Dot Plot: A graph for displaying a set of data. Each numerical value is represented by a dot placed above a horizontal number line.

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1 Introduction We continue our study of descriptive statistics with measures of dispersion, such as dot plots, stem and leaf displays, quartiles, percentiles, and box plots. Dot plots, a stem-and-leaf display, and box plots give additional insight into where the values are concentrated and dispersed and the general shape of the data. Finally we consider bivariate data where we observe two variables for each individual or observation selected. Dot Plots In Chapter 2 we grouped data in classes and constructed a histogram. When we organize the data into classes, we lose the exact value of the observations. Dot plots group data as little as possible, hence we do not lose the identity of the individual observations. Dot Plot: A graph for displaying a set of data. Each numerical value is represented by a dot placed above a horizontal number line. To develop a dot plot we display a dot for each observation along a horizontal number line indicating the value of each piece of data. For multiple observations we pile the dots on top of each other. The steps to follow in developing a dot plot graph are: 1. Sort the data from smallest to largest. 2. Draw and label a number line. 3. Place a dot for each observation. Length of Service (in years) As an example, the lengths of service, in years, of a sample of eighteen employees are given. Step 1:Sort the data from smallest to largest Step 2:Draw the number line and label it as shown. Step 3:Place a dot for each observation.

2 We note that the data range is from 2 to 10 years and that the data clusters around 6 years. Stem-and-Leaf Displays A stem-and-leaf display is a combination of sorting and graphing. Stem-and-Leaf Display: A statistical technique for displaying a set of data. Each numerical value is divided into two parts: The leading digit(s) become the stem, and the trailing digits the leaf. The stems are located along the main vertical axis, and the leaf for each observation along the horizontal axis. To develop a stem-and-leaf chart the first step is to locate the largest value and the smallest value. This will provide the range of the stem values. The stem is the leading digit or digits of the number, and the leaf is the trailing digit. For example, the number 15 has a stem value of 1 and a leaf value of 5. For another problem the number 231 has a stem value of 23 and a leaf value of 1. Shown at the right are the amounts spent (in dollars) in the grocery store by a sample of 12 people. $12 $28 $32 $24 $17 $6 $34 $18 $22 $42 $36 $26 The range of values is from $6 to $42. The first digit of each number is the stem and the second digit is the leaf. The first customer (upper left) spent $12. Hence, the stem value is 1 and the leaf value is 2. After each trailing digit is arranged from low to high, the completed display is shown at the right. Leading Digit Trailing Digit Other Measures of Dispersion The standard deviation is the most widely used measure of dispersion. However there are several others, which include Quartiles, Deciles, and Percentiles. Quartiles Recall that the median divides data that has been placed in order from smallest to largest, such that half the values are below the median and half are above the median. If we divide the lower and upper set of values into two equal parts, we have quartiles. Quartiles divide a set of data into four equal parts. First Quartile The point below which one-fourth or 25% of the ranked data values lie. (It is designated Q 1 ) Third Quartile The point below which three-fourths or 75% of the ranked data values lie. (It is designated Q 3 ) Logically the median is the Second Quartile (designated Q 2 ). The values corresponding to Q 1, Q 2 and Q 3

3 divide a set of data into four equal parts. Deciles and Percentiles Just as quartiles divide a distribution into 4 equal parts, deciles divide a distribution into ten equal parts; and percentiles divide a distribution into 100 equal parts. For example: If you were told that your Scholastic Aptitude Test score was in the 9 th decile, you could assume that 90 percent of those taking the test had a lower score than yours and that 10 percent had a higher score. A grade point average in the 55 th percentile means that 55 percent of students have a lower GPA than yours and that 45 percent have a higher GPA. The procedure for finding the quartile, decile, and a percentile for ungrouped data is to order the data from smallest to largest. Then use text formula [4-1]. Location of a Percentile Where: L p refers to the location of the desired percentile. n is the number of observations. P is the desired percentile Note that this is a generic formula for percentiles, deciles and quartiles. For example, if you had a set of data with 49 observations in ordered array and wanted to locate the 78 th percentile, then let P = 78 and n = 49 so observation.. Thus you would locate the 39 th If you wanted to locate the 6 th decile, then let P = 60 and locate the 30 th observation. Note that the 6 th decile equals the 60 percentile.. Thus you would Box Plots A box plot is a graphical display that helps us picture how a set of data is distributed relative to the quartiles. Box plot: A graphical display based on five statistics: the minimum value, Q 1 (the first quartile), Q 2 the median, Q 3 (the third quartile) and the maximum value. To construct a box plot we need five pieces of information. We need the minimum value, Q 1 (the first quartile), Q 2 the median, Q 3 (the third quartile) and the maximum value. The difference between Q 3 and Q 1 is called the interquartile range. The middle 50% of the data is contained within the interquartile range. The details for constructing and interpreting a box plot are found in Problem 4 of the Chapter Problems.

4 Skewness Another characteristic of a set of data is the shape of the distribution. There are four shapes commonly observed: symmetric, positively skewed, negatively skewed, and bimodal. The measures of location and the measures of dispersion are both descriptive characteristics of a set of data. A third characteristic of a distribution is its skewness. As noted before, a symmetric distribution has the same shape on either side of the median and it has no skewness. For a positively skewed distribution the long tail is to the right, the mean is larger than the median or the mode, and the mode appears at the highest point on the curve. For a negatively skewed distribution the mode is the largest value and is at the highest point of the curve, while the mean is the smallest. A bimodal distribution will have two or more peaks. The coefficient of skewness is used to describe how a distribution is skewed. Coefficient of skewness: A measure to describe the degree of skewness. Text Formula [4 2] is for Pearson's Coefficient of Skewness. Where: sk is the coefficient of skewness. is the mean. s is the standard deviation. Pearson's Coefficient of Skewness Characteristics of the coefficient of skewness are: 1. The coefficient of skewness, designated sk, measures the amount of skewness and may range from 3.0 to A value near 3, such as 2.57, indicates considerable negative skewness. 3. A value such as 1.63 indicates moderate positive skewness. 4. 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. This information is summarized in the chart.

5 Describing the Relationship Between Two Variables To summarize the distribution of a set of data, in Chapter 2 we used a histogram and in the first part of this chapter we used dot plots and stem-and-leaf displays. We studied a single variable sometimes called univariate data. When we study the relationship between two variables we refer to the data as bivariate. Bivariate data: A collection of paired data values. For example a realtor might want to study the relationship between the selling price of a home and the number of days the home is on the market. One technique used to study the relationship between variables is a scatter diagram. Scatter diagram: A graph in which paired data values are plotted on an X,Y Axis. The steps to follow in developing a scatter diagram are: 1. We need two variables. 2. We scale one variable (x) along the horizontal axis (X Axis) of a graph and the corresponding variable (y) along the vertical axis (Y Axis). 3. Place a dot for each (x, y) pair of observations. Usually one variable depends on another.

6 The scatter diagram shows the relationship between airfare and the total flight distance for a random sample of 20 airfares offered by a popular internet discount travel broker. It appears that as the flight distance increases, the cost of the airfare increases. Contingency Table When we study the relationship between two or more variables when one or both are nominal or ordinal scale, we tally the results into a two-way table. This two-way table is referred to as a contingency table. Contingency table: A table used to classify sample observations according to two or more identifiable characteristics. A contingency table is a cross tabulation that simultaneously summarizes two variables of interest and their relationship. A survey of 60 school children classified each as to gender and the number of times lunch was purchased at school during a four-week period. Each respondent is classified according to two criteria the number of times lunch was purchased and gender. Gender Bought Lunch Boys Girls Total 0 up to up to Total