Chpt 2 Frequency Distributions and Graphs 2-3 Histograms, Frequency Polygons, Ogives 1
Chpt 2 Homework 2-3 Read pages 48-57 p57 Applying the Concepts p58 2-4, 10, 14 2
Chpt 2 Objective Represent Data Graphically 3
Graphical Representations Now that you have organized the raw data into frequency distributions, what should you do with it? You draw a picture! All statistics should start with a picture of the data distribution. The most basic graphical representations of the data are: A dotplot (which we have seen) A histogram A Frequency Polygon (Line Graph). An Ogive (Cumulative Frequency Polygon, Cumulative Frequency Line graph) 4
Graphical Representations The following data represent the record high temperatures for each of the 50 states. We will use this data to demonstrate the various graphs. 112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 109 109 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 5
Dot plot A dot plot is a display of data using (surprise!) dots. A dot plot shows each item of data above a number line, or horizontal axis. Dot plots make it easy to see gaps and clusters in a data set, as well as how the data spreads along the axis. A dot plot is typically used for continuous, quantitative data, but may also be used for categorical data. The horizontal axis lists the values of the variable. Dots representing each instance of that value are stacked above the value. 6
112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 109 109 Objec&ve: Represent data graphically 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 The data represent the record high temperatures for each of the 50 states. Construct a dot plot of the data. The data range from a low of 100 to a high of 134. Each dot represents one of the values in the data set. 100 105 110 115 120 125 130 135 7
Histogram l The histogram is a graph that displays the data by using bins (bars) of various heights to represent the frequencies. Consists of bins (bars) that are contiguous (no gaps). Gaps between bars = bar graph. Horizontal-axis: the lower boundaries or lower limits align with the sides of the bins, the midpoints of data classes align with center of bar. Vertical-axis: The height of the bar represents the frequency of the class.
Histogram If we record the number of hours spent watching TV Hours of TV hours tally Frequency (f) 0 9 1 11 2 4 3 0 4 2 Frequency 12 9 6 3 0 0 1 2 3 4 Hours 9
112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 109 109 Objec&ve: Represent data graphically 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 The data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes. STEP 1 Determine the classes. Find the class width by dividing the range by the number of classes 7. Range = High Low = 134 100 = 34 Width = Range/7 = 34/7 = 5 (round up when remainder). 10
112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 109 109 Objec&ve: Represent data graphically 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 Class Limits Because it makes sense, we choose the lowest 100-104 data value, 100, for the first lower class limit. 105-109 The subsequent lower class limits are found by adding 110 115 120 125 130-114 - 119-124 - 129-134 the width to the previous lower class limits. The first upper class limit is one less than the next lower class limit. The subsequent upper class limits are found by adding the width to the previous upper limit. 11
112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 109 109 Objec&ve: Represent data graphically 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 The class boundary is midway between an upper class limit and a subsequent lower class limit. 104, 104.5, 105 Class Limits Class Boundaries Frequency Cumulative Frequency 100-104 105-109 110-114 115-119 120-124 125-129 130-134 99.5-104.5 104.5-109.5 109.5-114.5 114.5-119.5 119.5-124.5 124.5-129.5 129.5-134.5 12
112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 109 109 Objec&ve: Represent data graphically 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114 STEP 2: Tally the data. STEP 3: Find the frequencies. Class Limits Class Boundaries Frequency Cumulative Frequency 100-104 105-109 110-114 115-119 120-124 125-129 130-134 99.5-104.5 104.5-109.5 109.5-114.5 114.5-119.5 119.5-124.5 124.5-129.5 129.5-134.5 2 8 18 13 7 1 1 13
Histogram Record High Temperatures Class Limits Class Boundaries f 18 100-104 99.5-104.5 2 15 105-109 110-114 104.5-109.5 109.5-114.5 8 18 Frequency 12 9 115-119 120-124 125-129 130-134 114.5-119.5 119.5-124.5 124.5-129.5 129.5-134.5 13 7 1 1 6 3 0 100 105 110 115 120 125 130 135 99.5 104.5 109.5 114.5 119.5 124.5 129.5 134.5 Temperature F Note the limits/boundaries 14
Frequency Polygon l A frequency polygon displays the data by using lines that connect points plotted for frequencies at the midpoint of classes. The frequencies represent the heights of the midpoints. Horizontal axis: class midpoints Vertical axis: frequency for the class Each point (class midpoint, class frequency) Connect the dots A line graph suggests linear progression such as time. 15
For this example the frequencies are the same and each point is at the same height as the bars of a corresponding histogram. Hours of TV 12 9 6 3 0 0 1 2 3 4 hours tally Frequency (f) 0 9 1 11 2 4 3 0 4 2 16
Frequency Polygon 18 15 Record High Temperatures Objec&ve: Represent data graphically Class Limits Class Boundaries f 100-104 99.5-104.5 2 105-109 104.5-109.5 8 110-114 109.5-114.5 18 115-119 114.5-119.5 13 120-124 119.5-124.5 7 125-129 124.5-129.5 1 130-134 129.5-134.5 1 Frequency 12 9 class midpoints 6 (class midpoint, class frequency 3 Connect the dots 0 99.5 102 104.5 107 109.5 112 114.5 117 119.5 122 124.5 127 129.5 132 134.5 Temperature F Slide 27 17
Ogive l A cumulative frequency polygon or ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution. The upper class boundaries are represented on the horizontal axis. A line graph with points (class upper boundary, cumulative frequency) Horizontal axis = upper boundaries of classes Vertical axis = cumulative frequency
Ogive For the Ogive, the frequencies are cumulative and each point is increasing (or constant) in height. Hours of TV 26 20 13 7 0 0 1 2 3 4 hours tally Frequency (f) 0 9 1 11 2 4 3 0 4 2 19
Ogive Construct an ogive to represent the data for the record high temperatures for each of the 50 states. Ogives use upper class boundaries and cumulative frequency. Class Limits Class Boundaries f cf 100-104 105-109 110-114 115-119 120-124 125-129 130-134 99.5-104.5 104.5-109.5 109.5-114.5 114.5-119.5 119.5-124.5 124.5-129.5 129.5-134.5 2 8 18 13 7 1 1 2 10 28 41 48 49 50 Class Boundaries < 104.5 < 109.5 < 114.5 < 119.5 < 124.5 < 129.5 < 134.5 Cumulative Frequency 2 10 28 41 48 49 50 20
Cumulative Frequency 50 45 40 35 30 25 20 15 10 5 0 Class Boundaries Cumulative Frequency < 104.5 < 109.5 < 114.5 < 119.5 < 124.5 < 129.5 < 134.5 2 10 28 41 48 49 50 Objec&ve: Represent data graphically Record High Temperatures Cumulative frequencies class upper boundaries Connect the dots 99.5 104.5 109.5 114.5 119.5 124.5 129.5 134.5 Temperature F Slide 28 21
Graphical Representation Constructing Statistical Graphs 1: Draw and label the horizontal and vertical axes. 2: Choose a suitable scale for the frequencies or cumulative frequencies, and label it on the vertical axis. 3: Represent class boundaries for the histogram or ogive, or midpoints for the frequency polygon, on the horizontal axis. 4: Plot the points and then draw the bars or lines. 22
Relative Frequency If proportions are used instead of frequencies, the graphs are called relative frequency graphs. In place of absolute frequency of occurrence for our data, we can add a column of relative frequency. Relative frequency is a ratio of actual frequency to total occurrences. frequency of score total number of scores Relative frequency graphs are used when the proportion of data values that fall into a given class is more important than the actual number of data values that fall into that class. 23
Relative Frequency Class f RelaEve f CumulaEve rf 1-5 2 2/12 = 1/6 =.17.17 6-10 3 3/12 = 1/4 =.25.42 11-15 5 5/12 =.42.84 16-20 2 2/12 = 1/6 =.16 1.00 For a Relative Frequency Ogive we need the last column. 24
Relative Frequency Class Limits Class Boundaries f cf relative frequency cumulative rf 100-104 99.5-104.5 2 2 2/50 =.04.04 105-109 104.5-109.5 8 10 8/50 =.16.20 110-114 109.5-114.5 18 28 18/50 =.36.56 115-119 114.5-119.5 13 41 13/50 =.26.82 120-124 119.5-124.5 7 48 7/50 =.14.96 125-129 124.5-129.5 1 49 1/50 =.02.98 130-134 129.5-134.5 1 50 1/50 =.02 1.00 25
Relative Frequency Histogram Record High Temperatures relative frequency cumulative rf Relative Frequency Frequency.36 18.30 15.24 12.18 9.12 6.06 3 2/50 =.04 8/50 =.16 18/50 =.36 13/50 =.26 7/50 =.14 1/50 =.02 1/50 =.02.04.20.56.82.96.98 1.00 0 99.5 104.5 109.5 114.5 119.5 124.5 129.5 134.5 Temperature F 26
Objec&ve: Represent data graphically Relative Frequency Polygon Record High Temperatures Relative Frequency.36 class midpoints (class midpoint, class relative frequency).30.24.18.12.06 0 102 107 112 117 122 Temperature F 127 132 Slide 17 27
Relative Frequency Ogive Record High Temperatures Cumulative Relative Frequency 1.0.80.60.40.20 0 99.5 104.5 109.5 114.5 119.5 124.5 129.5 134.5 Temperature F Cumulative Relative frequencies class upper boundaries (class upper, class cumulative relative frequency) Slide 21 28
Shapes For histograms we can describe the shape of the distribution. 29
Shapes 8 6 4 2 Unimodal, Symmetric Distribution 0 1 2 3 4 5 6 7 The Infamous Bell Curve 30
Shapes 6 5 3 2 Uniform Distribution 0 1 2 3 4 5 6 7 31
Shapes 10 8 5 3 Positively (Right) Skewed 0 1 2 3 4 5 6 7 8 9 10 32
Shapes 10 8 5 3 Negatively (Left) Skewed 0 1 2 3 4 5 6 7 8 9 10 33
Shapes Bimodal 10 8 5 3 0 1 2 3 4 5 6 7 8 9 10 34
Wing Span Measure wing span from tip of fingers on right hand to tip of fingers on left hand with arms spread out to the side. From the class data create a dot plot, histogram, frequency polygon, ogive, cumulative frequency polygon, and cumulative frequency ogive. 35