Chapter 4 Measures of Central Tendency Measures of Variation Measures of Position Dot Plots Stem-and-Leaf Histograms

Median – A Measure of Center 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.

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 © 2012 Pearson Education, Inc. All rights reserved.

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. © 2012 Pearson Education, Inc. All rights reserved.

Example: Finding the Median The flight priced at $432 is no longer available. What is the median price of the remaining flights? © 2012 Pearson Education, Inc. All rights reserved.

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. © 2012 Pearson Education, Inc. All rights reserved.

CW 1 – 4: Median Separate Paper to Turn In Today! Find the median for each set of data 1)1, 2, 3, 4, 5, 6, 7 2)10, 11, 12, 13, 14, 15 3)1, 1, 2, 2, 3, 3, 4, 4 4)10, 11, 12, 13, 14, 15, 16

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) © 2012 Pearson Education, Inc. All rights reserved.

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) © 2012 Pearson Education, Inc. All rights reserved.

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 © 2012 Pearson Education, Inc. All rights reserved.

CW 5 – 8 : Range Turn In Today! Find the range for each set of data 5)1, 2, 3, 4, 5, 6, 7 6)10, 11, 12, 13, 14, 15 7)1, 1, 2, 2, 3, 3, 4, 4 8)10, 11, 12, 13, 14, 15, 16

Quartiles – Measure of Spread 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. © 2012 Pearson Education, Inc. All rights reserved.

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 © 2012 Pearson Education, Inc. All rights reserved.

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 fewer nuclear power plants; about one half have 18 or fewer; and about three fourths have 31 or fewer. © 2012 Pearson Education, Inc. All rights reserved.

CW 9 – 12: Quartiles Turn in Today! Find the 1 st and 3 rd Quartile for each set of data 9)1, 2, 3, 4, 5, 6, 7 10)10, 11, 12, 13, 14, 15 11)1, 1, 2, 2, 3, 3, 4, 4 12)10, 11, 12, 13, 14, 15, 16

Interquartile Range Interquartile Range (IQR) The difference between the third and first quartiles. IQR = Q 3 – Q 1 © 2012 Pearson Education, Inc. All rights reserved.

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. © 2012 Pearson Education, Inc. All rights reserved.

CW : IQR Turn In Today! Find the IQR for each set of data 13)1, 2, 3, 4, 5, 6, 7 14)10, 11, 12, 13, 14, 15 15)1, 1, 2, 2, 3, 3, 4, 4 16)10, 11, 12, 13, 14, 15, 16

Measures of Center & Spread (Option A) Median and IQR Based on POSITION of data Appropriate for any data

5 Number Summary  Min Q1Q1  Median Q3Q3  Max  Q 3  Median  Q 1  Min

Box-and-Whisker Plot Exploratory data analysis tool. Highlights important features of a data set. Requires (five-number summary) © 2012 Pearson Education, Inc. All rights reserved.

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 © 2012 Pearson Education, Inc. All rights reserved.

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

CW 17 – 20: Box & Whisker Turn In Today! Create the Box & Whisker Plot for each set of data 17)1, 2, 3, 4, 5, 6, 7 18)10, 11, 12, 13, 14, 15 19)1, 1, 2, 2, 3, 3, 4, 4 20)10, 11, 12, 13, 14, 15, 16

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: © 2012 Pearson Education, Inc. All rights reserved.

Mean – Notation for this Class & Book Mean (average) Our book uses Sample mean: © 2012 Pearson Education, Inc. All rights reserved.

Example: Finding a Sample Mean The prices (in dollars) for a sample of round-trip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights? © 2012 Pearson Education, Inc. All rights reserved.

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 $ © 2012 Pearson Education, Inc. All rights reserved.

CW Mean Turn In Today! Find the mean for each set of data 21)1, 2, 3, 4, 5, 6, 7 22)10, 11, 12, 13, 14, 15 23)1, 1, 2, 2, 3, 3, 4, 4 24)10, 11, 12, 13, 14, 15, 16

Comparing the Mean and Median Both 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). © 2012 Pearson Education, Inc. All rights reserved.

Example: Comparing the Mean and Median Find the mean and median of the sample ages of a class shown. Which best describes a typical entry of this data set? Are there any outliers? Ages in a class © 2012 Pearson Education, Inc. All rights reserved.

Solution: Comparing the Mean and Median Mean: Median: Ages in a class © 2012 Pearson Education, Inc. All rights reserved.

Solution: Comparing the Mean and Median Mean ≈ 23.8 years Median = 21.5 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. © 2012 Pearson Education, Inc. All rights reserved.

Measures of Center & Spread (Option B) Mean and Standard Deviation Based on Value of data Not as effective when data is strongly skewed or outliers are present

Measures of Center & Spread (Option A) Median and IQR Based on POSITION of data Appropriate for any data

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,

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

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

CW – Dot Plot Separate Paper to Turn In Today! Create the Dot Plot for each set of data 25)1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7 26)10, 11, 12, 12, 12, 13, 14, 14, 15, 17, 18, 19, 20, 20, 11, 12, 14, 16