Chpt 2. Frequency Distributions and Graphs. 2-3 Histograms, Frequency Polygons, Ogives / 35
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1 Chpt 2 Frequency Distributions and Graphs 2-3 Histograms, Frequency Polygons, Ogives 1
2 Chpt 2 Homework 2-3 Read pages p57 Applying the Concepts p58 2-4, 10, 14 2
3 Chpt 2 Objective Represent Data Graphically 3
4 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
5 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
6 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
7 Objec&ve: Represent data graphically 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
8 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.
9 Histogram If we record the number of hours spent watching TV Hours of TV hours tally Frequency (f) Frequency Hours 9
10 Objec&ve: Represent data graphically 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 = = 34 Width = Range/7 = 34/7 = 5 (round up when remainder). 10
11 Objec&ve: Represent data graphically Class Limits Because it makes sense, we choose the lowest data value, 100, for the first lower class limit The subsequent lower class limits are found by adding 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
12 Objec&ve: Represent data graphically 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
13 Objec&ve: Represent data graphically STEP 2: Tally the data. STEP 3: Find the frequencies. Class Limits Class Boundaries Frequency Cumulative Frequency
14 Histogram Record High Temperatures Class Limits Class Boundaries f Frequency Temperature F Note the limits/boundaries 14
15 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
16 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 hours tally Frequency (f)
17 Frequency Polygon Record High Temperatures Objec&ve: Represent data graphically Class Limits Class Boundaries f Frequency 12 9 class midpoints 6 (class midpoint, class frequency 3 Connect the dots Temperature F Slide 27 17
18 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
19 Ogive For the Ogive, the frequencies are cumulative and each point is increasing (or constant) in height. Hours of TV hours tally Frequency (f)
20 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 Class Boundaries < < < < < < < Cumulative Frequency
21 Cumulative Frequency Class Boundaries Cumulative Frequency < < < < < < < Objec&ve: Represent data graphically Record High Temperatures Cumulative frequencies class upper boundaries Connect the dots Temperature F Slide 28 21
22 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
23 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
24 Relative Frequency Class f RelaEve f CumulaEve rf /12 = 1/6 = /12 = 1/4 = /12 = /12 = 1/6 = For a Relative Frequency Ogive we need the last column. 24
25 Relative Frequency Class Limits Class Boundaries f cf relative frequency cumulative rf /50 = /50 = /50 = /50 = /50 = /50 = /50 =
26 Relative Frequency Histogram Record High Temperatures relative frequency cumulative rf Relative Frequency Frequency /50 =.04 8/50 =.16 18/50 =.36 13/50 =.26 7/50 =.14 1/50 =.02 1/50 = Temperature F 26
27 Objec&ve: Represent data graphically Relative Frequency Polygon Record High Temperatures Relative Frequency.36 class midpoints (class midpoint, class relative frequency) Temperature F Slide 17 27
28 Relative Frequency Ogive Record High Temperatures Cumulative Relative Frequency Temperature F Cumulative Relative frequencies class upper boundaries (class upper, class cumulative relative frequency) Slide 21 28
29 Shapes For histograms we can describe the shape of the distribution. 29
30 Shapes Unimodal, Symmetric Distribution The Infamous Bell Curve 30
31 Shapes Uniform Distribution
32 Shapes Positively (Right) Skewed
33 Shapes Negatively (Left) Skewed
34 Shapes Bimodal
35 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
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