Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Descriptive Statistics 2

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 2 Chapter Outline 2.1 Frequency Distributions and Their Graphs 2.2 More Graphs and Displays 2.3 Measures of Central Tendency 2.4 Measures of Variation 2.5 Measures of Position

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 3 Section 2.1 Frequency Distributions and Their Graphs

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 4 Section 2.1 Objectives How to construct a frequency distribution including limits, midpoints, relative frequencies, cumulative frequencies, and boundaries How to construct frequency histograms, frequency polygons, relative frequency histograms, and ogives

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 5 Frequency Distribution A table that shows classes or intervals of data with a count of the number of entries in each class. The frequency, f, of a class is the number of data entries in the class. ClassFrequency, f 1 – 55 6 – – – – – 304 Lower class limits Upper class limits Class width 6 – 1 = 5

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 6 Constructing a Frequency Distribution 1.Decide on the number of classes.  Usually between 5 and 20; otherwise, it may be difficult to detect any patterns. 2.Find the class width.  Determine the range of the data.  Divide the range by the number of classes.  Round up to the next convenient number.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 7 Constructing a Frequency Distribution 3.Find the class limits.  You can use the minimum data entry as the lower limit of the first class.  Find the remaining lower limits (add the class width to the lower limit of the preceding class).  Find the upper limit of the first class. Remember that classes cannot overlap.  Find the remaining upper class limits.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 8 Constructing a Frequency Distribution 4.Make a tally mark for each data entry in the row of the appropriate class. 5.Count the tally marks to find the total frequency f for each class.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 9 Example: Constructing a Frequency Distribution The following sample data set lists the prices (in dollars) of 30 portable global positioning system (GPS) navigators. Construct a frequency distribution that has seven classes

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 10 Solution: Constructing a Frequency Distribution 1.Number of classes = 7 (given) 2.Find the class width Round up to

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 11 Solution: Constructing a Frequency Distribution Lower limit Upper limit Class width = 56 3.Use 59 (minimum value) as first lower limit. Add the class width of 56 to get the lower limit of the next class = 115 Find the remaining lower limits.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 12 Solution: Constructing a Frequency Distribution The upper limit of the first class is 114 (one less than the lower limit of the second class). Add the class width of 56 to get the upper limit of the next class = 170 Find the remaining upper limits. Lower limit Upper limit Class width = 56

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 13 Solution: Constructing a Frequency Distribution 4.Make a tally mark for each data entry in the row of the appropriate class. 5.Count the tally marks to find the total frequency f for each class. ClassTallyFrequency, f 59 – 114 IIII – 170 IIII III – 226 IIII I – 282 IIII – 338 II – 394 I – 450 III 3

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 14 Determining the Midpoint Midpoint of a class ClassMidpointFrequency, f 59 – – – 2266 Class width = 56

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 15 Determining the Relative Frequency Relative Frequency of a class Portion or percentage of the data that falls in a particular class. ClassFrequency, fRelative Frequency 59 – – – 2266

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 16 Determining the Cumulative Frequency Cumulative frequency of a class The sum of the frequency for that class and all previous classes. ClassFrequency, fCumulative frequency 59 – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 17 Expanded Frequency Distribution ClassFrequency, fMidpoint Relative frequency Cumulative frequency 59 – – – – – – – Σf = 30

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 18 Graphs of Frequency Distributions Frequency Histogram A bar graph that represents the frequency distribution. The horizontal scale is quantitative and measures the data values. The vertical scale measures the frequencies of the classes. Consecutive bars must touch. data values frequency

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 19 Class Boundaries Class boundaries The numbers that separate classes without forming gaps between them. Class Boundaries Frequency, f 59 – – – 2266 The distance from the upper limit of the first class to the lower limit of the second class is 115 – 114 = 1. Half this distance is 0.5. First class lower boundary = 59 – 0.5 = 58.5 First class upper boundary = = – 114.5

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 20 Class Boundaries Class Class boundaries Frequency, f 59 – – – – – – – – – – – – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 21 Example: Frequency Histogram Construct a frequency histogram for the global positioning system (GPS) navigators. Class Class boundariesMidpoint Frequency, f 59 – – – – – – – – – – – – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 22 Solution: Frequency Histogram (using Midpoints)

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 23 Solution: Frequency Histogram (using class boundaries) You can see that more than half of the GPS navigators are priced below $

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 24 Graphs of Frequency Distributions Frequency Polygon A line graph that emphasizes the continuous change in frequencies. data values frequency

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 25 Example: Frequency Polygon Construct a frequency polygon for the GPS navigators frequency distribution. ClassMidpointFrequency, f 59 – – – – – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 26 Solution: Frequency Polygon You can see that the frequency of GPS navigators increases up to $ and then decreases. The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 27 Graphs of Frequency Distributions Relative Frequency Histogram Has the same shape and the same horizontal scale as the corresponding frequency histogram. The vertical scale measures the relative frequencies, not frequencies. data values relative frequency

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 28 Example: Relative Frequency Histogram Construct a relative frequency histogram for the GPS navigators frequency distribution. Class Class boundaries Frequency, f Relative frequency 59 – – – – – – – – – – – – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 29 Solution: Relative Frequency Histogram From this graph you can see that 20% of GPS navigators are priced between $ and $

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 30 Graphs of Frequency Distributions Cumulative Frequency Graph or Ogive A line graph that displays the cumulative frequency of each class at its upper class boundary. The upper boundaries are marked on the horizontal axis. The cumulative frequencies are marked on the vertical axis. data values cumulative frequency

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 31 Constructing an Ogive 1.Construct a frequency distribution that includes cumulative frequencies as one of the columns. 2.Specify the horizontal and vertical scales.  The horizontal scale consists of the upper class boundaries.  The vertical scale measures cumulative frequencies. 3.Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 32 Constructing an Ogive 4.Connect the points in order from left to right. 5.The graph should start at the lower boundary of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 33 Example: Ogive Construct an ogive for the GPS navigators frequency distribution. Class Class boundaries Frequency, f Cumulative frequency 59 – – – – – – – – – – – – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 34 Solution: Ogive From the ogive, you can see that about 25 GPS navigators cost $300 or less. The greatest increase occurs between $ and $

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 35 Section 2.1 Summary Constructed frequency distributions Constructed frequency histograms, frequency polygons, relative frequency histograms and ogives

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 36 Section 2.2 More Graphs and Displays

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 37 Section 2.2 Objectives How to graph and interpret quantitative data using stem-and-leaf plots and dot plots How to graph and interpret qualitative data using pie charts and Pareto charts How to graph and interpret paired data sets using scatter plots and time series charts

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 38 Graphing Quantitative Data Sets Stem-and-leaf plot Each number is separated into a stem and a leaf. Similar to a histogram. Still contains original data values. Data: 21, 25, 25, 26, 27, 28, 30, 36, 36,

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 39 Example: Constructing a Stem-and-Leaf Plot The following are the numbers of text messages sent last month by the cellular phone users on one floor of a college dormitory. Display the data in a stem-and-leaf plot

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 40 Solution: Constructing a Stem-and-Leaf Plot The data entries go from a low of 78 to a high of 159. Use the rightmost digit as the leaf.  For instance, 78 = 7 | 8 and 159 = 15 | 9 List the stems, 7 to 15, to the left of a vertical line. For each data entry, list a leaf to the right of its stem

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 41 Solution: Constructing a Stem-and-Leaf Plot Include a key to identify the values of the data. From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 42 Graphing Quantitative Data Sets Dot plot Each data entry is plotted, using a point, above a horizontal axis Data: 21, 25, 25, 26, 27, 28, 30, 36, 36,

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 43 Example: Constructing a Dot Plot Use a dot plot organize the text messaging data. So that each data entry is included in the dot plot, the horizontal axis should include numbers between 70 and 160. To represent a data entry, plot a point above the entry's position on the axis. If an entry is repeated, plot another point above the previous point

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 44 Solution: Constructing a Dot Plot From the dot plot, you can see that most values cluster between 105 and 148 and the value that occurs the most is 126. You can also see that 78 is an unusual data value

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 45 Graphing Qualitative Data Sets Pie Chart A circle is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 46 Example: Constructing a Pie Chart The numbers of earned degrees conferred (in thousands) in 2007 are shown in the table. Use a pie chart to organize the data. (Source: U.S. National Center for Educational Statistics) Type of degreeNumber (thousands) Associate’s728 Bachelor’s1525 Master’s604 First professional90 Doctoral60.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 47 Solution: Constructing a Pie Chart Find the relative frequency (percent) of each category. Type of degreeFrequency, fRelative frequency Associate’s 728 Bachelor’s 1525 Master’s 604 First professional 90 Doctoral

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 48 Solution: Constructing a Pie Chart Construct the pie chart using the central angle that corresponds to each category.  To find the central angle, multiply 360º by the category's relative frequency.  For example, the central angle for cars is 360(0.24) ≈ 86º.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 49 Solution: Constructing a Pie Chart Type of degreeFrequency, f Relative frequencyCentral angle Associate’s Bachelor’s Master’s First professional Doctoral º(0.02)≈7º 360º(0.24)≈86º 360º(0.51)≈184º 360º(0.20)≈72º 360º(0.03)≈11º.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 50 Solution: Constructing a Pie Chart Type of degree Relative frequency Central angle Associate’s0.2486º Bachelor’s º Master’s0.2072º First professional0.0311º Doctoral0.027º From the pie chart, you can see that most fatalities in motor vehicle crashes were those involving the occupants of cars..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 51 Graphing Qualitative Data Sets Pareto Chart A vertical bar graph in which the height of each bar represents frequency or relative frequency. The bars are positioned in order of decreasing height, with the tallest bar positioned at the left. Categories Frequency.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 52 Example: Constructing a Pareto Chart In a recent year, the retail industry lost $36.5 billion in inventory shrinkage. Inventory shrinkage is the loss of inventory through breakage, pilferage, shoplifting, and so on. The causes of the inventory shrinkage are administrative error ($5.4 billion), employee theft ($15.9 billion), shoplifting ($12.7 billion), and vendor fraud ($1.4 billion). Use a Pareto chart to organize this data. (Source: National Retail Federation and Center for Retailing Education, University of Florida).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 53 Solution: Constructing a Pareto Chart Cause$ (billion) Admin. error5.4 Employee theft 15.9 Shoplifting12.7 Vendor fraud1.4 From the graph, it is easy to see that the causes of inventory shrinkage that should be addressed first are employee theft and shoplifting..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 54 Graphing Paired Data Sets Paired Data Sets Each entry in one data set corresponds to one entry in a second data set. Graph using a scatter plot.  The ordered pairs are graphed as points in a coordinate plane.  Used to show the relationship between two quantitative variables. x y.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 55 Example: Interpreting a Scatter Plot The British statistician Ronald Fisher introduced a famous data set called Fisher's Iris data set. This data set describes various physical characteristics, such as petal length and petal width (in millimeters), for three species of iris. The petal lengths form the first data set and the petal widths form the second data set. (Source: Fisher, R. A., 1936).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 56 Example: Interpreting a Scatter Plot As the petal length increases, what tends to happen to the petal width? Each point in the scatter plot represents the petal length and petal width of one flower..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 57 Solution: Interpreting a Scatter Plot Interpretation From the scatter plot, you can see that as the petal length increases, the petal width also tends to increase..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 58 Graphing Paired Data Sets Time Series Data set is composed of quantitative entries taken at regular intervals over a period of time.  e.g., The amount of precipitation measured each day for one month. Use a time series chart to graph. time Quantitative data.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 59 Example: Constructing a Time Series Chart The table lists the number of cellular telephone subscribers (in millions) for the years 1998 through Construct a time series chart for the number of cellular subscribers. (Source: Cellular Telecommunication & Internet Association).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 60 Solution: Constructing a Time Series Chart Let the horizontal axis represent the years. Let the vertical axis represent the number of subscribers (in millions). Plot the paired data and connect them with line segments..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 61 Solution: Constructing a Time Series Chart The graph shows that the number of subscribers has been increasing since 1998, with greater increases recently..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 62 Section 2.2 Summary Graphed and interpreted quantitative data using stem- and-leaf plots and dot plots Graphed and interpreted qualitative data using pie charts and Pareto charts Graphed and interpreted paired data sets using scatter plots and time series charts.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 63 Section 2.3 Measures of Central Tendency.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 64 Section 2.3 Objectives How to find the mean, median, and mode of a population and of a sample How to find the weighted mean of a data set and the mean of a frequency distribution How to describe the shape of a distribution as symmetric, uniform, or skewed and how to compare the mean and median for each.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 65 Measures of Central Tendency Measure of central tendency A value that represents a typical, or central, entry of a data set. Most common measures of central tendency:  Mean  Median  Mode.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 66 Measure of Central Tendency: Mean Mean (average) The sum of all the data entries divided by the number of entries. Sigma notation: Σx = add all of the data entries (x) in the data set. Population mean: Sample mean:.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 67 Example: Finding a Sample Mean The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 68 Solution: Finding a Sample Mean The sum of the flight prices is Σx = = 3695 To find the mean price, divide the sum of the prices by the number of prices in the sample The mean price of the flights is about $

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 69 Measure of Central Tendency: Median Median The value that lies in the middle of the data when the data set is ordered. Measures the center of an ordered data set by dividing it into two equal parts. If the data set has an  odd number of entries: median is the middle data entry.  even number of entries: median is the mean of the two middle data entries..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 70 Example: Finding the Median The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 71 Solution: Finding the Median First order the data There are seven entries (an odd number), the median is the middle, or fourth, data entry. The median price of the flights is $427..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 72 Example: Finding the Median The flight priced at $432 is no longer available. What is the median price of the remaining flights?

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 73 Solution: Finding the Median First order the data There are six entries (an even number), the median is the mean of the two middle entries. The median price of the flights is $412..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 74 Measure of Central Tendency: Mode Mode The data entry that occurs with the greatest frequency. If no entry is repeated the data set has no mode. If two entries occur with the same greatest frequency, each entry is a mode (bimodal)..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 75 Example: Finding the Mode The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 76 Solution: Finding the Mode Ordering the data helps to find the mode The entry of 397 occurs twice, whereas the other data entries occur only once. The mode of the flight prices is $397..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 77 Example: Finding the Mode At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses? Political PartyFrequency, f Democrat34 Republican56 Other21 Did not respond9.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 78 Solution: Finding the Mode Political PartyFrequency, f Democrat34 Republican56 Other21 Did not respond9 The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 79 Comparing the Mean, Median, and Mode All three measures describe a typical entry of a data set. Advantage of using the mean:  The mean is a reliable measure because it takes into account every entry of a data set. Disadvantage of using the mean:  Greatly affected by outliers (a data entry that is far removed from the other entries in the data set)..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 80 Example: Comparing the Mean, Median, and Mode Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers? Ages in a class

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 81 Solution: Comparing the Mean, Median, and Mode Mean: Median: 20 years (the entry occurring with the greatest frequency) Ages in a class Mode:.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 82 Solution: Comparing the Mean, Median, and Mode Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years The mean takes every entry into account, but is influenced by the outlier of 65. The median also takes every entry into account, and it is not affected by the outlier. In this case the mode exists, but it doesn't appear to represent a typical entry..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 83 Solution: Comparing the Mean, Median, and Mode Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set. In this case, it appears that the median best describes the data set..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 84 Weighted Mean The mean of a data set whose entries have varying weights. where w is the weight of each entry x.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 85 Example: Finding a Weighted Mean You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A?.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 86 Solution: Finding a Weighted Mean SourceScore, xWeight, wx∙w Test Mean (0.50)= 43.0 Midterm (0.15) = 14.4 Final Exam (0.20) = 16.4 Computer Lab (0.10) = 9.8 Homework (0.05) = 5.0 Σw = 1Σ(x∙w) = 88.6 Your weighted mean for the course is You did not get an A..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 87 Mean of Grouped Data Mean of a Frequency Distribution Approximated by where x and f are the midpoints and frequencies of a class, respectively.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 88 Finding the Mean of a Frequency Distribution In Words In Symbols 1.Find the midpoint of each class. 2.Find the sum of the products of the midpoints and the frequencies. 3.Find the sum of the frequencies. 4.Find the mean of the frequency distribution..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 89 Example: Find the Mean of a Frequency Distribution Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session. ClassMidpointFrequency, f 7 – – – – – – –

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 90 Solution: Find the Mean of a Frequency Distribution ClassMidpoint, xFrequency, f(x∙f) 7 – ∙6 = – ∙10 = – ∙13 = – ∙8 = – ∙5 = – ∙6 = – ∙2 = n = 50Σ(x∙f) =

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 91 The Shape of Distributions Symmetric Distribution A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 92 The Shape of Distributions Uniform Distribution (rectangular) All entries or classes in the distribution have equal or approximately equal frequencies. Symmetric..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 93 The Shape of Distributions Skewed Left Distribution (negatively skewed) The “tail” of the graph elongates more to the left. The mean is to the left of the median..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 94 The Shape of Distributions Skewed Right Distribution (positively skewed) The “tail” of the graph elongates more to the right. The mean is to the right of the median..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 95 Section 2.3 Summary Found the mean, median, and mode of a population and of a sample Found the weighted mean of a data set and the mean of a frequency distribution Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 96 Section 2.4 Measures of Variation.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 97 Section 2.4 Objectives How to find the range of a data set How to find the variance and standard deviation of a population and of a sample How to use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation How to approximate the sample standard deviation for grouped data How to use the coefficient of variation to compare variation in different data sets.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 98 Range The difference between the maximum and minimum data entries in the set. The data must be quantitative. Range = (Max. data entry) – (Min. data entry).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 99 Example: Finding the Range A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries. Starting salaries (1000s of dollars)

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 100 Solution: Finding the Range Ordering the data helps to find the least and greatest salaries Range = (Max. salary) – (Min. salary) = 47 – 37 = 10 The range of starting salaries is 10 or $10,000. minimum maximum.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 101 Deviation, Variance, and Standard Deviation Deviation The difference between the data entry, x, and the mean of the data set. Population data set:  Deviation of x = x – μ Sample data set:  Deviation of x = x – x.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 102 Example: Finding the Deviation A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the deviation of the starting salaries. Starting salaries (1000s of dollars) Solution: First determine the mean starting salary..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 103 Solution: Finding the Deviation Determine the deviation for each data entry. Salary ($1000s), xDeviation: x – μ 4141 – 41.5 = – – 41.5 = – – 41.5 = – – 41.5 = – 41.5 = – 41.5 = – – 41.5 = – 41.5 = – – 41.5 = – – 41.5 = 0.5 Σx = 415 Σ(x – μ) = 0.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 104 Deviation, Variance, and Standard Deviation Population Variance Population Standard Deviation Sum of squares, SS x.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 105 Finding the Population Variance & Standard Deviation In Words In Symbols 1.Find the mean of the population data set. 2.Find deviation of each entry. 3.Square each deviation. 4.Add to get the sum of squares. x – μ (x – μ) 2 SS x = Σ(x – μ) 2.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 106 Finding the Population Variance & Standard Deviation 5.Divide by N to get the population variance. 6.Find the square root to get the population standard deviation. In Words In Symbols.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 107 Example: Finding the Population Standard Deviation A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the population variance and standard deviation of the starting salaries. Starting salaries (1000s of dollars) Recall μ =

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 108 Solution: Finding the Population Standard Deviation Determine SS x N = 10 Salary, xDeviation: x – μSquares: (x – μ) – 41.5 = –0.5(–0.5) 2 = – 41.5 = –3.5(–3.5) 2 = – 41.5 = –2.5(–2.5) 2 = – 41.5 = 3.5(3.5) 2 = – 41.5 = 5.5(5.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = 2.5(2.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = –4.5(–4.5) 2 = – 41.5 = 0.5(0.5) 2 = 0.25 Σ(x – μ) = 0 SS x = 88.5.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 109 Solution: Finding the Population Standard Deviation Population Variance Population Standard Deviation The population standard deviation is about 3.0, or $3000..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 110 Deviation, Variance, and Standard Deviation Sample Variance Sample Standard Deviation.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 111 Finding the Sample Variance & Standard Deviation In Words In Symbols 1.Find the mean of the sample data set. 2.Find deviation of each entry. 3.Square each deviation. 4.Add to get the sum of squares..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 112 Finding the Sample Variance & Standard Deviation 5.Divide by n – 1 to get the sample variance. 6.Find the square root to get the sample standard deviation. In Words In Symbols.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 113 Example: Finding the Sample Standard Deviation The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries. Starting salaries (1000s of dollars)

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 114 Solution: Finding the Sample Standard Deviation Determine SS x n = 10 Salary, xDeviation: x – μSquares: (x – μ) – 41.5 = –0.5(–0.5) 2 = – 41.5 = –3.5(–3.5) 2 = – 41.5 = –2.5(–2.5) 2 = – 41.5 = 3.5(3.5) 2 = – 41.5 = 5.5(5.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = 2.5(2.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = –4.5(–4.5) 2 = – 41.5 = 0.5(0.5) 2 = 0.25 Σ(x – μ) = 0 SS x = 88.5.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 115 Solution: Finding the Sample Standard Deviation Sample Variance Sample Standard Deviation The sample standard deviation is about 3.1, or $3100..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 116 Example: Using Technology to Find the Standard Deviation Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.) Office Rental Rates

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 117 Solution: Using Technology to Find the Standard Deviation Sample Mean Sample Standard Deviation.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 118 Interpreting Standard Deviation Standard deviation is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 119 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: About 68% of the data lie within one standard deviation of the mean. About 95% of the data lie within two standard deviations of the mean. About 99.7% of the data lie within three standard deviations of the mean..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 120 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) 68% within 1 standard deviation 34% 99.7% within 3 standard deviations 2.35% 95% within 2 standard deviations 13.5%.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 121 Example: Using the Empirical Rule In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64.3 inches, with a sample standard deviation of 2.62 inches. Estimate the percent of the women whose heights are between inches and 64.3 inches..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 122 Solution: Using the Empirical Rule Because the distribution is bell-shaped, you can use the Empirical Rule. 34% % = 47.5% of women are between and 64.3 inches tall..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 123 Chebychev’s Theorem The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: k = 2: In any data set, at least of the data lie within 2 standard deviations of the mean. k = 3: In any data set, at least of the data lie within 3 standard deviations of the mean..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 124 Example: Using Chebychev’s Theorem The age distribution for Florida is shown in the histogram. Apply Chebychev’s Theorem to the data using k = 2. What can you conclude?.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 125 Solution: Using Chebychev’s Theorem k = 2: μ – 2σ = 39.2 – 2(24.8) = (use 0 since age can’t be negative) μ + 2σ = (24.8) = 88.8 At least 75% of the population of Florida is between 0 and 88.8 years old..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 126 Standard Deviation for Grouped Data Sample standard deviation for a frequency distribution When a frequency distribution has classes, estimate the sample mean and standard deviation by using the midpoint of each class. where n= Σf (the number of entries in the data set).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 127 Example: Finding the Standard Deviation for Grouped Data You collect a random sample of the number of children per household in a region. Find the sample mean and the sample standard deviation of the data set. Number of Children in 50 Households

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 128 xfxf 0100(10) = (19) = (7) = (7) =21 424(2) = 8 515(1) = 5 646(4) = 24 Solution: Finding the Standard Deviation for Grouped Data First construct a frequency distribution. Find the mean of the frequency distribution. Σf = 50 Σ(xf )= 91 The sample mean is about 1.8 children..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 129 Solution: Finding the Standard Deviation for Grouped Data Determine the sum of squares. xf 0100 – 1.8 = –1.8(–1.8) 2 = (10) = – 1.8 = –0.8(–0.8) 2 = (19) = – 1.8 = 0.2(0.2) 2 = (7) = – 1.8 = 1.2(1.2) 2 = (7) = – 1.8 = 2.2(2.2) 2 = (2) = – 1.8 = 3.2(3.2) 2 = (1) = – 1.8 = 4.2(4.2) 2 = (4) =

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 130 Solution: Finding the Standard Deviation for Grouped Data Find the sample standard deviation. The standard deviation is about 1.7 children..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 131 Coefficient of Variation Coefficient of Variation (CV) Describes the standard deviation of a data set as a percent of the mean. Population data set:  Sample data set: .

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 132 Example: Comparing Variation in Different Data Sets The table shows the population heights (in inches) and weights (in pounds) of the members of a basketball team. Find the coefficient of variation for the heights and the weighs. Then compare the results..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 133 Solution: Comparing Variation in Different Data Sets The mean height is   72.8 inches with a standard deviation of   3.3 inches. The coefficient of variation for the heights is.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 134 Solution: Comparing Variation in Different Data Sets The mean weight is   pounds with a standard deviation of   17.7 pounds. The coefficient of variation for the weights is. The weights (9.4%) are more variable than the heights (4.5%).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 135 Section 2.4 Summary Found the range of a data set Found the variance and standard deviation of a population and of a sample Used the Empirical Rule and Chebychev’s Theorem to interpret standard deviation Approximated the sample standard deviation for grouped data Used the coefficient of variation to compare variation in different data sets.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 136 Section 2.5 Measures of Position.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 137 Section 2.5 Objectives How to find the first, second, and third quartiles of a data set, how to find the interquartile range of a data set, and how to represent a data set graphically using a box-and whisker plot How to interpret other fractiles such as percentiles and how to find percentiles for a specific data entry Determine and interpret the standard score (z-score).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 138 Quartiles Fractiles are numbers that partition (divide) an ordered data set into equal parts. Quartiles approximately divide an ordered data set into four equal parts.  First quartile, Q 1 : About one quarter of the data fall on or below Q 1.  Second quartile, Q 2 : About one half of the data fall on or below Q 2 (median).  Third quartile, Q 3 : About three quarters of the data fall on or below Q 3..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 139 Example: Finding Quartiles The number of nuclear power plants in the top 15 nuclear power-producing countries in the world are listed. Find the first, second, and third quartiles of the data set Solution: Q 2 divides the data set into two halves Q2Q2 Lower half Upper half.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 140 Solution: Finding Quartiles The first and third quartiles are the medians of the lower and upper halves of the data set Q2Q2 Lower half Upper half Q1Q1 Q3Q3 About one fourth of the countries have 10 or less, about one half have 18 or less; and about three fourths have 31 or less..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 141 Interquartile Range Interquartile Range (IQR) The difference between the third and first quartiles. IQR = Q 3 – Q 1.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 142 Example: Finding the Interquartile Range Find the interquartile range of the data set. Recall Q 1 = 10, Q 2 = 18, and Q 3 = 31 Solution: IQR = Q 3 – Q 1 = 31 – 10 = 21 The number of power plants in the middle portion of the data set vary by at most 21..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 143 Box-and-Whisker Plot Box-and-whisker plot Exploratory data analysis tool. Highlights important features of a data set. Requires (five-number summary):  Minimum entry  First quartile Q 1  Median Q 2  Third quartile Q 3  Maximum entry.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 144 Drawing a Box-and-Whisker Plot 1.Find the five-number summary of the data set. 2.Construct a horizontal scale that spans the range of the data. 3.Plot the five numbers above the horizontal scale. 4.Draw a box above the horizontal scale from Q 1 to Q 3 and draw a vertical line in the box at Q 2. 5.Draw whiskers from the box to the minimum and maximum entries. Whisker Maximum entry Minimum entry Box Median, Q 2 Q3Q3 Q1Q1.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 145 Example: Drawing a Box-and-Whisker Plot Draw a box-and-whisker plot that represents the 15 data set. Min = 6, Q 1 = 10, Q 2 = 18, Q 3 = 31, Max = 104, Solution: About half the scores are between 10 and 31. By looking at the length of the right whisker, you can conclude 104 is a possible outlier..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 146 Percentiles and Other Fractiles FractilesSummarySymbols QuartilesDivides data into 4 equal parts Q 1, Q 2, Q 3 DecilesDivides data into 10 equal parts D 1, D 2, D 3,…, D 9 PercentilesDivides data into 100 equal parts P 1, P 2, P 3,…, P 99.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 147 Example: Interpreting Percentiles The ogive represents the cumulative frequency distribution for SAT test scores of college-bound students in a recent year. What test score represents the 62 nd percentile? How should you interpret this? (Source: College Board).

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 148 Solution: Interpreting Percentiles The 62 nd percentile corresponds to a test score of This means that 62% of the students had an SAT score of 1600 or less..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 149 The Standard Score Standard Score (z-score) Represents the number of standard deviations a given value x falls from the mean μ..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 150 Example: Comparing z-Scores from Different Data Sets In 2009, Heath Ledger won the Oscar for Best Supporting Actor at age 29 for his role in the movie The Dark Knight. Penelope Cruz won the Oscar for Best Supporting Actress at age 34 for her role in Vicky Cristina Barcelona. The mean age of all Best Supporting Actor winners is 49.5, with a standard deviation of The mean age of all Best Supporting Actress winners is 39.9, with a standard deviation of Find the z-score that corresponds to the ages of Ledger and Cruz. Then compare your results..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 151 Solution: Comparing z-Scores from Different Data Sets Heath Ledger Penelope Cruz 1.49 standard deviations above the mean 0.42 standard deviations below the mean.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 152 Solution: Comparing z-Scores from Different Data Sets Both z-scores fall between  2 and 2, so neither score would be considered unusual. Compared with other Best Supporting Actor winners, Heath Ledger was relatively younger, whereas the age of Penelope Cruz was only slightly lower than the average age of other Best Supporting Actress winners..

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 153 Section 2.5 Summary Found the first, second, and third quartiles of a data set, how to find the interquartile range of a data set, and represented a data set graphically using a box-and whisker plot Interpreted other fractiles such as percentiles and percentiles for a specific data entry Determined and interpreted the standard score (z-score).