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2.2 Graphical Displays of Distributions Recall from the last section that there are two types of data: _____________ (categorical) and ______________ (numerical). It is important that you can differentiate between these types of variables. Quantitative variables can be further classified as either _______________ or _________. If the numerical data can take on any value on a given interval, the variable is _______________. Examples are height, weight, inflation rate, and _________________. If the numerical data cannot take on every value within their range, the variable is ___________. Examples are the number of cylinders in a car and the sizes of a collection of wrenches (only values that are a multiple of 1 16 of an inch). On the AP Statistics exam, interpretation of graphical display is crucial, but occasionally you are asked to make a plot as well as interpret it. Always remember to include scales and labels on all graphs. These are the standard plots used to display a single variable: Qualitative Variables Bar chart Quantitative Variables Dot Plot Stemplot (stem-and-leaf plot) Histogram Cumulative frequency plot (section 2.4) Boxplot (box-and-whiskers plot) (section 2.3) Bar Charts When to use: categorical data How to draw: 1. Write the category names below the horizontal axis at regularly spaced intervals. 2. Label the scale on the vertical axis using either frequency or relative frequency. 3. Center each bar over the category labels. Draw each bar with a blank space between it. All bars should have the same width so that the height and the area of the bar are proportional to frequency and relative frequency. The order of the bars, technically, is irrelevant. Example: page 53, P13 Dot Plots When to use: small numerical data sets How to Construct: 1. Draw a horizontal axis and mark it with an appropriate measurement scale. 2. Locate each value in the data set along the scale, and represent it by a dot. If there are two or more observations with the same value, stack the dots vertically. Example: page 52, P7. Stemplots When to use: numerical data sets with a small to moderate number of observations. How to construct: 1. Select one or more leading digits for the stem values. The trailing digits become the leaves. 2. List possible stem values in a vertical column. 3. Record the leaf for every observation as they appear beside the corresponding stem value. 4. Redo the plot, ordering the leaves. 5. Indicate the units for stems and leaves someplace in the display. Example: page 52, P11. Histogram When to use: continuous numerical data. Even works well for large data sets. How to construct: 1. Draw a horizontal scale, and mark the boundaries of the class intervals on the scale. 2. Draw a vertical scale, and mark it with either frequency or relative frequency. 3. Draw a rectangle for each class directly above the corresponding interval (so the edges are at the class boundaries). 4. The height of each rectangle is the corresponding frequency or relative frequency. Example: page 52, P8. Comparison of Graphical Displays These plots all give some indication of the shape, center, and spread of the data. You should choose the one that best meets the needs of the situation. Plot Type Bar Chart Dot Plot Stemplot Histogram Cumulative Frequency Plot Boxplot Advantages Only way to display categorical data (besides a pie chart) Shows the shape of the distribution nicely Can be used for large or small data sets Preserves at least two digits of the actual numerical values of the data Can be used for small to moderate data sets Can be made quickly by hand Most useful for moderate to large data sets Shows the relative standing of an individual observation Provides a quick summary of location, spread, and possible skewness Disadvantages Does not preserve the exact numerical values in the data set Best made by a computer or calculator Does not give detailed information on shape