpoint =Frequency, fClassMidpoint41 – 42.5Midpoint =
12 MidpointExample:Find the midpoints for the “Ages of Students” frequency distribution.50 – 5742 – 4934 – 4126 – 3323481318 – 25Frequency, fClassMidpointAges of Students= 4321.543 2 = 21.529.537.545.553.5
13 Relative FrequencyThe relative frequency of a class is the portion or percentage of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n.Class frequencySample sizeRelative frequency =ClassFrequency, fRelative Frequency1 – 440.222Relative frequency
14 Relative Frequency Example: Find the relative frequencies for the “Ages of Students” frequency distribution.50 – 5723481342 – 4934 – 4126 – 3318 – 25Frequency, fClassRelative FrequencyPortion of students0.4330.2670.1330.10.067
15 Cumulative FrequencyThe cumulative frequency of a class is the sum of the frequency for that class and all the previous classes.50 – 5723481342 – 4934 – 4126 – 3318 – 25Frequency, fClassCumulative FrequencyAges of Students13+21+25+28Total number of students+30
16 Frequency HistogramA frequency histogram is a bar graph that represents the frequency distribution of a data set.The horizontal scale is quantitative and measures the data values.The vertical scale measures the frequencies of the classes.Consecutive bars must touch.Class boundaries are the numbers that separate the classes without forming gaps between them.The horizontal scale of a histogram can be marked with either the class boundaries or the midpoints.
17 Class Boundaries Example: Find the class boundaries for the “Ages of Students” frequency distribution.Class Boundaries50 – 5723481342 – 4934 – 4126 – 3318 – 25Frequency, fClassAges of StudentsThe distance from the upper limit of the first class to the lower limit of the second class is 1.17.5 25.525.5 33.533.5 41.541.5 49.5Half this distance is 0.5.49.5 57.5
18 Frequency Histogram f Example: Draw a frequency histogram for the “Ages of Students” frequency distribution. Use the class boundaries.Ages of Students108642Age (in years)f121417.525.533.541.549.557.5138432Broken axis
19 Frequency PolygonA frequency polygon is a line graph that emphasizes the continuous change in frequencies.Broken axisAges of Students108642Age (in years)f121413.521.529.537.545.553.561.5Line is extended to the x-axis.Midpoints
20 Relative Frequency Histogram A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram.0.40.30.20.10.5Ages of StudentsAge (in years)Relative frequency(portion of students)17.525.533.541.549.557.50.4330.2670.1330.10.067
21 Cumulative Frequency Graph A cumulative frequency graph or ogive, is a line graph that displays the cumulative frequency of each class at its upper class boundary.17.5Age (in years)Ages of Students241812630Cumulative frequency(portion of students)25.533.541.549.557.5The graph ends at the upper boundary of the last class.