Ch 4: Describing Data:Displaying and Exploring Data Goals • Develop and interpret a stem-and-leaf display • Develop and interpret a: • Dot plot • Develop and interpret quartiles, deciles, and percentiles • Develop and interpret a: • Box plots • Compute and understand the: • Coefficient of Variation • Coefficient of Skewness • Draw and interpret a scatter diagram • Set up and interpret a contingency table
Stem-and-leaf Displays Stem-and-leaf display: A statistical technique for displaying a set of data. Each numerical value is divided into two parts: the leading digits become the stem and the trailing digits the leaf. Note:Advantages of the stem-and-leaf display over a frequency distribution: We do not lose the identity of each observation We can see the distribution
Stock prices on twelve consecutive days for a major publicly traded company
Stem and leaf display of stock prices Trailing digit(s) along horizontal axis Compare to: Leading digit(s) along vertical axis
Stem-and-Leaf – Example Suppose the seven observations in the 90 up to 100 class are: 96, 94, 93, 94, 95, 96, and 97. The stem value is the leading digit or digits, in this case 9. The leaves are the trailing digits. The stem is placed to the left of a vertical line and the leaf values to the right. The values in the 90 up to 100 class would appear as Then, we sort the values within each stem from smallest to largest. Thus, the second row of the stem-and-leaf display would appear as follows:
Stem-and-leaf: Another Example Listed in Table 4–1 is the number of 30-second radio advertising spots purchased by each of the 45 members of the Greater Buffalo Automobile Dealers Association last year. Organize the data into a stem-and-leaf display. Around what values do the number of advertising spots tend to cluster? What is the fewest number of spots purchased by a dealer? The largest number purchased?
Dot Plots • A dot plot groups the data as little as possible and the identity of an individual observation is not lost. • To develop a dot plot, each observation is simply displayed as a dot along a horizontal number line indicating the possible values of the data. • If there are identical observations or the observations are too close to be shown individually, the dots are “piled” on top of each other.
Dot Plot • Dot plots: • Report the details of each observation • Are useful for comparing two or more data sets
Percentage of women participating In the labor force for the 50 states. Percentage of men participating In the labor force for the 50 states.
Quartiles, Deciles, Percentiles(Measures Of Dispersion) • Quartiles divide a set of data into four equal parts (three points) • Each interval contains 1/4 of the scores • Deciles divide a set of data into ten equal parts (nine points) • Each interval contains 1/10 of the scores • Percentiles divide a set of data into 100 equal parts (99 points) • Each interval contains 1/100 of the scores • GPA in the 43rd percentile means that 43% of the students have a GPA lower and 57% of the students have a GPA higher
Location Of A Percentile In An Ordered Array If there are an even number of observation, your Lp may be between two numbers In this case, you must estimate the number that you will report as the percentile If Lp = 4.25, and the distance between the two numbers is 37, Lower number + .25(37) = percentile
Compute And Interpret Quartiles • Quartiles • Quartiles divide a set of data into four equal parts • Value Q1, Value Q2(Median), Value Q3, are the three marking points that divide the data into four parts • 25% of the values occur below Value Q1 • 50% of the values occur below Value Q2 • 75% of the values occur below Value Q3 • Value Q1is the median of the lower half of the data • Value Q2is the median of all the data • Value Q3is the median of the upper half of the data
Example 2: Using twelve stock prices, we can find the median, 25th, and 75th percentiles as follows: Quartile 3 Median Quartile 1
75th percentile Price at 9.75 observation = 88 + .75(91-88) = 90.25 12 11 10 9 8 7 6 5 4 3 2 1 96 92 91 88 86 85 84 83 82 79 78 69 Q4 Q3 50th percentile: Median Price at 6.50 observation = 85 + .5(85-84) = 84.50 Q2 25th percentile Price at 3.25 observation = 79 + .25(82-79) = 79.75 Q1
A box plot is a graphical display, based on quartiles, that helps to picture a set of data. Five pieces of data are needed to construct a box plot: • Minimum Value • First Quartile • Median • Third Quartile • Maximum Value
Outliers • Outlier > Q3 + 1.5*(Q3 –Q1) • Outlier < 0
Coefficient Of Variation • Coefficient Of Variation converts the standard deviation to standard deviations per unit of mean • Use Coefficient Of Variation to compare: • Data in different units • Data in the same units, but the means are far apart • s = Standard Deviation • Xbar = Sample Mean
Skewness - Formulas for Computing The coefficient of skewness can range from -3 up to 3. • Measures the lack of symmetry in a distribution • A value near -3, such as -2.57, indicates considerable negative skewness. • A value such as 1.63 indicates moderate positive skewness. • A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that there is no skewness present. In Excel use the SKEW function
Skewness – An Example • Following are the earnings per share for a sample of 15 software companies for the year 2005. The earnings per share are arranged from smallest to largest. • Compute the mean, median, and standard deviation. Find the coefficient of skewness using Pearson’s estimate. What is your conclusion regarding the shape of the distribution?
Skewness – An Example Using Pearson’s Coefficient The skew is moderately positive. This means that a few large values are pulling the mean up, above the median and mode.
Describing Relationship between Two Variables • One graphical technique we use to show the relationship between variables is called a scatter diagram. • To draw a scatter diagram we need two variables. We scale one variable along the horizontal axis (X-axis) of a graph and the other variable along the vertical axis (Y-axis).
Describing Relationship between Two Variables – Scatter Diagram Examples
Scatter Diagram • Independent Variable (X) • The Independent Variable provides the basis for estimation • It is the predictor variable • Dependent Variable • The Dependent Variable is the variable being predicted or estimated • Scatter Diagrams • Visual portrayal of the relationship between two variables • A chart that portrays the relationship between the two variables • X axis (properly labeled – name and units) • Y axis (properly labeled– name and units) • Scatter diagram requires both variables to be at least interval scale
Describing Relationship between Two Variables – Scatter Diagram Excel Example In the Introduction to Chapter 2 we presented data from AutoUSA. In this case the information concerned the prices of 80 vehicles sold last month at the Whitner Autoplex lot in Raytown, Missouri. The data shown include the selling price of the vehicle as well as the age of the purchaser. Is there a relationship between the selling price of a vehicle and the age of the purchaser? Would it be reasonable to conclude that the more expensive vehicles are purchased by older buyers?
Describing Relationship between Two Variables – Scatter Diagram Excel Example
Contingency Tables • A scatter diagram requires that both of the variables be at least interval scale. • What if we wish to study the relationship between two variables when one or both are nominal or ordinal scale? In this case we tally the results in a contingency table.
A contingency table is used to classify observations according to two identifiable characteristics. Contingency tables are used when one or both variables are nominally or ordinally scaled. A contingency table is a cross tabulation that simultaneously summarizes two variables of interest.
Contingency Tables – Example 1 Weight Loss 45 adults, all 60 pounds overweight, are randomly assigned to three weight loss programs. Twenty weeks into the program, a researcher gathers data on weight loss and divides the loss into three categories: less than 20 pounds, 20 up to 40 pounds, 40 or more pounds. Here are the results.
Contingency Tables – Example 1 Compare the weight loss under the three plans.
Contingency Tables – Example 2 A manufacturer of preassembled windows produced 50 windows yesterday. This morning the quality assurance inspector reviewed each window for all quality aspects. Each was classified as acceptable or unacceptable and by the shift on which it was produced. Thus we reported two variables on a single item. The two variables are shift and quality. The results are reported in the following table.