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Statistical Reasoning. 5. Statistical Tables and Graphs. Frequency Tables. A basic frequency table has two columns: The first column lists the categories of data.
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Statistical Reasoning 5 Statistical Tablesand Graphs
Frequency Tables A basic frequency table has two columns: • The first column lists the categories of data. • The second column lists the frequency of each category, which is the number of times each category appears in the data set.
Relative Frequency The relative frequency of any category is the fraction (or percentage) of the data values that fall in that category: • relative frequency = frequency in category total frequency The cumulative frequency of any category is the number of data values in that category and all preceding categories.
Data Types and Binning Qualitative data describe qualities or categories. Quantitative data represent counts or measurements. When dealing with quantitative data categories, it is often useful to group, or bin, the data into categories that cover a range of possible values.
Example Classify each of the following types of data as either qualitative or quantitative. a. Brand names of shoes in a consumer survey qualitative b. Heights of students quantitative
Example Classify each of the following types of data as either qualitative or quantitative. c. Audience ratings of a film on a scale of 1 to 5, where 5 means excellent Although the film rating categories involve numbers, the numbers represent subjective opinions about a film, not counts or measurements. These data are therefore qualitative, despite being stated as numbers.
Summarizing Raw Data Consider the following 20 scores from a 100-point exam: • 80 78 76 94 75 98 77 84 88 81 72 91 72 74 86 79 88 72 75 Determine appropriate bins and make a frequency table including columns for relative and cumulative frequency.
Bar and Pie Graphs A bar chart shows each category with a bar whose length corresponds to its frequency or relative frequency. Pie charts are used primarily for relative frequencies, because the total pie must always represent the total relative frequency of 100%. The size of each wedge is proportional to the relative frequency of the category it represents.
Bar and Pie Graphs The bar chart and pie chart below both show the data from table 5.1. Grade Data Grade Data
Important Labels for Graphs Title/caption: The graph should have a title or caption (or both) that explains what is being shown and, if applicable, lists the source of the data. Vertical scale and title: Numbers along the vertical axis should clearly indicate the scale. The numbers should line up with the tick marks. Include a label that describes the variable. Horizontal scale and title: The categories should be clearly indicated along the horizontal axis. (Tick marks may not be necessary for qualitative data, but should be included for quantitative data.) Include a label that describes the variable. Legend: If multiple data sets are displayed on a single graph, include a legend or key to identify the individual data sets.
Definitions • A histogram is a bar graph for quantitative data categories. The bars have a natural order and the bar widths have specific meaning. • A line chart shows the data value for each category as a dot, and the dots are connected with lines. For each dot, the horizontal position is the center of the bin it represents and the vertical position is the data value for the bin. • A time-series diagram is a histogram or line chart in which the horizontal axis represents time.
Histogram and Line Chart The histogram and line chart below both show the same data.
Example The table shows the ages (at the time when they won the award) of all Academy Award–winning actresses through 2013. Make a histogram and a line chart to display these data. Discuss the results.
Example (cont) The data are quantitative and organized in 10-year bins. The figure on the next slide shows the data as both a histogram and a line chart; the line chart overlays the histogram so you can see how the two diagrams compare. Note that the histogram bars touch one another because there are no gaps between the categories. The data show that most actresses win the award at a fairly young age, which stands in contrast to the older ages of most male winners of Best Actor (see Exercise 31). Many actresses believe this difference arises because Hollywood producers rarely make movies that feature older women in strong character roles.
Reading Time-Series Diagrams The figure shows a time-series graph of homicide rates in the United States. Briefly summarize what it shows.
Example (cont) The graph shows how the homicide rate per 100,000 people has changed since 1960. We see that the homicide rate rose dramatically—more than doubling— from a minimum around 1962 to a first peak around 1974. It then remained high, with some variations, through about 1993. After 1993, it fell dramatically to the year 2000, then stayed nearly constant until a slight drop from 2008 through 2012. The decrease in the homicide rate during the 1990s has been attributed to tougher enforcement of drug laws and a crackdown on gangs.
