Displaying and Summarizing Quantitative Data 90 min

Example Here are the scores for the first test.

 A histogram is a graphical display of a frequency distribution of quantitative variable, shown as adjacent rectangles (like a bar graph)  Here is a histogram of earthquake magnitudes

 First, slice up the entire span of values covered by the quantitative variable into equal-width piles called bins.  Count the number of values of each bin, and the counts in each bin will be the heights of the bars.  The bins and the corresponding counts give the distribution of the quantitative variable.

 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.  Here is a relative frequency histogram of earthquake magnitudes:

 Stem-and-leaf displays or stemplots show the distribution of a quantitative variable, like histograms do, while preserving the individual values.  Stem-and-leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution.

Solution

 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?

 First, cut each data value into leading digits (“stems”) and trailing digits (“leaves”).  Use the stems to label the bins.  Use only one digit for each leaf—either round or truncate the data values to one decimal place after the stem.

 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.  You might see a dotplot displayed horizontally or vertically.

Example John wanted to add a new DVD player to his home theater system. He used the Internet to shop and went to pricewatch.com. There he found 16 quotes on different brands and styles of DVD players. Construct a dotplot for these data.

Solution

1. Humps 2. Symmetry 3. Unusual features

Does the histogram have a single, central hump or several separated humps? ◦ Humps in a histogram are called modes. ◦ A histogram with one main peak is called unimodal ◦ Histograms with two peaks are bimodal ◦ Histograms with three or more peaks are called multimodal.

 A bimodal histogram has two apparent peaks: Diastolic Blood Pressure

 A histogram that doesn’t appear to have any mode and in which all the bars are approximately the same height is called uniform: Proportion of Wins

Is the histogram symmetric? ◦ 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. 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

Do any unusual features stick out? ◦ Sometimes it is 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.

 The following histogram has outliers—there are three cities in the leftmost bar:

The figure on the next slide displays a relative- frequency histogram for the heights of the 3264 female students who attend a midwestern college. Also included is a smooth curve that approximates the overall shape of the distribution. Both the histogram and the smooth curve show that this distribution of heights is bell shaped, but the smooth curve makes seeing the shape a little easier.

Common distribution shapes

Example The relative-frequency histogram for household size in the U. S. shown in the figure is based on data contained in Current Population Reports, a publication of the U.S. Census Bureau. Identify the distribution shape for sizes of U.S. households. Solution - Skewed right

Find the frequency of each value in the data set.  If no value occurs more than once, then the data set has no mode.  Otherwise, any value that occurs with the greatest frequency is a mode of the data set. Mode of a data set is the data value with the greatest frequency.

 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.

Arrange the data in increasing order.  If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list.  If the number of observations is even, then the median is the average of the two middle observations in the ordered list.

 Mean = average of a data set  The formula says that to find the mean, we add up the numbers and divide by n.

 The mean feels like the center because it is the point where the histogram balances:

 Because the median considers only the order of values, it’s resistant to values that are extraordinarily large or small; it simply notes that they are one of the “big ones” or “small ones” and ignores their distance from center.  To choose between the mean and median, start by looking at the data. ◦ If the histogram is symmetric and there are no outliers, use the mean. ◦ However, if the histogram is skewed or with outliers, you are better off with the median.

Example Find the mean, median, mode for each team’s heights

 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.

Team I has range 6 inches, Team II has range 17 inches.

 Quartiles divide the data into four equal sections. ◦ One quarter of the data lies below the lower quartile Q 1 ◦ One quarter of the data lies above the upper quartile Q 3.  The difference between the quartiles is the interquartile range (IQR), so IQR = Q 3 – Q 1  The interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data.

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.

Arrange the data in increasing order and determine the median.  The second quartile is the median of the entire data set.  The first quartile is the median of the part of the entire data set that lies at or below the median of the entire data set.  The third quartile is the median of the part of the entire data set that lies at or above the median of the entire data set.

 The 5-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum)  The 5-number summary for the recent tsunami earthquake Magnitudes looks like this:

 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.

 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!

 The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data.

Example Data set II has greater variation, and hence has greater standard deviation.

Data set II has greater variation and the visual clearly shows that it is more spread out. Data Set I Data Set II

 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.

 Choose a bin width appropriate to the data. ◦ Changing the bin width changes the appearance of the histogram:

 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.  We’ve learned how to summarize distributions of quantitative variables numerically. ◦ Measures of center for a distribution include the median and mean. ◦ Measures of spread include the range, IQR, and standard deviation. ◦ Use the median and IQR if the distribution is skewed. ◦ Use the mean and standard deviation if the distribution is symmetric.

Example A pediatrician tested the cholesterol levels of several young patients and was alarmed to find that many had levels higher than 200 mg per 100 mL. The following table presents the readings of 20 patients with high levels. Construct a stem-and-leaf diagram for these data by using a. one line per stem.b. two lines per stem.

Solution

Data Set I Data Set II Example A firm employed a few senior consultants, who made between $800 and $1050 per week; a few junior consultants, who made between $400 and $450 per week; and several clerical workers, who made $300 per week. The firm required more employees during the first half of the summer than the second half. The tables list typical weekly earnings for the two halves of the summer.

Solution Interpretation: The employees who worked in the first half of the summer earned more, on average (a mean salary of $483.85), than those who worked in the second half (a mean salary of $474.00). Comparing Distribution  Ch.5

 Page :  Problem # 7, 9, 13, 15, 17, 19, 23, 25, 29, 31, 33, 37, 39, 41, 43, 47.