Objectives: Students will: Chapter 4 1. Be able to identify an appropriate display for any quantitative variable: stem leaf plot, time plot, histogram and dotplot given a set of quantitative data. 2. Be able to describe the distribution of a quantitative variable in terms of shape, center, spread and outliers given a graph. 3. Be able to compare the distributions of two or more groups given two graphs. Displaying Quantitative Data Dealing With a Lot of Numbers pg 45 Summarizing the data will help us when we look at large sets of quantitative data. Without summaries of the data, it s hard to grasp what the data tell us. The best thing to do is to make a picture Aug 27 10:16 PM Here is a histogram of the monthly price changes in Enron pg 46 stock: pg 47 A relative frequency histogram displays the percentage of cases in each bin instead of the count. In this way, relative frequency histograms are faithful to the area principle. Histograms are useful when working with large sets of data, and they can easily be constructed using a graphing calculator. A disadvantage of histograms is that they do not show individual values. Be sure to choose an appropriate bin width when constructing a histogram. As a general rule of thumb, your histogram should contain about 10 bars. Here is a relative frequency histogram of the monthly price changes in Enron stock: Aug 23 10:39 PM 1
Histograms 1. For quantitative variables that take many values 2. Divide the possible values into class intervals (we will only consider equal widths) 3. Count how many observations fall in each interval (may change to percents) 4. Draw picture representing distribution Histograms: Steps How many intervals? 1.One rule is to calculate the square root of the sample size, and round up. Size of intervals? 2. Divide range of data (max min) by number of intervals desired, and round to convenient number 3.Pick intervals so each observation can only fall in exactly one interval (no overlap) Aug 27 10:21 PM Aug 27 10:23 PM Example #1 (look at data from class survey) Stem and Leaf Displays pg 47 A stem and leaf plot is similar to a histogram, but it shows individual values rather than bars. It may be necessary to split stems if the range of data values is small. Stem and leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution. Aug 23 8:49 PM 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. pg 48 Stem and Leaf Example Compare the histogram and stem and leaf display for the pulse rates of 24 women at a health clinic. Which graphical display do you prefer and why? (Look at histogram in page 47) pg 47 Use only one digit for each leaf either round or truncate the data values to one decimal place after the stem. 2
Quiz scores are given for a 25 students in a math class: 1, 4, 10, 9, 1, 10,9, 5, 4, 17, 16, 9, 16, 20, 15, 7, 14, 5, 15, 13, 13, 13, 20, 20, 18 Homework: count how many pairs of shoes you own. Record your data on the board. Aug 23 8:54 PM Aug 27 10:36 PM The stems of the stem and leaf plot correspond to the bins of a histogram. You may only use one digit for the leaves. Round or truncate your values if necessary. Number of pairs of shoes owned Male Female. 0 0 1 1 2 2 3 3 Stem and leaf plots are useful when working with sets of data that are small to moderate in size, and when you want to display individual values. How would you setup the following stem and leaf plots? v quiz scores (out of 100) v student GPA s v student weights v SAT scores v weights of cattle (1000 2000 pounds) Pair Share Aug 23 8:58 PM Aug 23 9:00 PM Dotplots pg 49 A dotplot is a simple display. It just places a dot along an axis for each case in the data. The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot. Dot plots may also be used to display quantitative variables. Dot plots are useful when working with small sets of data. Pair Share Create a dot plot Number of Siblings You might see a dotplot displayed horizontally or vertically. Aug 23 9:09 PM 3
Think Before You Draw, Again Describing shape of the distribution Remember the Make a picture rule? Now that we have options for data displays, you need to Think carefully about which type of display to make. Before making a stem and leaf display, a histogram, or a dotplot, check the Quantitative Data Condition: The data are values of a quantitative variable whose units are known. Does the histogram have a single, central hump or several separated bumps? Humps in a histogram are called modes. A histogram with one main peak is dubbed unimodal histograms with two peaks are bimodal histograms with three or more peaks are called multimodal. Humps and Bumps (cont.) A bimodal histogram has two apparent peaks: A histogram that doesn t appear to have any mode and in which all the bars are approximately the same height is called uniform: Symmetry pg 51 If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric. The (usually) thinner ends of a distribution are called the tails. pg 51 If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail. If the tail of the histogram extends to the left is said to be skewed left, If the tail of the histogram extends to the right is said to be skewed right. 4
Do any unusual features stick out? Sometimes it s the unusual features that tell us something interesting or exciting about the data. pg 51 The following histogram has outliers there are three cities in pg 52 the leftmost bar: 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. Where is the Center of the Distribution? If you had to pick a single number to describe all the data what would you pick? pg 52 It s easy to find the center when a histogram is unimodal and symmetric it s right in the middle. On the other hand, it s not so easy to find the center of a skewed histogram or a histogram with more than one mode. For now, we will eyeball the center of the distribution. In the next chapter we will find the center numerically What is the Shape of the Distribution? Does the histogram have a single, central hump or several separated bumps? Is the histogram symmetric? Do any unusual features stick out? Where is the center of the distribution? Pair Share How Spread Out is the Distribution? Variation matters, and Statistics is about variation. pg 53 Warm up What are the four things you must mention when describing a quantitative distribution? Are the values of the distribution tightly clustered around the center or more spread out? In the next two chapters, we will talk about spread See Step by step pg 53 TI Tips pg 55 Sep 6 8:54 AM 5
Comparing Distributions pg 56 Step by step Often we would like to compare two or more distributions instead of looking at one distribution by itself. Think pair share Compare the following distributions of ages for female and male heart attack patients:using shape, center and spread. When looking at two or more distributions, it is very important that the histograms have been put on the same scale. Otherwise, we cannot really compare the two distributions. When we compare distributions, we talk about the shape, center, and spread of each distribution.and compare any unusual features. Timeplots: pg 57 For some data sets, we are interested in how the data behave over time. In these cases, we construct timeplots of the data. *Re expressing Skewed Data to Improve Symmetry pg 58 Figure 4.11 *Re expressing Skewed Data to Improve Symmetry One way to make a skewed distribution more symmetric is to re express or transform the data by applying a simple function (e.g., logarithmic function). Note the change in skewness from the raw data (Figure 4.11) to the transformed data (Figure 4.12): pg 59 Book work pg 65 #3,4,6,7,10,11,12,14,16,20,23,36 Figure 4.12 Sep 3 10:06 PM 6
What Can Go Wrong? pg 59 Don t make a histogram of a categorical variable bar charts or pie charts should be used for categorical data. Don t look for shape, center, and spread of a bar chart. What Can Go Wrong? (cont.) pg 60 Don t use bars in every display save them for histograms and bar charts. Below is a badly drawn timeplot and the proper histogram for the number of eagles sighted in a collection of weeks: What Can Go Wrong? (cont.) pg 61 Choose a bin width appropriate to the data. Changing the bin width changes the appearance of the histogram: What Can Go Wrong? (cont.) pg 61 Avoid inconsistent scales, either within the display or when comparing two displays. Label clearly so a reader knows what the plot displays. Good intentions, bad plot: What have we learned? We ve learned how to make a picture for quantitative data to help us see the story the data have to Tell. We can display the distribution of quantitative data with a histogram, stem and leaf display, or dotplot. Tell about a distribution by talking about shape, center, spread, and any unusual features. We can compare two quantitative distributions by looking at side by side displays (plotted on the same scale). Trends in a quantitative variable can be displayed in a timeplot. pg 62 Aug 27 10:15 PM 7