Chapter 5 Describing Distributions Numerically Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-1 Finding the Center: The Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center of symmetry. Finding the Center: The Median (cont.) As a measure of center, the midrange (the average of the minimum and maximum values) is very sensitive to skewed distributions and outliers. The median is a more reasonable choice for center than the midrange. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-3 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-4 Finding the Center: The Median (cont.) The median is the value with exactly half the data values below it and half above it. It is the middle data value (once the data values have been ordered) that divides the histogram into two equal areas. It has the same units as the data. 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 difference between the maximum and minimum values: Range = max min A disadvantage of the range is that a single extreme value can make it very large and, thus, not representative of the data overall. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-5 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-6 1
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 quartiles are Spread: The Interquartile Range (cont.) Quartiles divide the data into four equal sections. The lower quartile is the median of the half of the data below the median. The upper quartile is the median of the half of the data above the median. The difference between the quartiles is the IQR, so IQR = upper quartile lower quartile Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-7 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-8 Spread: The Interquartile Range (cont.) The Five-Number Summary The lower and upper quartiles are the 25 th and 75 th percentiles of the data, so The IQR contains the middle 50% of the values of the distribution, as shown in Figure 5.3 from the text: The five-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum). Example: The fivenumber summary for the ages at death for rock concert goers who died from being crushed is Max 47 years Q3 22 Median 19 Q1 17 Min 13 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-9 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-10 Rock Concert Deaths: Making Boxplots A boxplot is a graphical display of the five-number summary. Boxplots are particularly useful when comparing groups. Constructing Boxplots 1. Draw a single vertical axis spanning the range of the data. Draw short horizontal lines at the lower and upper quartiles and at the median. Then connect them with vertical lines to form a box. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-11 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-12 2
Constructing Boxplots (cont.) 2. Erect fences around the main part of the data. The upper fence is 1.5 IQRs above the upper quartile. The lower fence is 1.5 IQRs below the lower quartile. Note: the fences only help with constructing the boxplot and should not appear in the final display. Constructing Boxplots (cont.) 3. Use the fences to grow whiskers. Draw lines from the ends of the box up and down to the most extreme data values found within the fences. If a data value falls outside one of the fences, we do not connect it with a whisker. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-13 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-14 Constructing Boxplots (cont.) 4. Add the outliers by displaying any data values beyond the fences with special symbols. We often use a different symbol for far outliers that are farther than 3 IQRs from the quartiles. Rock Concert Deaths: Making Boxplots (cont.) Compare the histogram and boxplot for rock concert deaths: How does each display represent the distribution? Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-15 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-16 Comparing Groups With Boxplots The following set of boxplots compares the effectiveness of various coffee containers: Summarizing Symmetric Distributions Medians do a good job of identifying the center of skewed distributions. When we have symmetric data, the mean is a good measure of center. We find the mean by adding up all of the data values and dividing by n, the number of data values we have. What does this graphical display tell you? Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-17 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-18 3
Summarizing Symmetric Distributions (cont.) The distribution of pulse rates for 52 adults is generally symmetric, with a mean of 72.7 beats per minute (bpm) and a median of 73 bpm: The Formula for Averaging The formula for the mean is given by The formula says that to find the mean, we add up the numbers and divide by n. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-19 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-20 Mean or Median? Regardless of the shape of the distribution, the mean is the point at which a histogram of the data would balance: Mean or Median? (cont.) In symmetric distributions, the mean and median are approximately the same in value, so either measure of center may be used. For skewed data, though, it s better to report the median than the mean as a measure of center. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-21 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-22 What About Spread? The Standard Deviation A more powerful measure of spread than the IQR is the standard deviation, which takes into account how far each data value is from the mean. A deviation is the distance that a data value is from the mean. Since adding all deviations together would total zero, we square each deviation and find an average of sorts for the deviations. What About Spread? The Standard Deviation (cont.) The variance, notated by s 2, is found by summing the squared deviations and (almost) averaging them: The variance will play a role later in our study, but it is problematic as a measure of spread it is measured in squared units! Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-23 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-24 4
What About Spread? The Standard Deviation (cont.) The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data. Thinking About Variation Since Statistics is about variation, spread is an important fundamental concept of Statistics. Measures of spread help us talk about what we don t know. When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small. When the data values are scattered far from the center, the IQR and standard deviation will be large. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-25 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-26 Shape, Center, and Spread When telling about a quantitative variable, always report the shape of its distribution, along with a center and a spread. If the shape is skewed, report the median and IQR. If the shape is symmetric, report the mean and standard deviation and possibly the median and IQR as well. What About Outliers? 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. The differences may be quite revealing. Note: The median and IQR are not likely to be affected by the outliers. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-27 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-28 What Can Go Wrong? Don t forget to do a reality check don t let technology do your thinking for you. Don t forget to sort the values before finding the median or percentiles. Don t compute numerical summaries of a categorical variable. Watch out for multiple modes multiple modes might indicate multiple groups in your data. What Can Go Wrong? (cont.) Be aware of slightly different methods different statistics packages and calculators may give you different answers for the same data. Beware of outliers. Make a picture (make a picture, make a picture). Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-29 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-30 5
What Can Go Wrong? (cont.) Be careful when comparing groups that have very different spreads. Consider these side-by-side boxplots of cotinine levels: *Re-expressing to Equalize the Spread of Groups Here are the side-by-side boxplots of the log(cotinine) values: Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-31 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-32 What have we learned? We can now summarize distributions of quantitative variables numerically. The 5-number summary displays the min, Q1, median, Q3, and max. Measures of center include the mean and median. Measures of spread include the range, IQR, and standard deviation. We know which measures to use for symmetric distributions and skewed distributions. What have we learned? (cont.) We can also display distributions with boxplots. While histograms better show the shape of the distribution, boxplots reveal the center, middle 50%, and any outliers in the distribution. Boxplots are useful for comparing groups. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-33 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5-34 6