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.Data set 17,19,13,14,14,16,9,12,12,6,6,8,6,7,2,4,3,1ClassFrequency, f1 – 445 – 859 – 12313 – 1617 – 202Lower Class LimitsUpper Class LimitsFrequencies
6 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: range = 20 – 1 = 19.
7 Constructing a Frequency Distribution GuidelinesDecide on the number of classes to include. The number of classes should (not a rule) 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.
8 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.
9 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.IMPORTANT: Even if the computed class width comes out an integer, still add 1!Continued.
10 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, 50.The upper class limits are 25, 33, 41, 49, 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.
11 Constructing a Frequency Distribution Example continued:Number of studentsAgesAges of StudentsTallyFrequency, fClass18 – 251326 – 33834 – 41442 – 493Check that the sum equals n, the number in the sample.50 – 572
12 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
13 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.Do this for each class.(Lower class limit) + (Upper class limit)2Midpoint =Frequency, fClassMidpoint41 – 42.5Midpoint =
14 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
15 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
16 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
17 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
18 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.
19 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
20 Midpoint Midpoints for the “Ages of Students” frequency distribution. 50 – 5742 – 4934 – 4126 – 3323481318 – 25Frequency, fClassMidpointAges of Students21.529.537.545.553.5
21 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
22 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
23 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.
24 More Graphs and Displays § 2.2More Graphs and Displays
25 Stem-and-Leaf PlotIn a stem-and-leaf plot, each number is separated into a stem (usually the entry’s leftmost digits) and a leaf (usually the rightmost digit). This is an example of exploratory data analysis.Example:The following data represents the ages of 30 students in a statistics class. Display the data in a stem-and-leaf plot.Ages of Students1820212729193032342437382239443346544951Continued.
26 Stem-and-Leaf Plot Ages of Students Key: 1|8 = 18 1 8 8 8 9 9 9 2 3 45Most of the values lie between 20 and 39.4 6 91 4This graph allows us to see the shape of the data as well as the actual values.
27 Stem-and-Leaf Plot Example: Construct a stem-and-leaf plot that has two lines for each stem.Ages of StudentsKey: 1|8 = 18123457 9 9From this graph, we can conclude that more than 50% of the data lie between 20 and 34.7 8 946 91 4
28 Dot PlotIn a dot plot, each data entry is plotted, using a point, above a horizontal axis.Example:Use a dot plot to display the ages of the 30 students in the statistics class.Ages of Students182021192322242930323354Continued.
29 Dot Plot Ages of Students 151824454821513054394233362757From this graph, we can conclude that most of the values lie between 18 and 32.
30 Pie ChartA pie chart is a circle that is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category.Accidental Deaths in the USA in 2002TypeFrequencyMotor Vehicle43,500Falls12,200Poison6,400Drowning4,600Fire4,200Ingestion of Food/Object2,900Firearms1,400(Source: US Dept. of Transportation)Continued.
31 Pie ChartTo create a pie chart for the data, find the relative frequency (percent) of each category.TypeFrequencyRelative FrequencyMotor Vehicle43,5000.578Falls12,2000.162Poison6,4000.085Drowning4,6000.061Fire4,2000.056Ingestion of Food/Object2,9000.039Firearms1,4000.019n = 75,200Continued.
32 Pie ChartNext, find the central angle. To find the central angle, multiply the relative frequency by 360°.TypeFrequencyRelative FrequencyAngleMotor Vehicle43,5000.578208.2°Falls12,2000.16258.4°Poison6,4000.08530.6°Drowning4,6000.06122.0°Fire4,2000.05620.1°Ingestion of Food/Object2,9000.03913.9°Firearms1,4000.0196.7°Continued.
33 Pie Chart Ingestion 3.9% Firearms 1.9% Fire 5.6% Drowning 6.1% Poison 8.5%Motor vehicles57.8°Falls16.2°
34 Pareto ChartA Pareto chart is a vertical bar graph is which the height of each bar represents the frequency. The bars are placed in order of decreasing height, with the tallest bar to the left.Accidental Deaths in the USA in 2002TypeFrequencyMotor Vehicle43,500Falls12,200Poison6,400Drowning4,600Fire4,200Ingestion of Food/Object2,900Firearms1,400(Source: US Dept. of Transportation)Continued.
35 Ingestion of Food/Object Pareto ChartAccidental Deaths4500040000350003000025000200001500010000Poison5000Motor VehiclesFallsPoisonDrowningFireIngestion of Food/ObjectFirearms
36 Scatter PlotWhen each entry in one data set corresponds to an entry in another data set, the sets are called paired data sets.In a scatter plot, the ordered pairs are graphed as points in a coordinate plane. The scatter plot is used to show the relationship between two quantitative variables.The following scatter plot represents the relationship between the number of absences from a class during the semester and the final grade.Continued.
37 Scatter Plot Final grade (y) Absences (x) 825121596y78929058437481Finalgrade(y)246810121416405060708090Absences (x)100From the scatter plot, you can see that as the number of absences increases, the final grade tends to decrease.
38 Time Series ChartA data set that is composed of quantitative data entries taken at regular intervals over a period of time is a time series. A time series chart is used to graph a time series.Example:The following table lists the number of minutes Robert used on his cell phone for the last six months.MonthMinutesJanuary236February242March188April175May199June135Construct a time series chart for the number of minutes used.Continued.
39 Times Series Chart Robert’s Cell Phone Usage Month Minutes 200 150 100 250MinutesMonthJanFebMarAprMayJune