2 Useful screencast/videos: Video on creating a frequency distribution by hand:Video on using Excel 2007 to create frequency distributions:Video on using Excel 2007 to create a histogramLarson/Farber 4th ed.
3 Frequency Distributions and Their Graphs Section 2.1Frequency Distributionsand Their GraphsLarson/Farber 4th ed.
4 Frequency Distribution - Terminology 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, f1 – 556 – 10811 – 15616 – 2021 – 2526 – 304Larson/Farber 4th ed.
5 Determining the Relative Frequency Relative Frequency of a classPortion or percentage of the data that falls in a particular class.ClassFrequency, fRelative Frequency7 – 18619 – 301031 – 4213Larson/Farber 4th ed.
6 Example: Constructing a Frequency Distribution The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes.Video on computing frequency distribution using this data:Larson/Farber 4th ed.
8 Graphs of Frequency Distributions Frequency HistogramA 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 valuesfrequencyLarson/Farber 4th ed.
9 Solution: Frequency Histogram (using Midpoints) Larson/Farber 4th ed.
10 Graphs of Frequency Distributions Relative Frequency HistogramHas the same shape and the same horizontal scale as the corresponding frequency histogram.The vertical scale measures the relative frequencies, not frequencies.data valuesrelative frequencyLarson/Farber 4th ed.
11 Solution: Relative Frequency Histogram From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online.Larson/Farber 4th ed.
12 More Graphs and Displays Section 2.2More Graphs and DisplaysLarson/Farber 4th ed.
13 Graphing Quantitative Data Sets Stem-and-leaf plotEach number is separated into a stem and a leaf.Similar to a histogram.Still contains original data values.262345Data: 21, 25, 25, 26, 27, 28, , 36, 36, 45Larson/Farber 4th ed.
14 Graphing Qualitative Data Sets Pie ChartA circle is divided into sectors that represent categories.The area of each sector is proportional to the frequency of each category.Larson/Farber 4th ed.
15 Measures of Central Tendency Section 2.3Measures of Central TendencyLarson/Farber 4th ed.
16 Measures of Central Tendency Measure of central tendencyA value that represents a typical, or central, entry of a data set.Most common measures of central tendency:MeanMedianModeLarson/Farber 4th ed.
17 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:Larson/Farber 4th ed.
18 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?Larson/Farber 4th ed.
19 Solution: Finding a Sample Mean The sum of the flight prices isΣx = = 3695To find the mean price, divide the sum of the prices by the number of prices in the sampleThe mean price of the flights is about $Larson/Farber 4th ed.
20 Measure of Central Tendency: 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 anodd number of entries: median is the middle data entry.even number of entries: median is the mean of the two middle data entries.Larson/Farber 4th ed.
21 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 pricesLarson/Farber 4th ed.
22 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.Larson/Farber 4th ed.
23 Example: Finding the Median The flight priced at $432 is no longer available. What is the median price of the remaining flights?Larson/Farber 4th ed.
24 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.Larson/Farber 4th ed.
25 Measure of Central Tendency: 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).Larson/Farber 4th ed.
26 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 pricesLarson/Farber 4th ed.
27 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.Larson/Farber 4th ed.
28 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, fDemocrat34Republican56Other21Did not respond9Larson/Farber 4th ed.
29 Solution: Finding the Mode Political PartyFrequency, fDemocrat34Republican56Other21Did not respond9The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation.Larson/Farber 4th ed.
30 Section 2.4Measures of VariationLarson/Farber 4th ed.
31 Deviation, Variance, and Standard Deviation The difference between the data entry, x, and the mean of the data set.Population data set:Deviation of x = x – μSample data set:Deviation of x = x – xLarson/Farber 4th ed.
32 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.Larson/Farber 4th ed.
33 Solution: Finding the Deviation Determine the deviation for each data entry.Salary ($1000s), xDeviation: x – μ4141 – 41.5 = –0.53838 – 41.5 = –3.53939 – 41.5 = –2.54545 – 41.5 = 3.54747 – 41.5 = 5.54444 – 41.5 = 2.53737 – 41.5 = –4.54242 – 41.5 = 0.5Σx = 415Σ(x – μ) = 0Larson/Farber 4th ed.
34 Deviation, Variance, and Standard Deviation Population VariancePopulation Standard DeviationSum of squares, SSxLarson/Farber 4th ed.
35 Deviation, Variance, and Standard Deviation Sample VarianceSample Standard DeviationLarson/Farber 4th ed.
36 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 Rates35.0033.5037.0023.7526.5031.2536.5040.0032.0039.2537.5034.7537.7537.2536.7527.0035.7526.0029.0040.5024.5033.0038.00Larson/Farber 4th ed.
37 Solution: Using Technology to Find the Standard Deviation Sample MeanSample Standard DeviationLarson/Farber 4th ed.
38 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.Larson/Farber 4th ed.
39 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.Larson/Farber 4th ed.
40 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) 99.7% within 3 standard deviations2.35%95% within 2 standard deviations13.5%68% within 1 standard deviation34%Larson/Farber 4th ed.
41 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 inches, with a sample standard deviation of inches. Estimate the percent of the women whose heights are between 64 inches and inches.Larson/Farber 4th ed.
42 Solution: Using the Empirical Rule Because the distribution is bell-shaped, you can use the Empirical Rule.34%13.5%55.8758.5861.296466.7169.4272.1334% % = 47.5% of women are between 64 and inches tall.Larson/Farber 4th ed.
45 Important Formulas Range = Maximum value – Minimum value Population VariancePopulation Standard DeviationSample VarianceSample Standard Deviation
46 Using the Empirical Rule 1. The mean value of homes on a street is $125 thousand with a standard deviation of $5 thousand. The data set has a bell shaped distribution. Estimate the percent of homes between $120 and $135 thousand.125130135120140145115110105$120 thousand is 1 standard deviation belowthe mean and $135 thousand is 2 standarddeviations above the mean.68% % = 81.5%
47 2. An instructor recorded the average number of absences for his students in one semester. For a random sample the data are:Calculate the mean, the median, and the mode, using the appropriate notation. [Hint: is this a sample or a population?]