1.2. Pictorial and Tabular Methods in Descriptive Statistics

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1 1.2. Pictorial and Tabular Methods in Descriptive Statistics Section Objectives. 1. Stem-and-Leaf displays. 2. Dotplots. 3. Histogram. Types of histogram shapes. Common notation. Sample size n : the number of observations in a single sample. x1, x2,..., x n : individual observations on some variable x Stem - and - Leaf Display. Stem-and-Leaf Display is a visual presentation of grouped discrete data. Consider numerical data set x1, x2,..., x n, where each x i consist of at least two digits Constructing Stem - and Leaf Display. 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 each observation beside the corresponding stem value. 4. Indicate the units for stems and leaves. Example How long is a Minute? How well people can judge when a minute has elapsed? There are 26 responses to the Question

2 Create a STEM Sort our data values from smallest to largest (this will make creating a display easier). 33, 39, 42, 52, 52, 54, 55, 56, 57, 57, 59, 59, 60, 63, 64, 65, 66, 67, 68, 70, 72, 75, 79, 82, 86, 89 Stem- and -Leaf plot does show the individual observations Questions. 1) What is the range of this data? 2) How many estimates are between 33 and 89 seconds (inclusive)? 3) How many estimates are between 52 and 68 seconds (inclusive)? 4) How many estimates were more than five seconds away from one minute? 5) How many estimates were within 10 seconds of one minute? 6) What observations in the set have highest frequency of occurrence? 7) How can be computed the average of responses? Analysis of Stem - and - Leaf plot can reveal patterns in the responses that may not be apparent in the original list of numbers. 2

3 Constructing a stem-and-leaf chart in Minitab (Manual p. 45) Graph->Stem-and-Leaf->Select the Column, click OK Back-to-Back Stem - and - Leaf Display. For comparison of two sets of data can be used Back-to-Back Stem - and - Leaf Display. Example Starbucks prides itself on its low line-up times in order to be served. A new coffee house in town has also boasted that it will have your order in your hands and have you on your way quicker than the Starbucks. The following data was collected for the line-up times (in minutes) for both coffee houses: Starbucks: 20, 26, 26, 27, 19, 12, 12, 16, 12, 15, 17, 20, 8, 8, 18 Just Us Coffee: 17, 16, 15, 10, 16, 10, 10, 29, 20, 22, 22, 12, 13, 24, 15 Construct a two-sided stem-and-leaf plot for the data. Determine the median and mode using the two-sided stem-and-leaf plot. What can you conclude from the distributions? Starbucks Just Us Coffee Analysis of two-sided stem-and-leaf plot. 3

4 Dotplots. A dotplot is a summary of small numerical data set with a few distinct data values. The observations are placed on a horizontal axis and each observation is presented by a dot above the corresponding location. Example Dotplot Corresponding Frequency Distribution. Value Frequency Example The following Dot Plot represents the number of tornadoes in South Africa each year from 2001 to

5 Answer the following questions. 1) Which year had the lowest number of tornadoes? 2006 (none at all). 2) Which year had the highest number of tornadoes? 2002 (9 tornadoes). 3) Were there years with the same number of tornadoes? Yes (2005 and 2007). 4) How many tornadoes happened in year 2004? Definitions of Discrete and Continuous Variables. A numerical variable is discrete if its set of possible values either is finite or else can be listed in an infinite series. A numerical variable is continuous if its possible values consist of an entire interval on the number line Frequency and Relative Frequency of a Value of a Discrete Variable. Consider data consisting of observations on a discrete variable x. Definition(s). The frequency of any particular x value is the number of times that value occurs in the data set. The relative frequency of a value is the fraction or proportion of times the value occurs. Relative frequency of a value number of times the value occurs number of observations in the data set A frequency distribution is a tabulation of frequencies and/or relative frequencies. 5

6 Histogram. A histogram is a graphical method for displaying the shape of a distribution. It is particularly useful when there are a large number of observations Constructing a Histogram for a Discrete Data Set. Determine frequency and relative frequency for each x value. Mark possible x values on a horizontal axis. Above each value, draw a rectangle whose height is frequency (or relative frequency) of that value. Example An audit of twenty tax returns revealed 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, and 0 mistakes in arithmetic. 1. Construct a table showing the number of tax returns with 0, 1, 2, and 3, mistakes in arithmetic. Value Frequency 2. Draw a dot diagram displaying the same information. 3. Construct the histogram for the data. 6

7 Example ( Create a histogram for the scores of 642 students on a psychology test. The test consists of 197 items each graded as "correct" or "incorrect." The students' scores ranged from 46 to The first step is to create a frequency table. Because of large number of observations, we group the scores. By placing the limits of the class intervals midway between two numbers (e.g., 49.5), we prevent the scores being on the boundary between intervals. 2. Make rectangles with bases on horizontal axis and heights equal to intervals frequencies. Mention, that as the heights of intervals can be used relative frequencies. The area of each bar is proportional to frequency (relative frequency) of the corresponding interval. 7

8 Histogram for Continuous Data. Constructing a histogram for continuous data (measurements) entails subdividing the measurement axis into a suitable number of class intervals or classes, such that each observation is contained in exactly one class. Occasionally an observation can fall on the class boundary. To prevent such complications, we can use the following rule: If an observation is on the boundary of the class, then it has to be placed to the (class) interval on the right. This is how the MINITAB constructs the histograms. Constructing a Histogram for a Continuous Data Set: Equal Class Width Determine frequency and relative frequency for each class. Mark the class boundaries on a horizontal measurement axis. Above each class interval, draw a rectangle whose height is corresponding relative frequency (or frequency). Example ( How much is that puppy growing? Each month Jane measure how much weight her puppy has gained and got these results: 0.5, 0.5, 0.3, 0.2, 1.6, 0, 0.1, 0.1, 0.6, 0.4 The ordered data is 0.2, 0, 0.1, 0.1, 0.3, 0.4, 0.5, 0.5, 0.6, 1.6. The histogram with the groups width of 0.5 is shown in the right. Intervals Frequency

9 Histogram Shapes. 9

10 Qualitative Data. Qualitative data is sometimes called categorical data is information about qualities; information that can't be measured by numbers. This is data can be organized into mutually exclusive categories. For qualitative data can be made frequency table and visual presentations, bar chart and pie chart Examples of qualitative data. a) freshmen, sophomores, juniors, seniors, graduate students (with natural order). b) Catholic, Jewish, Protestant (arbitrary order). Example Frequency table and bar chart for a qualitative data. The colors of students' backpacks are recorded as follows: red, green, red, blue, black, blue, blue, blue, red, blue, blue, black Frequency table. Bar Chart 10

11 1.3. Measures of Location Definition of the Mean. For a given set of numbers x1, x2,..., x n, the mean is arithmetic average of the set. The mean is the measure of center. Common Notation for the mean. Population Mean Sample Mean x Formula for Sample Mean. The sample mean of observations x1, x2,..., xn is given by x x x x n 1 2 n i 1 n n x i Median - center of a distribution of data. One of the big benefits of the median, is that it is not as affected by outliers and something like the mean is. The median represents the middle value of the data set. The median of a data set is the number that, when the set is put into increasing order, divides the data into two equal parts. If a data set has an odd number of data points, then the median is the middle data value (when the data is in increasing order). If a data set has an even number of data points, then the median is the mean of the two middle data values (when the data is in increasing order). Example a) Find the median of the data set {3, 7, 8, 5, 12, 14, 21, 13, 18}. b) Find the median of the data set {3, 7, 8, 5, 12, 14, 21, 13, 14}. 11

12 First Quartile and Third Quartile (Lower fourth and Upper Fourth) Definitions: The lower half of a data set is the set of all values that are to the left of the median value when the data has been put into increasing order. The upper half of a data set is the set of all values that are to the right of the median value when the data has been put into increasing order. The first quartile, denoted by Q 1, is the median of the lower half of the data set. This means that about 25% of the numbers in the data set lie below Q 1 and about 75% lie above Q 1. The third quartile, denoted by Q 3, is the median of the upper half of the data set. This means that about 75% of the numbers in the data set lie below Q 3 and about 25% lie above Q 3. Example Find the first and third quartiles of the data set {3, 7, 8, 5, 12, 14, 21, 13, 18}. First, we write data in increasing order: 3, 5, 7, 8, 12, 13, 14, 18, 21. 3, 5, 7, 8, 12, 13, 14, 18, 21 The first quartile, Q 1, is the median of {3, 5, 7, 8}. Since there is an even number of values, we need the mean of the middle two values to find the first quartile: Q Similarly, the upper half of the data is: {13, 14, 18, 21}, so Q

13 Five-Number Summary Definitions: The minimum value of a data set is the least value in the set. The maximum value of a data set is the greatest value in the set. The range of a data set is the distance between the maximum and minimum values. Range = maximum minimum. The interquartile range (IQR) or fourth spread f s of a data set is the distance between the two quartiles Q3 and Q 1 Interquartile range = Q 3 Q 1. IQR measures the variability only among the middle 50% of the distribution. Example Find the range and interquartile range of the set {3, 7, 8, 5, 12, 14, 21, 13, 18}. First, we write the data in increasing order: 3, 5, 7, 8, 12, 13, 14, 18, 21. Range = max min = 21 3 = 18. Because Q 1 = 6 and Q 3 = 16. Interquartile range = Q 3 Q 1 = 16 6 = Illustration of Quartiles. 13