Honors Statistics Chapter 4 Part 3 Displaying and Summarizing Quantitative Data

Learning Goals Know how to display the distribution of a quantitative variable with a histogram, a stem-and-leaf display, or a dotplot. Know how to display the relative position of quantitative variable with a Cumulative Frequency Curve and analysis the Cumulative Frequency Curve. Be able to describe the distribution of a quantitative variable in terms of its shape. Be able to describe any anomalies or extraordinary features revealed by the display of a variable.

Learning Goals Be able to determine the shape of the distribution of a variable by knowing something about the data. Know the basic properties and how to compute the mean and median of a set of data. Understand the properties of a skewed distribution. Know the basic properties and how to compute the standard deviation and IQR of a set of data.

Learning Goals Understand which measures of center and spread are resistant and which are not. Be able to select a suitable measure of center and a suitable measure of spread for a variable based on information about its distribution. Be able to describe the distribution of a quantitative variable in terms of its shape, center, and spread.

Quantitative Data Dotplots

Learning Goal 1: Dotplots What is a dot plot? A dot plot is a plot that displays a dot for each value in a data set along a number line. If there are multiple occurrences of a specific value, then the dots will be stacked vertically.

Learning Goal 1: Dotplots A dotplot is a simple display. It just places a dot along an axis for each case in the data. The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot. You may see a dotplot displayed horizontally or vertically.

Learning Goal 1: Dotplots To construct a dot plot Draw a horizontal line. Label it with the name of the variable. Mark regular values of the variable (scale) on it. For each observation, place a dot above its value on the number line. Sodium in Cereals

Learning Goal 1: Dotplots - Example: The following data shows the length of 50 movies in minutes. Construct a dot plot for the data. 64, 64, 69, 70, 71, 71, 71, 72, 73, 73, 74, 74, 74, 74, 75, 75, 75, 75, 75, 75, 76, 76, 76, 77, 77, 78, 78, 79, 79, 80, 80, 81, 81, 81, 82, 82, 82, 83, 83, 83, 84, 86, 88, 89, 89, 90, 90, 92, 94, 120. Figure 2-5 Length of 50 Movies

Learning Goal 1: Dotplots – Frequency Table Data The following frequency distribution shows the number of defectives observed by a quality control officer over a 30 day period. Construct a dot plot for the data.

Learning Goal 1: Dotplots – Solution

Learning Goal 1: Dotplots – Your Turn One of Professor Weiss’s sons wanted to add a new DVD player to his home theater system. He used the Internet to shop and went to pricewatch.com. There he found 16 quotes on different brands and styles of DVD players. Construct a dotplot for these data. Change to page 61 Insert Table 2.14 alongside figure 2.4

Learning Goal 1: Think Before You Draw Remember the “Make a picture” rule? Now that we have options for data displays, you need to Think carefully about which type of display to make. Before making a stem-and-leaf display, a histogram, or a dotplot, check the Quantitative Data Condition: The data are values of a quantitative variable whose units are known.

Learning Goal 2 Know how to display the relative position of quantitative variable with a Cumulative Frequency Curve and analysis the Cumulative Frequency Curve.

Ogive - Cumulative Frequency Curve Quantitative Data Ogive - Cumulative Frequency Curve

Learning Goal 2: Cumulative Frequency and the Ogive Histogram displays the distribution of a quantitative variable. It tells little about the relative standing (percentile, quartile, etc.) of an individual observation. For this information, we use a Cumulative Frequency graph, called an Ogive (pronounced O-JIVE).

Learning Goal 2: Measures of Relative Standing How many measurements lie below the measurement of interest? This is measured by the pth percentile. x (100-p) % p % p-th percentile

Learning Goal 2: Percentile The pth percentile is a value such that p percent of the observations fall below or at that value.

Learning Goal 2: Special Percentiles – Deciles and Quartiles Deciles and quartiles are special percentiles. Deciles divide an ordered data set into 10 equal parts. Quartiles divide the ordered data set into 4 equal parts. We usually denote the deciles by D1, D2, D3, … , D9. We usually denote the quartiles by Q1, Q2, and Q3.

Learning Goal 2: Special Percentiles – Deciles and Quartiles There are 9 deciles and 3 quartiles. Q1 = first quartile = P25 Q2 = second quartile = P50 Q3 = third quartile = P75 D1 = first decile = P10 D2 = second decile = P20 . . . D9 = ninth decile = P90

Learning Goal 2: Percentile - Examples 90% of all men (16 and older) earn more than $319 per week. BUREAU OF LABOR STATISTICS 10% 90% $319 is the 10th percentile. $319 50th Percentile 25th Percentile 75th Percentile = Median = Lower Quartile (Q1) = Upper Quartile (Q3)

Learning Goal 2: Calculating Percentile The percentile corresponding to a given data value, say x, in a set is obtained by using the following formula.

Learning Goal 2: Calculating Percentile - Example Example: The shoe sizes, in whole numbers, for a sample of 12 male students in a statistics class were as follows: 13, 11, 10, 13, 11, 10, 8, 12, 9, 9, 8, and 9. What is the percentile rank for a shoe size of 12?

Learning Goal 2: Calculating Percentile - Solution Solution: First, we need to arrange the values from smallest to largest. The ordered array is given below: 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 13, 13. Observe that the number of values at or below the value of 12 is 10.

Learning Goal 2: Calculating Percentile - Solution Solution (continued): The total number of values in the data set is 12. Thus, using the formula, the corresponding percentile is: The value of 12 corresponds to approximately the 83rd percentile.

Learning Goal 2: Calculating Percentile - Example Example: The data given below represents the 19 countries with the largest numbers of total Olympic medals – excluding the United States, which had 101 medals – for the 1996 Atlanta games. Find the 65th percentile for the data set. 63, 65, 50, 37, 35, 41, 25, 23, 27, 21, 17, 17, 20, 19, 22, 15, 15, 15, 15.

Learning Goal 2: Calculating Percentile - Solution Solution: First, we need to arrange the data set in order. The ordered set is: . 15, 15, 15, 15, 17, 17, 19, 20, 21, 22, 23, 25, 27, 35, 37, 41, 50, 63, 65. Next, compute the position of the percentile. Here n = 19, k = 65. Thus, c = (19  65)/100 = 12.35. We need to round up to a value 13.

Learning Goal 2: Calculating Percentile - Solution Solution (continued): Thus, the 13th value in the ordered data set will correspond to the 65th percentile. That is P65 = 27.

Learning Goal 2: Cumulative Frequency What is a cumulative frequency for a class? The cumulative frequency for a specific class in a frequency table is the sum of the frequencies for all values at or below the given class.

Learning Goal 2: Cumulative Frequency Tables Cumulative frequencies for a class are the sums of all the frequencies up to and including that class. Example

Learning Goal 2: Cumulative Frequency Tables Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Cumulative Frequency Cumulative Percentage Class Frequency Percentage 10 - 20 3 15 3 15 20 - 30 6 30 9 45 30 - 40 5 25 14 70 40 - 50 4 20 18 90 50 - 60 2 10 20 100 Total 20 100

Learning Goal 2: Cumulative Frequency Curve - Ogive A line graph that depicts cumulative frequencies. Used to Find Quartiles and Percentiles.

Learning Goal 2: Constructing an Ogive Make a frequency table and add a cumulative frequency column. To fill in the cumulative frequency column, add the counts in the frequency column that fall in or below the current class interval. Label and scale the axes and title the graph. Horizontal axis “classes” and vertical axis “cumulative frequency or relative cumulative frequency”. Begin the ogive at zero on the vertical axis and lower boundary of the first class on the horizontal axis. Then graph each additional Upper class boundary vs. cumulative frequency for that class.

Learning Goal 2: Ogive - Example

Learning Goal 2: Cumulative Frequency Curve – Example The frequencies of the scores of 80 students in a test are given in the following table. Complete the corresponding cumulative frequency table. A suitable table is as follows:

Learning Goal 2: Cumulative Frequency Curve – Example The information provided by a cumulative frequency table can be displayed in graphical form by plotting the cumulative frequencies given in the table against the upper class boundaries, and joining these points with a smooth. Construct the Cumulative Frequency Curve. The cumulative frequency curve corresponding to the data is as follows:

Learning Goal 2: Cumulative Frequency Curve – Class Problem The results obtained by 200 students in a mathematics test are given in the following table. Draw a cumulative frequency curve and use it to estimate. The median mark. The number of students who scored less than 22 marks. The pass mark if 120 students passed the test. The min. mark required to obtain an A grade if 10% of the students received an A grade.