Honors Statistics Chapter 4 Part 1 Displaying and Summarizing Quantitative Data

Learning Goals Know how to display the distribution of a quantitative variable with a histogram, a stem-and-leaf display, or a dotplot. Know how to display the relative position of quantitative variable with a Cumulative Frequency Curve and analysis the Cumulative Frequency Curve. Be able to describe the distribution of a quantitative variable in terms of its shape. Be able to describe any anomalies or extraordinary features revealed by the display of a variable.

Learning Goals Be able to determine the shape of the distribution of a variable by knowing something about the data. Know the basic properties and how to compute the mean and median of a set of data. Understand the properties of a skewed distribution. Know the basic properties and how to compute the standard deviation and IQR of a set of data.

Learning Goals Understand which measures of center and spread are resistant and which are not. Be able to select a suitable measure of center and a suitable measure of spread for a variable based on information about its distribution. Be able to describe the distribution of a quantitative variable in terms of its shape, center, and spread.

Learning Goal 1 Know how to display the distribution of a quantitative variable with a histogram, a stem-and-leaf display, or a dotplot

Learning Goal 1: Ways to Graph Quantitative Data Histograms and Stemplots These are summary graphs for a single variable. They are very useful to understand the pattern of variability in the data. Dotplots Quick and easy graph for small data sets. Cumulative Frequency Curves (Ogive) Used to compare relative standings of the data. Line Graphs: Time Plots Use when there is a meaningful sequence, like time. The line connecting the points helps emphasize any change over time.

Learning Goal 1: Dealing With a Lot of Numbers… 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… We can’t use bar charts or pie charts for quantitative data, since those displays are for categorical variables.

Learning Goal 1: Tabulating Numerical Data What is a Frequency Distribution (table)? A frequency distribution is a list or a table … containing class groupings (ranges within which the data fall) ... and the corresponding frequencies with which data fall within each grouping or class.

Learning Goal 1: Why Use a Frequency Distribution? It is a way to summarize numerical data. It condenses the raw data into a more useful form. It allows for a quick visual interpretation of the data.

Quantitative Data Histogram

Learning Goal 1: Histograms A Histogram is a graph that uses bars to portray the frequencies or the relative frequencies of the possible outcomes for a quantitative variable.

Learning Goal 1: Histograms The most common graph used to display one variable quantitative data.

Learning Goal 1: Histograms To make a histogram we first need to organize the data using a quantitative frequency table. Two types of quantitative data Discrete – use ungrouped frequency table to organize. Continuous – use grouped frequency table to organize.

Learning Goal 1: Quantitative Frequency Tables – Ungrouped What is an ungrouped frequency table? An ungrouped frequency table simply lists the data values with the corresponding frequency counts with which each value occurs. Commonly used with discrete quantitative data.

Learning Goal 1: Quantitative Frequency Tables – Ungrouped Example: The at-rest pulse rate for 16 athletes at a meet were 57, 57, 56, 57, 58, 56, 54, 64, 53, 54, 54, 55, 57, 55, 60, and 58. Summarize the information with an ungrouped frequency distribution.

Learning Goal 1: Quantitative Frequency Tables – Ungrouped Class (pulse rate) Frequency, f 53 1 54 3 55 2 56 57 4 58 59 60 61 62 63 64 Total N =16 Example continued: 57, 57, 56, 57, 58, 56, 54, 64, 53, 54, 54, 55, 57, 55, 60, 58. Note: The (ungrouped) classes are the observed values themselves.

Learning Goal 1: Quantitative Relative Freq. Tables - Ungrouped Class (pulse rate) Frequency, f Relative Frequency 53 1 0.0625 54 3 0.1875 55 2 0.1250 56 57 4 0.2500 58 59 60 61 62 63 64 Total N =16 Note: The relative freq. for a class is obtained by computing f/n.

Learning Goal 1: Relative Freq. Tables – Your Turn TVs per Household Trends in Television, published by the Television Bureau of Advertising, provides information on television ownership. The table gives the number of TV sets per household for 50 randomly selected households. Use classes based on a single value to construct a ungrouped-data relative frequency table for these data. Change to page 52

Learning Goal 1: Relative Freq. Tables – Solution Change to page 52

Learning Goal 1: Quantitative Frequency Tables – Grouped What is a grouped frequency table? A grouped frequency table is obtained by constructing classes (or intervals) for the data, and then listing the corresponding number of values (frequency counts) in each interval. Commonly used with continuous quantitative data.

Learning Goal 1: Quantitative Frequency Tables – Grouped Class: an interval of values. Example: 61  x  70. Frequency: the number of data values that fall within a class. “Five data fall within the class 61  x  70”. Relative Frequency: the proportion of data values that fall within a class. “18% of the data fall within the class 61  x  70”.

Learning Goal 1: Grouped Frequency Tables – Example A frequency table organizes quantitative data. partitions data into classes (intervals). shows how many data values are in each class. Test Score Number of Students 61-70 4 71-80 8 81-90 15 91-100 7

Learning Goal 1: Grouped Frequency Table Terminology Class - non-overlapping intervals the data is divided into. Class Limits –The smallest and largest observed values in a given class. Class Boundaries – Fall halfway between the upper class limit for the smaller class and the lower class limit for larger class. Used to close the gap between classes. Class Width – The difference between the class boundaries for a given class. Class Midpoint or Mark – The midpoint of a class.

Learning Goal 1: Grouped Frequency Tables – Classes A grouped frequency table should have a minimum of 5 classes and a maximum of 20 classes. For small data sets, one can use between 5 and 10 classes. For large data sets, one can use up to 20 classes.

Learning Goal 1: Number of Classes Same data set Too Many Classes - Not summarized enough.

Learning Goal 1: Number of Classes Same data set Too Few Classes – summarized too much.

Learning Goal 1: Number of Classes Same data set Correct Number of Classes – 5 to 10.

Learning Goal 1: Class Limits Lower Class Limits are the smallest numbers that can actually belong to different classes. Lower Class Limits

Learning Goal 1: Class Limits Upper Class Limits are the largest numbers that can actually belong to different classes. Upper Class Limits

Learning Goal 1: Class Boundaries Class Boundaries are the numbers used to separate classes, but without the gaps created by class limits. Class boundaries split the gap, created by the class limits between two consecutive classes, in half. Half of the gap is given to the upper class and half given to the lower class. Thus, bringing the bars of the two consecutive classes together, with no gap.

Learning Goal 1: Structure of a Data Class A “class” is basically an interval on a number line. It has: A lower limit a and an upper limit b. A width. A lower boundary and an upper boundary (integer data). A midpoint. (b + 0.5) - (a - 0.5)

Learning Goal 1: Structure of a Data Class - Problem (b + 0.5) - (a - 0.5) If a = 60 and b = 69 for integer data, what is the value of the lower boundary? a). 60 b). 59.5 c). 9 d). 64.5

Learning Goal 1: Class Boundaries Class Boundaries are the number separating classes. Class Boundaries - 0.5 99.5 199.5 299.5 399.5 499.5

Learning Goal 1: Class Midpoints or Class Mark Class Midpoint or Class Mark is the midpoint of each class. Class midpoints can be found by adding the lower class limit to the upper class limit and dividing the sum by two.

Learning Goal 1: Class Midpoints Class Midpoint is the midpoint of each class. Class Midpoints 49.5 149.5 249.5 349.5 449.5

Learning Goal 1: Class Width Class Width is the difference between two consecutive lower class limits or two consecutive lower class boundaries Class Width 100

Learning Goal 1: Constructing A Frequency Table 1. Decide on the number of classes (should be between 5 and 20) . 2. Calculate (round up). 3. Starting point: Begin by choosing a lower limit of the first class. 4. Using the lower limit of the first class and class width, proceed to list the lower class limits. 5. List the lower class limits in a vertical column and proceed to enter the upper class limits. 6. Go through the data set putting a tally in the appropriate class for each data value. class width  (highest value) – (lowest value) number of classes

Learning Goal 1: Constructing A Frequency Table - Example A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature. 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

Learning Goal 1: Constructing A Frequency Table - Example (continued) Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: 58 - 12 = 46 Select number of classes: 5 (usually between 5 and 10) Compute class interval (width): 10 (46/5 then round up) Determine lower class (limits): 10, 20, 30, 40, 50. List in a vertical column. Compute upper class limits 19, 29, 39, 49, 59, and then class midpoints: 14.5, 24.5, 34.5, 44.5, 54.5. Count observations & assign to classes

Learning Goal 1: Constructing A Frequency Table - Example (continued) Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Relative Frequency Class Frequency Percentage 10 - 19 3 .15 15% 20 - 29 6 .30 30% 30 - 39 5 .25 25% 40 - 49 4 .20 20% 50 - 59 2 .10 10% Total 20 1.00 100%

Learning Goal 1: Tip for Constructing A Frequency Table Use Tally marks to count the data in each class. Record the frequencies (and relative frequencies if desired) on the table.

Learning Goal 1: Histogram Then to make the Histogram, graph the Frequency Table data.

Learning Goal 1: Making a Histogram Make a frequency table. Choose appropriate scale for vertical axis (freq. or relative freq.) and horizontal axis (based on classes). Label both axis. Place class boundaries on horizontal axis. Place frequencies on vertical axis. For each class, draw a bar with height equal to the class frequency and width equal to the class width. Title the graph.

Learning Goal 1: Making a Histogram Class Midpoint Class Frequency 10 - 19 15 3 20 - 29 25 6 30 - 39 35 5 40 - 49 45 4 50 - 59 55 2 (No gaps between bars) Temperatures (degrees)

Learning Goal 1: Frequency Table From a Histogram There are several procedures that one can use to construct a grouped frequency tables. However, because of the many statistical software packages (MINITAB, SPSS etc.) and graphing calculators (TI-84 etc.) available today, it is not necessary to try to construct such distributions using pencil and paper.

Learning Goal 1: Frequency Table From a Histogram The weights of 30 female students majoring in Physical Education on a college campus are as follows: 143, 113, 107, 151, 90, 139, 136, 126, 122, 127, 123, 137, 132, 121, 112, 132, 133, 121, 126, 104, 140, 138, 99, 134, 119, 112, 133, 104, 129, and 123. Summarize the data with a frequency distribution using seven classes.

Learning Goal 1: Frequency Table From a Histogram The MINITAB statistical software was used to generate the histogram (similar to the histogram on our TI-84) in the next slide. The histogram has seven classes. Classes for the weights are along the x-axis and frequencies are along the y-axis. The number at the top of each rectangular box, represents the frequency for the class.

Learning Goal 1: Frequency Table From a Histogram Histogram with 7 classes for the weights.

Learning Goal 1: Frequency Table From a Histogram Observations From the histogram, the classes (intervals) are 85 – 95, 95 – 105,105 – 115 etc. with corresponding frequencies of 1, 3, 4, etc. We will use this information to construct the group frequency distribution.

Learning Goal 1: Frequency Table From a Histogram Observations (continued) Observe that the upper class limit of 95 for the class 85 – 95 is listed as the lower class limit for the class 95 – 105. Since the value of 95 cannot be included in both classes, we will use the convention that the upper class limit is not included in the class.

Learning Goal 1: Frequency Table From a Histogram Observations (continued) That is, the class 85 – 95 should be interpreted as having the values 85 and up to 95 but not including the value of 95. Using these observations, the grouped frequency distribution is constructed from the histogram and is given on the next slide.

Learning Goal 1: Frequency Table From a Histogram Class (weight) Frequency 85 – 95 1 95 – 105 3 105 – 115 4 115 – 125 6 125 – 135 9 135 – 145 145 – 155 Total n = 30

Learning Goal 1: Using the TI-84 to Make Histograms Start by entering data into a list (STAT / Edit / L1). Example: Enter the presidential data on the next slide into list L1.

Learning Goal 1: Using the TI-84 to Make Histograms

Learning Goal 1: Using the TI-84 to Make Histograms Choose 2nd: Stat Plot to choose a histogram plot. Caution: Watch out for other plots that might be “turned on” or equations that might be graphed.

Learning Goal 1: Using the TI-84 to Make Histograms Turn the plot “on”, Choose the histogram plot. Xlist should point to the location of the data.

Learning Goal 1: Using the TI-84 to Make Histograms Under the “Zoom” menu, choose option 9: ZoomStat

Learning Goal 1: Using the TI-84 to Make Histograms The result is a histogram where the calculator has decided the width and location of the ranges. You can use the Trace key to get information about the ranges and the frequencies.

Learning Goal 1: Using the TI-84 to Make Histograms You can change the size and location of the ranges by using the Window button. Use the Xscl to change the class width on the graph. Press the Graph button to see the results

Learning Goal 1: Using the TI-84 to Make Histograms Voila! Of course, you can still change the ranges if you don’t like the results. And you can construct a frequency table from the histogram.

Learning Goal 1: Using the TI-84 to Make Histograms – Your Turn Using the data given, on sodium in cereals, construct a histogram on your TI – 84 and then using your histogram construct a frequency/relative frequency table. Use 8 classes, with a lower class limit of 0. Sodium Data: 0 210 260 125 220 290 210 140 220 200 125 170 250 150 170 70 230 200 290 180

Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Class Limits Frequency 350 to < 450 450 to < 550 550 to < 650 650 to < 750 750 to < 850 850 to < 950 11 10 2 1 Same as raw data, using the class midpoint to represent the class.

Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Enter the data into 2 lists. L1 is the classes (class midpoint) and L2 is the frequency.

Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Turn on Stats Plot1 and select the histogram. Xlist is L1 the classes and Freq is L2 the frequencies.

Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Select ZoomStat to graph the histogram.

Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Adjust the WINDOW to improve the picture and/or make the values better.

Learning Goal 1: TI-84 to Make Histogram Using Freq. Table Data Use the Trace Key to determine values on the graph.

Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram Frequency Histogram - a bar graph in which the horizontal scale represents the classes of data values and the vertical scale represents the frequencies.

Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram Relative Frequency Histogram - has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies.

Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram They look the same with the exception of the vertical axis scale.

Learning Goal 1: Freq. Histogram vs Relative Freq. Histogram - Example Move Page 59, Figure 2.2 to Slide 15 Insert Page 59 Definition 2.4 this slide 16

Learning Goal 1: Histograms - Facts Histograms are useful when the data values are quantitative. A histogram gives an estimate of the shape of the distribution of the population from which the sample was taken. If the relative frequencies were plotted along the vertical axis to produce the histogram, the shape will be the same as when the frequencies are used.

Learning Goal 1: Anatomy of a Histogram Title Note that there are no spaces between bars. (continuous data) Number of observations. Height of each bar represents the frequency in each class. Number of occurrences (frequencies) are shown on the vertical axis. Empty Class: No data were recorded between 75 and 80. The numbers shown on the horizontal axis are the boundaries of each class. Each bar represents a class. The number of classes is usually between 5 and 20. Here, there are 17 classes. The width of each class is determined by dividing the range of the data set by the number of classes, and rounding up. In this data set, the range is 82. 82/17 = 4.8, rounded up to 5. This class goes from 5 to 10. Label both horizontal and vertical axes. NOTE: Sometimes the numbers shown on the horizontal axis are the midpoints of each class. (A class midpoint is also referred to as the mark of the class.)