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# Displaying quantitative data

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### Displaying quantitative data

1. 1. Displaying Quantitative Data
2. 2. How do we examine the distribution of a quantitative variable? Dotplot  Stemplot  Histogram  Ogive
3. 3. To interpret graphs of quantitative data: • Look for an overall pattern and for notable departures from the pattern.
4. 4. Keep these features in mind:  Shape – mode?, skewed left/right, symmetric  Outlier - unusual features  Center – middle value  Spread - smallest to largest values
5. 5. Shape • Is the distribution approximately bell shape? • Skewed left? • Skewed right? • Is there a mode?
6. 6. Outlier • Are there any values that fall outside the overall pattern?
7. 7. Center • Find a value that divides the observations so that about half take larger values and about half take smaller values. • Median
8. 8. Spread • Give the smallest and largest values. • Range
9. 9. Dotplot Things to remember:  You only need a properly labeled horizontal axis  You need to be neat  Title the graph  Each dot represents a count of 1  Works well with a small data set
10. 10. Random Values: 3,7,4,6,0,7,2,0,7,5,8,2,4,0,3,1,9,9,3,4,2,3,7,9,2,4,6,7, 7,1,3,9,2,9,4,1,6,7,9,9,2,5,4,9,2,3,5,6,8,7
11. 11. What is the shape, center, and spread of the dot plot? Check For Understanding - 1
12. 12. Stemplot Things to remember:  Separate each piece of data into a stem (all but the rightmost digit) and a leaf (the final digit) Write the stems vertically in increasing order from top to bottom.  Be very neat and make sure you leave the same amount of space in between leaves.  Title your graph  Include a key identifying what the stem and leaves represent.  Works well with a small data set
13. 13. Random Values: 35,40,43,49,51,54,57,58,58,64,68,68,75,77 Stemplot of Random Values stem leaves 3 5 4 0 3 9 5 1 4 7 8 8 6 4 8 8 7 5 7 Key: 3 5 represent 35│
14. 14.  Splitting the stem makes it easier to see the distribution. Stemplot of random numbers 2 12 2 577 3 44 3 4 01 4 6788 5 233344 5 55567788899 key:2 1= 21
15. 15. The data below give the amount of caffeine content (in milligrams) for an 8-ounce serving of popular soft drinks. 20 15 23 29 23 15 23 31 28 35 37 27 24 26 47 28 24 28 28 16 38 36 35 37 27 33 37 25 47 27 29 26 43 43 28 35 31 25 (a) Construct a stemplot. (b) Construct a split stemplot. Check For Understanding - 2
16. 16. Histogram Things to remember:  It is the most common graph of a quantitative variable.  The x-axis is continuous, so there should be no gaps between the bars (unless a class has zero observations).  Title your graph.
17. 17. More about histograms  Be sure to choose classes that are all the same width.  Use your judgement in choosing classes to display the shape.
18. 18. How to make a histogram: STEP 1: Divide the data into "classes“ of equal size. STEP 2: Find the frequencies. Count the number of individuals in each class. STEP 3: Draw your histogram. Classes go on the horizontal and frequencies go on the vertical.
19. 19. Histogram of Random Values Frequency Table Class Frequency 1 - 5 4 6 - 10 30 11 - 15 26 16 - 20 50 21 - 25 12
20. 20. Interpreting a Histogram Describe the Histogram: • Shape: roughly symmetric and unimodal • Outlier: no outliers • Center: midpoint around 110 • Spread: about 80 to 150
21. 21. Check For Understanding - 3 The following data represents scores of 50 students on a calculus test. 72 72 93 70 59 78 74 65 73 80 57 67 72 57 83 76 74 56 68 67 74 76 79 72 61 72 73 76 67 49 71 53 67 65 99 83 69 61 72 68 65 51 75 68 75 66 77 61 64 74 (a) Construct a frequency histogram for this data set. (b) Describe the shape, center, and spread of the distribution of test scores.
22. 22. Cumulative Frequency Plot (Ogive) The Ogive is plotted by graphing a dot at each class end point for the cumulative frequency value and connecting the dots.
23. 23. Check For Understanding - 4 The following data represents the percentage of people aged 65 or older in each state. (a) Construct a relative cumulative frequency graph (ogive) for these data. (b) Use your ogive to answer the following questions: • In what percentage of states was the percentage of “65 or older” less than 15%? • What is the 40th percentile of this distribution, and what does it tell us? • What percentile is associated with Missouri?
24. 24. Time Plots  Plots each observation against the time at which it was measured. Always mark the time scale on the horizontal axis and the variable being measured on the y axis.  Trend: A common overall pattern.  Seasonal Variations: A pattern that repeats itself at regular time intervals
25. 25. Identify any trends and describe the time plot. Check For Understanding - 5
26. 26. Comparing Two Data Sets  If you want to compare two data sets, then make sure that the two graphic displays are as alike as possible—the scale of the axes, the groups for the histogram, etc. If you are using stemplots, use the same set of stems for both sets (thus creating a back-to-back stemplot).  When describing a single distribution, you want to comment on its center, spread and shape. When comparing two distributions, you will want to compare center, spread and shape! Don't forget to make the comparisons in context.
27. 27. Back-to-Back Stemplot of Random Values • A back-to-back stemplot is used to compare two data sets. Key: 2 0 = 20
28. 28. Check For Understanding - 6 During the early part of the 1994 baseball season, many sports fans and baseball players noticed that the number of home runs being hit seemed to be unusually large. Here are the data on the number of home runs hit by American and National League teams: American League 35, 40, 43, 49, 51, 54, 57, 58, 58, 64, 68, 68, 75, 77 National League 29, 31, 42, 46, 47, 48, 48, 53, 55, 55, 55, 63, 63, 67 (a) Construct a back-to-back stemplot to compare the number of home runs hit in the two leagues. (b) Write a few sentences comparing the distributions of home runs in the two leagues.