10 Wyner Statistics Fall 2013

Transcription

1 1 Wyner Statistics Fall 213 CHAPTER TWO: GRAPHS Summary Terms Objectives For research to be valuable, it must be shared. The fundamental aspect of a good graph is that it makes the results clear at a glance. Graphs often fail to reach this standard due to poor labeling, being more complicated than necessary, or being an inappropriate type for the data. Types of graphs discussed in this chapter are circle graphs, bar charts, time plots, and scatter plots. A special type of bar graph, a histogram, is used to show numerical distributions. 2-A Types of Graphs Tuesday 9/24 time plot scatter plot bar graph circle graph ➊ Choose an appropriate type of graph for a data set. ➋ Make a circle graph. ➌ Make a time plot, scatter plot, or bar chart. 2-B Histograms Friday 9/27 distribution class frequency histogram relative frequency histogram uniform bimodal skewed ➊ Make a frequency histogram. ➋ Make a relative frequency histogram. ➌ Describe the shape of a distribution graphed in a histogram. Review Friday 9/27 Test Friday 1/4

2 11 Wyner Statistics Fall A Types of Graphs Different types of graphs are used for different types of data. A TIME Plot is used when the independent variable is time on a timeline. A SCATTER Plot is used when the data are x-y coordinates and the independent variable is not time on a timeline. A BAR Graph (bar chart) is used to compare averages, totals, or frequencies of two or more categories. When comparing frequencies of two or more nominal categories that must total 1%, a CIRCLE Graph (pie chart) may be used instead. A HISTOGRAM (see 2-B) is a bar graph in which the x-axis has equal-width numerical ranges that connect (e.g., -1, 1-2, 2-3) and the y-axis shows the frequency or proportion of data in each range. ➊ Choose an appropriate type of graph for a data set. 1. If the independent variable is categorical, such as in an experiment or quasi-experiment, use a bar graph. 2. To compare nominal categories with proportions that must total 1%, use a bar graph or a circle graph. 3. If the data can be sorted into distinct numerical ranges and there is no independent variable, use a histogram. 4. To view change over time, such as in science class or for a stock chart, use a time plot. In a time plot, the independent variable is time or date and the x-axis is a timeline. 5. If each participant or data point has an x value and a y value that are numerical, use a scatter plot. ➊ Choose and justify an appropriate type of graph to display results for each of the following surveys. a) What are kids favorite seasons? 2. Use a bar graph or a circle graph because the categories (winter, spring, summer, fall) are nominal and the proportions must total 1%. b) What movies did high school students watch this summer? 1. Use a bar graph because the independent variable (movie) is categorical. Do not use a circle graph because some people watched more than one move so the total will not necessarily be 1%. c) How much do adults pay for food each month? 3. Use a histogram because the data can be sorted into numerical ranges such as $1 to $2, $2 to $3, etc., and there is no independent variable (everyone is an adult). d) How much has Facebook membership increased over the past five years? 4. Use a time plot because it will show the trend in membership over time. e) How much can football players bench based on their weight? 5. Use a scatterplot because each football player has his own numerical x value (weight) and y value (amount benched).

3 12 Wyner Statistics Fall 213 The most important aspect of a graph is that it makes the results clear at a glance. This can be achieved by using a type appropriate for the data, precisely labeling it, and keeping it neat and not unnecessarily complicated. ➋ Make a circle graph. 1. Multiply each proportion by Use a protractor to measure out each segment. 3. Title the graph and label the slices. ➋ Show in a circle graph that out of 7 kids, 31 preferred chocolate, 26 preferred strawberry, and 13 preferred vanilla. chocolate: 31 / 7 36 = 159 strawberry: 26 / 7 36 = 134 vanilla: 13 / 7 36 = 67 ➌ Make a time plot, scatter plot, or bar graph Label and scale the independent variable on the x-axis. For a bar graph, each level of the independent variable should be labeled. 2. Label and scale the dependent variable on the y-axis. The scale should represent an appropriate range for the variable, starting at zero unless there is a specific reason to do otherwise, and should be in natural, equal increments. 3. For a bar chart, draw the bars. For a time plot or scatter plot, plot the points. For a time plot, connect the points. 4. Title the graph. ➌ Make up realistic data and create each of the following: a) a time plot showing Alec s height through age 15 b) a scatter plot showing how long 25 kids of different ages could hold their breath c) a bar chart comparing the average height of boys versus girls at age 15 Alec s Height Growing Up 1 How Long Kids Held Their Breath 2 Kids Favorite Ice Cream Flavors Vanilla Chocolate Strawberry Average Heights of 15-Year-Olds Height (cm) 12 8 Seconds 6 4 Height (cm) Boys Girls Age Age Gender

4 13 Wyner Statistics Fall B Histograms A Frequency DISTRIBUTION shows how a data set is distributed, that is, how many data fall into each of several ranges called CLASSES. A histogram is a graph of a frequency distribution. The x-axis is a numerical scale, and each bar represents one class. ➊ Make a frequency histogram. 1. Decide on classes. They must all be the same size with no space between them. 2. Draw the x-axis and label the class boundaries. 3. Count how many data points fall into each class. If a data point falls on a boundary, include it in the higher class. 4. Draw and scale the y-axis, starting at zero. 5. Draw a bar for each class representing how many data are in the class. 6. Label the axes and title the graph. ➊ Make a frequency histogram for the following chapter three homework scores:, 11, 12, 16, 25, 25, 29, 3, 31, 34, 35, 35, 38, 38, 38, 39, 41, 41, 41, 41, 42, 42, 42, 43, 43, 43, 44, 45, 45, 45. A RELATIVE Frequency Histogram is a histogram in which the y-axis represents proportion of the total rather than raw frequencies. Other than labeling, it is exactly the same as a frequency histogram. ➋ Make a relative frequency histogram. 1. Do steps 1-3 for a frequency histogram (see ➊). 4. Count the data to get the sample size n. 5. Find the proportion of data in each class by dividing each class frequency by n. 6. Draw the y-axis and scale it using percentages or decimals, starting with and ending with a value equal to or slightly higher than the highest proportion in step Draw a bar for each class representing the proportions in step Label the axes and title the graph. ➋ Make a relative frequency histogram for the homework data in ➊. 4. n = / 3 3% 3 / 3 = 1% 9 / 3 = 3% 14 / 3 47% Points Scored In some histograms, the categories are labeled instead of the boundaries. This will not apply until Chapter 5, which deals with discrete variables. Number of Students Percentage of Students % 4% 3% 2% 1% Chapter Three Homework Scores Chapter Three Homework Scores % Points Scored

5 14 Wyner Statistics Fall 213 A distribution can be various shapes. Some general shapes are of particular importance. A UNIFORM Distribution has the same frequency of data in every class. A NORMAL Distribution curves down from its peak at the same rate in each direction. A BIMODAL Distribution has two classes that are each higher than those around them. A SKEWED Distribution has one or more data values on only one side that are far away from the mean. The side (right or left) with these extreme data values is the direction of the skew. ➌ Describe the shape of a distribution graphed in a histogram. 1. If all the bars are equal height, it is a uniform distribution. 2. If there are two separate peaks, it is a bimodal distribution. 3. If it is somewhat symmetrical and mound shaped, it is approximately a normal distribution. 4. If there is a longer tail on one side, it is skewed in that direction. ➌ Describe the following distributions. a) b) c) d) e) uniform bimodal skewed left skewed right approximately normal