Algebra II Descriptive Statistics 1 Larson/Farber 4th ed.

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 Larson/Farber 4th ed. 2

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. Larson/Farber 4th ed. 3 data values frequency

Solution: Frequency Histogram Larson/Farber 4th ed. 4

Frequency Histogram Larson/Farber 4th ed You can see that more than half of the subscribers spent between 19 and 54 minutes on the Internet during their most recent session.

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. Larson/Farber 4th ed. 6 Data: 21, 25, 25, 26, 27, 28, 30, 36, 36,

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. Larson/Farber 4th ed

Solution: Constructing a Stem-and-Leaf Plot Larson/Farber 4th ed. 8 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

Solution: Constructing a Stem-and-Leaf Plot Larson/Farber 4th ed. 9 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.

Graphing Quantitative Data Sets Dot plot Each data entry is plotted, using a point, above a horizontal axis Larson/Farber 4th ed. 10 Data: 21, 25, 25, 26, 27, 28, 30, 36, 36,

Example: Constructing a Dot Plot Use a dot plot organize the text messaging data. Larson/Farber 4th ed. 11 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

Solution: Constructing a Dot Plot Larson/Farber 4th ed. 12 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

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. Larson/Farber 4th ed. 13 x y

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) Larson/Farber 4th ed. 14

Example: Interpreting a Scatter Plot As the petal length increases, what tends to happen to the petal width? Larson/Farber 4th ed. 15 Each point in the scatter plot represents the petal length and petal width of one flower.

Solution: Interpreting a Scatter Plot Larson/Farber 4th ed. 16 Interpretation From the scatter plot, you can see that as the petal length increases, the petal width also tends to increase.

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 Larson/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 Larson/Farber 4th ed. 20 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 $

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. 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 prices Larson/Farber 4th ed. 22

Solution: Finding the Median Larson/Farber 4th ed. 23 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.

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 Larson/Farber 4th ed. 25 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.

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). 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 prices Larson/Farber 4th ed. 27

Solution: Finding the Mode Larson/Farber 4th ed. 28 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.

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? Larson/Farber 4th ed. 29 Political PartyFrequency, f Democrat34 Republican56 Other21 Did not respond9

Solution: Finding the Mode Larson/Farber 4th ed. 30 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.

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). Larson/Farber 4th ed. 31

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? Larson/Farber 4th ed. 32 Ages in a class

Solution: Comparing the Mean, Median, and Mode Larson/Farber 4th ed. 33 Mean: Median: 20 years (the entry occurring with the greatest frequency) Ages in a class Mode:

Solution: Comparing the Mean, Median, and Mode Larson/Farber 4th ed. 34 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.

Solution: Comparing the Mean, Median, and Mode Larson/Farber 4th ed. 35 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.

The Shape of Distributions Larson/Farber 4th ed. 36 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.

The Shape of Distributions Larson/Farber 4th ed. 37 Uniform Distribution (rectangular) All entries or classes in the distribution have equal or approximately equal frequencies. Symmetric.

The Shape of Distributions Larson/Farber 4th ed. 38 Skewed Left Distribution (negatively skewed) The “tail” of the graph elongates more to the left. The mean is to the left of the median.

The Shape of Distributions Larson/Farber 4th ed. 39 Skewed Right Distribution (positively skewed) The “tail” of the graph elongates more to the right. The mean is to the right of the median.

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) Larson/Farber 4th ed. 40

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) Larson/Farber 4th ed. 41

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. Larson/Farber 4th ed. 42 minimum maximum

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.) Larson/Farber 4th ed. 43 Office Rental Rates

Solution: Using Technology to Find the Standard Deviation Larson/Farber 4th ed. 44 Sample Mean Sample Standard Deviation

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. 45

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: Larson/Farber 4th ed. 46 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.

Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) Larson/Farber 4th ed % within 1 standard deviation 34% 99.7% within 3 standard deviations 2.35% 95% within 2 standard deviations 13.5%

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 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and inches. Larson/Farber 4th ed. 48

Solution: Using the Empirical Rule Larson/Farber 4th ed % 13.5% Because the distribution is bell-shaped, you can use the Empirical Rule. 34% % = 47.5% of women are between 64 and inches tall.

Quartiles 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. Larson/Farber 4th ed. 50

Example: Finding Quartiles The test scores of 15 employees enrolled in a CPR training course are listed. Find the first, second, and third quartiles of the test scores Larson/Farber 4th ed. 51 Solution: Q 2 divides the data set into two halves Q2Q2 Lower half Upper half

Solution: Finding Quartiles The first and third quartiles are the medians of the lower and upper halves of the data set Larson/Farber 4th ed. 52 Q2Q2 Lower half Upper half Q1Q1 Q3Q3 About one fourth of the employees scored 10 or less, about one half scored 15 or less; and about three fourths scored 18 or less.

Interquartile Range Interquartile Range (IQR) The difference between the third and first quartiles. IQR = Q 3 – Q 1 Larson/Farber 4th ed. 53

Example: Finding the Interquartile Range Find the interquartile range of the test scores. Recall Q 1 = 10, Q 2 = 15, and Q 3 = 18 Larson/Farber 4th ed. 54 Solution: IQR = Q 3 – Q 1 = 18 – 10 = 8 The test scores in the middle portion of the data set vary by at most 8 points.

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 Larson/Farber 4th ed. 55

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. Larson/Farber 4th ed. 56 Whisker Maximum entry Minimum entry Box Median, Q 2 Q3Q3 Q1Q1

Example: Drawing a Box-and-Whisker Plot Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q 1 = 10 Q 2 = 15 Q 3 = 18 Max = 37 Larson/Farber 4th ed Solution: About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier.