Presentation on theme: "Frequency Distributions"— Presentation transcript:
1 Frequency Distributions Section 2.1Frequency Distributionsand Their GraphsLarson/Farber 4th ed.
2 Section 2.1 Objectives Construct frequency distributions Construct frequency histograms, frequency polygons, relative frequency histograms, and ogivesLarson/Farber 4th ed.
3 Frequency Distribution A table that shows classes or intervals of data with a count of the number of entries in each class.The frequency, f, of a class is the number of data entries in the class.ClassFrequency, f1 – 556 – 10811 – 15616 – 2021 – 2526 – 304Class width 6 – 1 = 5Lower classlimitsUpper classlimitsLarson/Farber 4th ed.
4 Constructing a Frequency Distribution Decide on the number of classes.Usually between 5 and 20; otherwise, it may be difficult to detect any patterns.Find the class width.Determine the range of the data.Divide the range by the number of classes.Round up to the next convenient number.Larson/Farber 4th ed.
5 Constructing a Frequency Distribution Find the class limits.You can use the minimum data entry as the lower limit of the first class.Find the remaining lower limits (add the class width to the lower limit of the preceding class).Find the upper limit of the first class. Remember that classes cannot overlap.Find the remaining upper class limits.Larson/Farber 4th ed.
6 Constructing a Frequency Distribution 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.Larson/Farber 4th ed.
7 Example: Constructing a Frequency Distribution The following sample data set lists the amount spent on books for a semester.Construct a frequency distribution that has seven classes.91472279249530376188341266199142273189130489248101375486190398269433012735484Larson/Farber 4th ed.
8 Solution: Constructing a Frequency Distribution 91472279249530376188341266199142273189130489248101375486190398269433012735484Number of classes = 7 (given)Find the class widthRound up to 72
9 Solution: Constructing a Frequency Distribution Use 30 (minimum value) as first lower limit. Add the class width of 72 to get the lower limit of the next class.= 102Find the remaining lower limits.Lower limitUpper limit30102174246318390462Class width = 72
10 Solution: Constructing a Frequency Distribution The upper limit of the first class is 101 (one less than the lower limit of the second class).Add the class width of 72 to get the upper limit of the next class.= 173Find the remaining upper limits.Lower limitUpper limit30101102173174245246317318389390461462533Class width = 72Larson/Farber 4th ed.
11 Solution: Constructing a Frequency Distribution 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.ClassTallyFrequency, flllll5lll3lllll ll7llll4l1Σf = 29Larson/Farber 4th ed.
12 Determining the Midpoint Midpoint of a classClassMidpointFrequency, f65.55137.53209.5281.57353.54425.51497.5Class width = 72
13 Determining the Relative Frequency Relative Frequency of a classPortion or percentage of the data that falls in a particular class.Larson/Farber 4th ed.
14 Determining the Relative Frequency continued ClassMidpointFrequency, fRelativeFrequency65.550.172137.530.103209.5281.570.241353.540.138425.510.034497.5Σf =291.000Sum of Rel. Freq.
15 Determining the Cumulative Frequency Cumulative frequency of a classThe sum of the frequency for that class and all previous classes.ClassMidpointFrequency, fCumulativeFrequency65.55137.538209.513281.5720353.5424425.5125497.529Σf =Sum of class and previous class.
16 Expanded Frequency Distribution ClassMidpointFrequency, fRelativeFrequencyCumulative65.550.172137.530.1038209.513281.570.24120353.540.13824425.510.03425497.529Σ =
17 Graphs of Frequency Distributions Frequency HistogramA bar graph that represents the frequency distribution.The horizontal scale is quantitative and measures the data values.The vertical scale measures the frequencies of the classes.Consecutive bars must touch.data valuesfrequencyLarson/Farber 4th ed.
18 Class Boundaries Class boundaries The numbers that separate classes without forming gaps between them.The distance from the upper limit of the first class to the lower limit of the second class is 102 – 101 = 1.Half this distance is 0.5.ClassBoundaries29.5 – 101.5First class lower boundary = 30 – 0.5 = 29.5First class upper boundary = = 101.5Larson/Farber 4th ed.
19 Class Boundaries Class Class boundaries Frequency, f 30 - 101 53741Larson/Farber 4th ed.
20 Example: Frequency Histogram Construct a frequency histogram for the book costs frequency distribution.ClassClass boundariesMidpointFrequency, f65.55137.53209.5281.57353.54425.51497.5Larson/Farber 4th ed.21
21 Solution: Frequency Histogram (using Midpoints)
22 Graphs of Frequency Distributions Frequency PolygonA line graph that emphasizes the continuous change in frequencies.data valuesfrequencyLarson/Farber 4th ed.
23 Example: Frequency Polygon Construct a frequency polygon for the Books costs frequency distribution.ClassMidpointFrequency, f65.55137.53209.5281.57353.54425.51497.5Larson/Farber 4th ed.
24 Solution: Frequency Polygon The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.This is bull! In most cases it is better to just show the data and not have false markers!
25 Graphs of Frequency Distributions Relative Frequency HistogramHas the same shape and the same horizontal scale as the corresponding frequency histogram.The vertical scale measures the relative frequencies, not frequencies.data valuesrelative frequencyLarson/Farber 4th ed.
26 Graphs of Frequency Distributions Cumulative Frequency Graph or OgiveA line graph that displays the cumulative frequency of each class at its upper class boundary.The upper boundaries are marked on the horizontal axis.The cumulative frequencies are marked on the vertical axis.data valuescumulative frequencyLarson/Farber 4th ed.
27 Constructing an OgiveConstruct a frequency distribution that includes cumulative frequencies as one of the columns.Specify the horizontal and vertical scales.The horizontal scale consists of the upper class boundaries or upper limit.The vertical scale measures cumulative frequencies.Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.Larson/Farber 4th ed.
28 Constructing an Ogive Connect the points in order from left to right. The graph should start at the lower boundary of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size).Larson/Farber 4th ed.
29 Example: OgiveConstruct an ogive for the book cost frequency distribution.ClassMidpointFrequency, fCumulativeFrequency65.55137.538209.513281.5720353.5424425.5125497.529
30 Solution: OgiveFrom the ogive, you can see that about 25 students spent $461 or less. The greatest increase in in cost occurs between $245 and $389.Larson/Farber 4th ed.
31 Section 2.1 Summary Constructed frequency distributions Constructed frequency histograms, frequency polygons, relative frequency histograms and ogivesLarson/Farber 4th ed.