MATH& 146 Lesson 10. Section 1.6 Graphing Numerical Data

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1 MATH& 146 Lesson 10 Section 1.6 Graphing Numerical Data 1

2 Graphs of Numerical Data One major reason for constructing a graph of numerical data is to display its distribution, or the pattern of variability displayed by the data of a variable. Three popular methods for displaying distributions of numerical data are the dotplot, the histogram, and the box plot. 2

3 Dotplots The dotplot displays the data of a sample by representing each data value with a dot positioned along a scale, either horizontally or vertically. The frequency of the dotplot is represented along the other scale. 3

4 Example 1 Create a dotplot of the following exam scores. 4

5 Histograms For much of the work you do in this course, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. 5

6 Histograms Unlike dotplots, histograms use ranges of values instead of individual values. These ranges of values are represented by bars (called classes), with the heights equal to the frequency of each class. 6

7 Constructing histograms The basic steps to construct a histogram are as follows: 1) Find the minimum and maximum values of the data. 2) Create classes by slicing data into intervals of equal width (choose "nice" numbers). 3) Make a table (called a frequency table) to count the number of values in each class. 4) Make a bar for each class, using the heights to determine the height of each bar. 7

8 Example 2 The following are the scores on a measure of sensitivity to smell taken by 13 chefs attending a national conference: 96, 83, 59, 64, 73, 74, 80, 68, 87, 67, 64, 92, 76 Make a histogram of the data. 8

9 Shape of a Distribution When describing the shape of a distribution (the outline of a histogram), you should answer the following three questions: 1) Does it have a single, central peak or several separated peaks, or none at all? 2) Is it symmetric or is it skewed one way or the other? 3) Do any unusual features (e.g. outliers) stick out? 9

10 Peaks 1) Does the distribution have a single, central peak or several separated peaks, or none at all? A distribution with one main peak is dubbed unimodal. 10

11 Peaks A distribution with two peaks is bimodal. A distribution with no peaks and shaped more or less like a rectangle is called uniform. 11

12 Bimodal Distributions Bimodal distributions usually occur when the data of two separate groups are combined. Diastolic Blood Pressure 12

13 Uniform Distributions A distribution that doesn't appear to have any mode and in which all the bars are approximately the same height (in the "real world," the bars will never be exactly the same) is called uniform: Proportion of Wins 13

14 Symmetry 2) Is the distribution symmetric? Essentially, a distribution is symmetric if you can fold the distribution along a vertical line through the middle and have the edges match pretty closely. 14

15 Skewness The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail. skewed left skewed right 15

16 Skewness Symmetric graphs are ideal for inferential statistics, though skewed graphs can also work, provided the sample size is large enough. Generally, the more skewed the graph, the larger the sample size is needed to be. skewed left skewed right 16

17 Outliers 3) Do any unusual features stick out? Sometimes it's the unusual features that tell us something interesting or exciting about the data. You should always mention any stragglers, or outliers, that stand off away from the body of the distribution. 17

18 Outliers Often, not always, outliers are due to mistakes (such as writing 5,000 instead of 50). Other outliers may indicate that something unusual is happening. If you see an outlier, proceed carefully. 18

19 Example 3 What can be said about the following histogram? 19

20 Example 4 What can be said about the following histogram? 20

21 Example 5 What can be said about the following histogram? 21

22 Example 6 What can be said about the following histogram? 22

23 Box Plots Box plots, or box-and-whisker plots, give a graphical image of the concentration of the data. The box plot is constructed from five values, called the five-number summary: 23

24 The Five-Number Summary The five-number summary includes: The minimum The lower quartile, Q 1 The median The upper quartile, Q 3 The maximum These numbers divide the data into four more or less equal pieces. 25% 25% 25% 25% Min Q 1 Med Q 3 Max 24

25 The Interquartile Range The middle fifty percent of all data is represented by the box. The length of this box is the Interquartile Range. In other words: IQR Q Q 3 1 The length of the entire boxplot is the Range. Range Max Min IQR Range 25

26 Construct the Box Plot To construct a box plot, use a number line and mark each of the five numbers: minimum, first quartile, median, third quartile, and maximum (use a dotted tick mark for the median). Draw a top and bottom around the middle three numbers to make a box, and then draw lines connecting the box to the minimum and maximum. 26

27 Example 7 Construct a box plot and find the range and interquartile range. 91, 96, 84, 100, 92, 23, 84 27

28 Comparing Groups Boxplots are ideal when it comes to comparing two or more groups or categories. 28

29 Outliers Box Plots can be used to show extreme values by using dots or asterisks ( or *) to represent potential outliers. Any potential outlier should be examined carefully in your data analysis. 29

30 Example 8 The boxplots below show the number of millionaires by state per 1000 households, as reported by Netscape.com in

31 Example 8 continued a) List the regions from lowest to highest in terms of the median rate of millionaires in that region. b) Which region has the smallest interquartile range? c) Which region has potential outliers? 31

32 Example 9 The following box plot shows the U.S. population for a) Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? b) 12.6% are age 65 and over. Approximately what percent of the population are of working age adults (above age 17 to age 65)? 32

33 Example 10 Match each histogram with its boxplot. X Y Z 33