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Chapter 4 Displaying and Summarizing Quantitative Data Dealing With a Lot of Numbers… Summarizing the data will help us when we look at large sets of quantitative data. The best thing to do is We can’t use bar charts or pie charts for quantitative data, since those displays are for categorical variables. Slide 4- 2 Excel Data Sets and Software Excel data sets for some examples used in the book: http://media.pearsoncmg.com/aw/aw_deveaux_introstats _3/datasets/stat3dv_datasets_excel.html Could use add-in for Excel – DDXL comes with the CD in book – to produce most of the graphs in presentations. Slide 4- 3 Histograms: Earthquake Magnitudes US National Geophysical Data Center data set Who: 2410 earthquakes known to have caused tsunamis 2 What: Magnitude (Richter scale ), depth (m), date, location, and other variables When: From 2000 B.C.E. to the present All over the earth Slide 4- 4 Histograms: Earthquake Magnitudes (cont.) First, slice up the entire span of values covered by the quantitative variable into equal-width piles called bins: Min=3.0 Max=9.2 Each bin has a width of 0.2 The bins and the counts in each bin give the distribution of the quantitative variable and provide the building blocks for the histogram. Slide 4- 5 Histograms: Earthquake Magnitudes (cont.) A histogram plots Here is a histogram of earthquake magnitudes Slide 4- 6 Histograms: Earthquake magnitudes (cont.) A relative frequency histogram displays the Faithful to the area principle. Here is a relative frequency histogram of earthquake magnitudes: Slide 4- 7 Stem-and-Leaf Displays Stem-and-leaf displays show the distribution of a quantitative variable while preserving the individual values. Contain all the information found in a Satisfies the Shows the distribution Slide 4- 8 Stem-and-Leaf Example Compare the histogram and stem-and-leaf display for the pulse rates of 24 women at a health clinic. Slide 4- 9 Constructing a Stem-and-Leaf Display First, cut each data value into leading digits (“stems”) and trailing digits (“leaves”). Use the stems to label the bins. Use only for each leaf If working with whole numbers, use the one’s digit for the leaf If working with decimals, round to the tenths place (1st decimal) and use that for the leaf Slide 4- 10 Dotplots A dotplot places a dot along an axis for each Can be horizontal or vertical. The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot. Slide 4- 11 Think Before You Draw, Again Now we have options for data displays Think carefully about which : The data are values of a quantitative variable whose units are known. If satisfied, create a stem-and-leaf display, a histogram, or a dotplot Slide 4- 12 Shape, Center, and Spread When describing a distribution, make sure to always tell about three things: , , and … Slide 4- 13 What is the Shape of a Distribution? Does the histogram have a single, central hump or several separated humps? Is the histogram symmetric? Do any unusual features stick out? Slide 4- 14 Humps • Does the histogram have a single, central hump or several separated bumps? • Humps in a histogram are called • One main peak: • Two peaks: • Three or more peaks: Slide 4- 15 Humps (cont.) A bimodal histogram has two apparent peaks: Diastolic Blood Pressure Slide 4- 16 Symmetry • Is the histogram symmetric? If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, Slide 4- 17 Symmetry (cont.) The thinner ends of a distribution are called the tails If one tail stretches out farther than the other, the histogram is said to be In the figure below, the histogram on the left is Slide 4- 18 Anything Unusual? 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. Are there any gaps in the distribution? If so, we might have data from more than one group. Slide 4- 19 Anything Unusual? (cont.) The following histogram has are three cities in the leftmost bar: —there Slide 4- 20 Center of a Distribution – Median The is the value with exactly half the data values below it and half above it. It is the (once the data values have been ordered) that divides the histogram into two equal areas. It has the as the data. Slide 4- 21 Spread: Home on the Range Always report a measure of spread along with a measure of center when describing a distribution numerically. The range of the data is the Disadvantage: A single extreme value can make it very large and not representative of the data overall. Slide 4- 22 Spread: The Interquartile Range The interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data. To find the IQR, we first need to know what are… Slide 4- 23 Spread: The Interquartile Range (cont.) Quartiles divide the data into four equal sections. One quarter of the data lies below the lower quartile, One quarter of the data lies above the upper quartile, The difference between the quartiles is the interquartile range (IQR), so IQR = Slide 4- 24 Spread: The Interquartile Range (cont.) Lower quartile = 25th percentile Upper quartile = 75th percentile of the data The IQR contains the of the values of the distribution, as shown in figure: Slide 4- 25 5-Number Summary The 5-number summary of a distribution reports the The 5-number summary for the recent tsunami earthquake Magnitudes is: Slide 4- 26 Summarizing Symmetric Distributions – The Mean When the data is symmetric, use the instead of the We use the Greek letter sigma to mean “sum” and write: The formula says that to find the mean, Slide 4- 27 Summarizing Symmetric Distributions – The Mean (cont.) The mean feels like the center because it is the point where the histogram balances: Slide 4- 28 Summarizing Symmetric Distributions – The Mean (cont) Median is resistant to values that are extraordinarily large or small Mean or median? If the histogram is and there are no outliers, use the However, if the histogram is or with outliers, you are better off with the Slide 4- 29 What About Spread? The Standard Deviation The takes into account how far each data value is from the mean. A is the distance that Since adding all deviations together would total zero, we square each deviation and find a type of average for the deviations. Slide 4- 30 What About Spread? The Standard Deviation (cont.) The variance, notated by , is found by summing the squared deviations and (almost) averaging them: Slide 4- 31 What About Spread? The Standard Dev (cont.) The variance is measured in squared units! The is just the square root of the variance and is measured in the same units as the original data. Slide 4- 32 Thinking About Variation Recall: Statistics is about variation When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation When the data values are scattered far from the center, the IQR and standard deviation Slide 4- 33 Tell - Draw a Picture When telling about quantitative variables, start by making a histogram or stem-andleaf display and discuss the shape of the distribution. Slide 4- 34 Tell - Shape, Center, and Spread Next, always report the distribution, along with a If the shape is If the shape is of its and a , report the Slide 4- 35 Tell - What About Unusual Features? If there are multiple modes, try to understand why If there are any clear outliers and you are reporting the mean and standard deviation, report them with the outliers present and with the outliers removed Slide 4- 36 What Can Go Wrong? Don’t make a histogram of a categorical variable— use a Don’t look for of a bar chart Slide 4- 37 What Can Go Wrong? (cont.) Choose a bin width appropriate to the data. Changing the bin width changes the appearance of the histogram. Slide 4- 38 What Can Go Wrong? (cont.) Don’t forget to sort the values before finding the median or percentiles. Don’t worry about small differences when using different methods. Don’t compute numerical summaries of a categorical variable. Don’t round in the middle of a calculation. Watch out for multiple modes. Beware of outliers. Slide 4- 39 What have we learned? Make a picture!! With quantitative data use a histogram, stem-andleaf display, or dotplot. Summarize distributions of quantitative variables numerically. Measures of center for a distribution include the Measures of spread include Slide 4- 40 What have we learned? (cont.) Use the median and IQR when the distribution is Use the mean and standard deviation if the distribution is Think about the type of variable we are summarizing. The Quantitative Data Condition serves as a check that the data are, in fact, quantitative. Slide 4- 41 Class Exercise a. b. c. d. e. Number of days six patients of heart transplant survived: 3, 64, 623, 15, 46, 64 Find the median Find the mean Find the mode Find the quartiles Find the range and inter-quartile range (IQR) of the data set. Slide 4- 42 Class Exercise (cont.) Data: 3, 64, 623, 15, 46, 64 a. What is the median survival time? Slide 4- 43 Class Exercise (cont.) Ordered Data: 3, 15, 46, 64, 64, 623 b. What is the mean survival time? 3 15 46 64 64 623 815 y 1358 . 6 6 Slide 4- 44 Class Exercise (cont.) Ordered Data: 3, 15, 46, 64, 64, 623 c. What is the mode of the survival times? Slide 4- 45 Class Exercise (cont.) Ordered Data: 3, 15, 46, 64, 64, 623 d. What are the quartiles of the survival times? Slide 4- 46 Class Exercise (cont.) Ordered Data: 3, 15, 46, 64, 64, 623 e. Find the range and inter-quartile range (IQR) of the survival times Slide 4- 47 Class Exercise Find the standard deviation of the following data set: 5 4 6 5 5 3 7 Slide 4- 48 Class Exercise (cont.) Ordered data: 3, 4, 5, 5, 5, 6, 7 (obs – mean) (obs – mean)2 5 0 (0)2 = 0 5 0 (0)2 = 0 6 1 (+1)2 = 1 7 2 (+2)2 = 4 Observation 3 4 5 Summation: Slide 4- 49 Class Exercise (cont.) Recall: Thus s2 = Taking the square root of the variance we obtain the standard deviation: SD = Slide 4- 50