1. 1 ES Chapter 1 & 2 ~ Descriptive Analysis & Presentation of Single-Variable Data Mean: 60.07 inches Median: 62.50 inches Range: 42 inches Variance: 117.681 Standard deviation: 10.85 inches Minimum: 36 inches Maximum: 78 inches First quartile: 51.63 inches Third quartile: 67.38 inches Count: 58 bears Sum: 3438.1 inches 0 10 20 304050607080 Frequency Length in Inches Black Bears

2. 2 ES Chapter 1 & 2 : Goals Learn how to present and describe sets of data Learn measures of central tendency, measures of dispersion (spread), measures of position, and types of distributions Learn how to interpret findings so that we know what the data is telling us about the sampled population

3. 3 ES Graphic Presentation of Data Use initial exploratory data-analysis techniques to produce a pictorial representation of the data Resulting displays reveal patterns of behavior of the variable being studied The method used is determined by the type of data and the idea to be presented No single correct answer when constructing a graphic display

4. 4 ES Circle Graphs & Bar Graphs Circle Graphs and Bar Graphs: Graphs that are used to summarize attribute data Circle graphs (pie diagrams) show the amount of data that belongs to each category as a proportional part of a circle Bar graphs show the amount of data that belongs to each category as proportionally sized rectangular areas

5. 5 ES Example Example:The table below lists the number of automobiles sold last week by day for a local dealership. Describe the data using a circle graph and a bar graph: DayNumber Sold Monday15 Tuesday23 Wednesday35 Thursday11 Friday12 Saturday42

6. 6 ES Circle Graph Solution Automobiles Sold Last Week

7. 7 ES Bar Graph Solution Automobiles Sold Last Week Frequency

8. 8 ES Key Definitions Quantitative Data: One reason for constructing a graph of quantitative data is to examine the distribution - is the data compact, spread out, skewed, symmetric, etc. Distribution: The pattern of variability displayed by the data of a variable. The distribution displays the frequency of each value of the variable. Dotplot Display: Displays the data of a sample by representing each piece of data with a dot positioned along a scale. This scale can be either horizontal or vertical. The frequency of the values is represented along the other scale.

9. 9 ES Example: A random sample of the lifetime (in years) of 49 home washing machines is given below: Note:Notice how the data is “bunched” near the lower extreme and more “spread out” near the higher extreme Example 2.58.912.24.118.11.612.2 16.92.53.50.42.62.24.0 4.56.42.93.34.49.24.1 0.914.54.00.97.25.21.8 1.50.73.74.26.915.321.8 17.87.36.83.37.04.018.3 8.51.47.44.70.710.43.6 The figure below is a dot plot for the 49 lifetimes:

10. 10 ES Stem-and-Leaf Display: Pictures the data of a sample using the actual digits that make up the data values. Each numerical data is divided into two parts: The leading digit(s) becomes the stem, and the trailing digit(s) becomes the leaf. The stems are located along the main axis, and a leaf for each piece of data is located so as to display the distribution of the data. Stem & Leaf Display Background: –The stem-and-leaf display has become very popular for summarizing numerical data –It is a combination of graphing and sorting –The actual data is part of the graph –Well-suited for computers

11. 11 ES Example Example:A city police officer, using radar, checked the speed of cars as they were traveling down the main street in town. Construct a stem-and-leaf plot for this data: 41 31 33 35 36 37 39 49 33 19 26 27 24 32 40 39 16 55 38 36 Solution: All the speeds are in the 10s, 20s, 30s, 40s, and 50s. Use the first digit of each speed as the stem and the second digit as the leaf. Draw a vertical line and list the stems, in order to the left of the line. Place each leaf on its stem: place the trailing digit on the right side of the vertical line opposite its corresponding leading digit.

12. 12 ES 20 Speeds --------------------------------------- 1 | 6 9 2 | 4 6 7 3 | 1 2 3 3 5 6 6 7 8 9 9 4 | 0 1 9 5 | 5 ---------------------------------------- Example The speeds are centered around the 30s Note:The display could be constructed so that only five possible values (instead of ten) could fall in each stem. What would the stems look like? Would there be a difference in appearance?

13. 13 ES Remember! 1.It is fairly typical of many variables to display a distribution that is concentrated (mounded) about a central value and then in some manner be dispersed in both directions. (Why?) 2.A display that indicates two “mounds” may really be two overlapping distributions 3.A back-to-back stem-and-leaf display makes it possible to compare two distributions graphically 4.A side-by-side dotplot is also useful for comparing two distributions

14. 14 ES Frequency Distributions & Histograms Stem-and-leaf plots often present adequate summaries, but they can get very big, very fast Need other techniques for summarizing data Frequency distributions and histograms are used to summarize large data sets

15. 15 ES Frequency Distributions Frequency Distribution: A listing, often expressed in chart form, that pairs each value of a variable with its frequency 1.A table that summarizes data by classes, or class intervals 2.In a typical grouped frequency distribution, there are usually 5-12 classes of equal width 3.The table may contain columns for class number, class interval, tally (if constructing by hand), frequency, relative frequency, cumulative relative frequency, and class midpoint 4.In an ungrouped frequency distribution each class consists of a single value Ungrouped Frequency Distribution: Each value of x in the distribution stands alone Grouped Frequency Distribution: Group the values into a set of classes

16. 16 ES Frequency Distribution Guidelines for constructing a frequency distribution: 1.All classes should be of the same width 2.Classes should be set up so that they do not overlap and so that each piece of data belongs to exactly one class 3.For problems in the text, 5-12 classes are most desirable. The square root of n is a reasonable guideline for the number of classes if n is less than 150. 4.Use a system that takes advantage of a number pattern, to guarantee accuracy 5.If possible, an even class width is often advantageous

17. 17 ES Frequency Distributions Procedure for constructing a frequency distribution: 1.Identify the high (H) and low (L) scores. Find the range. Range = H - L 2.Select a number of classes and a class width so that the product is a bit larger than the range 3.Pick a starting point a little smaller than L. Count from L by the width to obtain the class boundaries. Observations that fall on class boundaries are placed into the class interval to the right.

18. 18 ES Example 6.5 5.0 5.6 7.6 4.8 8.0 7.5 7.9 8.0 9.2 6.4 6.0 5.6 6.0 5.7 9.2 8.1 8.0 6.5 6.6 5.0 8.0 6.5 6.1 6.4 6.6 7.2 5.9 4.0 5.7 7.9 6.0 5.6 6.0 6.2 7.7 6.7 7.7 8.2 9.0 Example:The hemoglobin test, a blood test given to diabetics during their periodic checkups, indicates the level of control of blood sugar during the past two to three months. The data in the table below was obtained for 40 different diabetics at a university clinic that treats diabetic patients: 1)Construct a grouped frequency distribution using the classes 3.7 - <4.7, 4.7 - <5.7, 5.7 - <6.7, etc. 2)Which class has the highest frequency?

19. 19 ES Class Frequency Relative Cumulative Class Boundaries f Frequency Rel. Frequency Midpoint, x --------------------------------------------------------------------------------------- 3.7 - <4.710.0250.025 4.2 4.7 - <5.760.1500.1755.2 5.7 - <6.7160.4000.5756.2 6.7 - <7.740.1000.6757.2 7.7 - <8.7100.2500.9258.2 8.7 - <9.730.0751.0009.2 Solutions 2)The class 5.7 - <6.7 has the highest frequency. The frequency is 16 and the relative frequency is 0.40 1)

20. 20 ES Histogram Histogram: A bar graph representing a frequency distribution of a quantitative variable. A histogram is made up of the following components: 1.A title, which identifies the population of interest 2.A vertical scale, which identifies the frequencies in the various classes 3.A horizontal scale, which identifies the variable x. Values for the class boundaries or class midpoints may be labeled along the x-axis. Use whichever method of labeling the axis best presents the variable. Notes: The relative frequency is sometimes used on the vertical scale It is possible to create a histogram based on class midpoints

21. 21 ES Example:Construct a histogram for the blood test results given in the previous example Example 9.28.27.26.25.24.2 15 10 5 0 Frequency Blood Test Solution: The Hemoglobin Test

22. 22 ES Age Frequency Class Midpoint ------------------------------------------------------------ 20 up to 303425 30 up to 405835 40 up to 507645 50 up to 6018755 60 up to 7025465 70 up to 8024175 80 up to 9014785 Example:A recent survey of Roman Catholic nuns summarized their ages in the table below. Construct a histogram for this age data: Example

23. 23 ES Solution 85756555453525 200 100 0 Frequency Age Roman Catholic Nuns

24. 24 ES Terms Used to Describe Histograms Symmetrical: Both sides of the distribution are identical mirror images. There is a line of symmetry. Uniform (Rectangular): Every value appears with equal frequency Skewed: One tail is stretched out longer than the other. The direction of skewness is on the side of the longer tail. (Positively skewed vs. negatively skewed) J-Shaped: There is no tail on the side of the class with the highest frequency Bimodal: The two largest classes are separated by one or more classes. Often implies two populations are sampled. Normal: A symmetrical distribution is mounded about the mean and becomes sparse at the extremes

25. 25 ES  The mode is the value that occurs with greatest frequency (discussed in Section 2.3) Important Reminders  The modal class is the class with the greatest frequency  A bimodal distribution has two high-frequency classes separated by classes with lower frequencies  Graphical representations of data should include a descriptive, meaningful title and proper identification of the vertical and horizontal scales

26. 26 ES Cumulative Frequency Distribution Cumulative Frequency Distribution: A frequency distribution that pairs cumulative frequencies with values of the variable The cumulative frequency for any given class is the sum of the frequency for that class and the frequencies of all classes of smaller values The cumulative relative frequency for any given class is the sum of the relative frequency for that class and the relative frequencies of all classes of smaller values

27. 27 ES Class Relative Cumulative Cumulative BoundariesFrequency Frequency FrequencyRel. Frequency ------------------------------------------------------------------------------------- 0 up to 440.0840.08 4 up to 880.16120.24 8 up to 1280.16200.40 12 up to 16200.40400.80 16 up to 2060.12460.92 20 up to 2430.06490.98 24 up to 2810.02501.00 Example:A computer science aptitude test was given to 50 students. The table below summarizes the data: Example