Chapter 3 Lecture 3 Sections 3.4 – 3.5. Measure of Position We would like to compare values from different data sets. We will introduce a “ z – score”

Slides:



Advertisements
Similar presentations
Statistics: 2.5 – Measures of Position
Advertisements

Chapter 2 Exploring Data with Graphs and Numerical Summaries
Probabilistic & Statistical Techniques
SECTION 3.3 MEASURES OF POSITION Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Section 3-4.
Z-value SamplePopulation The z-value tells us how many standard deviations above or below the mean our data value x is. Positive z-values are above the.
Numerical Representation of Data Part 3 – Measure of Position
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by.
Measures of Relative Standing and Boxplots
Basics of z Scores, Percentiles, Quartiles, and Boxplots 3-4 Measures of Relative Standing.
Section 2.5 Measures of Position.
Section 2.5 Measures of Position Larson/Farber 4th ed.
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series.
M08-Numerical Summaries 2 1  Department of ISM, University of Alabama, Lesson Objectives  Learn what percentiles are and how to calculate quartiles.
1 Measure of Center  Measure of Center the value at the center or middle of a data set 1.Mean 2.Median 3.Mode 4.Midrange (rarely used)
Section 2.5 Measures of Position.
Measures of Position and Outliers. z-score (standard score) = number of standard deviations that a given value is above or below the mean (Round z to.
Section 2.5 Measures of Position Larson/Farber 4th ed. 1.
Section 2.5 Measures of Position Larson/Farber 4th ed. 1.
Slide 1 Statistics Workshop Tutorial 6 Measures of Relative Standing Exploratory Data Analysis.
Chapter 3 Numerically Summarizing Data 3.4 Measures of Location.
Chapter 6 1. Chebychev’s Theorem The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: 2 k = 2: In any data.
Section 3.3 Measures of Relative Position HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems,
Measures of Relative Standing Percentiles Percentiles z-scores z-scores T-scores T-scores.
 z – Score  Percentiles  Quartiles  A standardized value  A number of standard deviations a given value, x, is above or below the mean  z = (score.
Section 3.4 Measures of Relative Standing
Lecture 16 Sec – Mon, Oct 2, 2006 Measuring Variation – The Five-Number Summary.
1 Measure of Center  Measure of Center the value at the center or middle of a data set 1.Mean 2.Median 3.Mode 4.Midrange (rarely used)
Measures of Variability Percentile Rank. Comparison of averages is not enough. Consider a class with the following marks 80%, 80%, 80%, 90%, 20%, 70%,
Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Descriptive Statistics 2.
Chapter 2 Section 5 Notes Coach Bridges
1 Measures of Center. 2 Measure of Center  Measure of Center the value at the center or middle of a data set 1.Mean 2.Median 3.Mode 4.Midrange (rarely.
Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 4 – Slide 1 of 23 Chapter 3 Section 4 Measures of Position.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by.
Measures of Position. Determine the quartiles of a data set Determine the interquartile range of a data set Create a box-and-whisker plot Interpret.
Descriptive Statistics Chapter 2. § 2.5 Measures of Position.
Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Numerically Summarizing Data 3.
Measures of Position Section 3-3.
Section 3-4 Measures of Relative Standing and Boxplots.
Foundations of Math I: Unit 3 - Statistics Arithmetic average Median: Middle of the data listed in ascending order (use if there is an outlier) Mode: Most.
Section 2.5 Measures of Position.
Chapter 3.3 – 3.4 Applications of the Standard Deviation and Measures of Relative Standing.
The Normal Distributions.  1. Always plot your data ◦ Usually a histogram or stemplot  2. Look for the overall pattern ◦ Shape, center, spread, deviations.
Descriptive Statistics Chapter 2. § 2.5 Measures of Position.
CHAPTER 4 NUMERICAL METHODS FOR DESCRIBING DATA What trends can be determined from individual data sets?
Measures of Relative Standing and Boxplots
Measures of Position – Quartiles and Percentiles
Chapter 2 Descriptive Statistics.
Relative Standing and Boxplots
Measures of Position Section 2-6
Lecture Slides Elementary Statistics Twelfth Edition
Elementary Statistics
STATISTICS ELEMENTARY MARIO F. TRIOLA Section 2-6 Measures of Position
Lecture Slides Essentials of Statistics 5th Edition
Lecture Slides Elementary Statistics Twelfth Edition
Lecture Slides Elementary Statistics Twelfth Edition
Measures of Position.
The lengths (in minutes) of a sample of cell phone calls are shown:
Five Number Summary and Box Plots
Chapter 2 Descriptive Statistics.
Measuring Variation – The Five-Number Summary
Descriptive Statistics
Section 3:3 Answers page #4-14 even
Statistics and Data (Algebraic)
Measures of Position Section 3.3.
Five Number Summary and Box Plots
Statistics Review MGF 1106 Fall 2011
Measures of Relative Standing
Lecture Slides Elementary Statistics Eleventh Edition
Box Plot Lesson 11-4.
Presentation transcript:

Chapter 3 Lecture 3 Sections 3.4 – 3.5

Measure of Position We would like to compare values from different data sets. We will introduce a “ z – score” or “standard score”. This measures how many standard deviation from the mean a given number x is. We use the following: At UCLA in a specific quarter, I took two classes that were graded on a curve. In Math, the class had a mean of 80 and standard deviation of 11. In Economics, the class mean was 46 with a standard deviation of 5. I received a grade 90 in Math and a grade of 54 in Economics. Does my grade in Math among the class exceed my grade in Econ among the class? If the value of x is smaller than the mean, then z will be negative.

Math Standard Score Econ Standard Score This means that my score in my Economics class is relatively higher when compared to the class than that of my Math class. Ordinary Values: –2 ≤ z-score ≤ 2, Unusual Values: z-score 2 How many standard deviation is a score of 30 in the economics class?

Example: Men have heights with a mean of 69.0in. and a standard deviation of 2.8in.; women have heights with a mean of 63.6 with a standard deviation of 2.5in. If a man is 74in. tall and a women is 70in. tall, who is relatively taller?

Percentiles Recall that the median of 56, 66, 70, 77, 80, 86, 99 was 77. This means is that 50% of the values are equal to or less than the median and 50% of the values are equal to or greater than the median. In other words, it separates the top 50% form the bottom 50%. We are also able to fine other values that separate data. After arranging the data in increasing order: Q 1 =First Quartile: Separates the bottom 25% from the top 75%. This is the same as the P 25 =25 th Percentile. Q 2 =Second Quartile: Same as the median. This is the same as the P 50 =50 th Percentile. Q 3 =Third Quartile: Separates the bottom 75% from the top 25%. This is the same as the P 75 =75 th Percentile.

There are other ways to separate the data. P 1 =First Percentile: Separates the bottom 1% from the top 99%. P 10 =Tenth Percentile: Separates the bottom 10% from the top 90%. This is the same as the D 1 =1 st Decile. P 20 =20 th Percentile: Separates the bottom 20% from the top 80%. This is the same as the D 2 =2 nd Decile. P 66 =66 th Percentile: Separates the bottom 66% from the top 34%. P 95 =95 th Percentile: Separates the bottom 95% from the top 5%. P k =k th Percentile: Separates the bottom k% from the top (100-k)%. This is the general form of percentiles. Just to name a few.

Finding Percentiles Finding a percentile that corresponds to a particular value “x” of the data set is as follows: Example: The following data represents the final 50 percentages of last semesters Algebra class arranged in increasing order

Find the percentile that corresponds to the value of 22. This tells us that 22 is the 10 th percentile (P 10 = D 1 ). We conclude that 10% of the students are below or equal to 22 and 90% of the class is above or equal to. If a student received a score of 78, what percentile does the student fall in? This tells us that 78 is the 76 th percentile (P 76 ). We conclude that 76% of the students are below or equal to 78 and 24% of the class is above or equal to 78.

Lets find the value of the 70 th percentile (P 70 ). We will need to use the following formula. Where k is the percentile, n is the total # of values, and L is the Locator that tells us where the value we are looking for is. Since L = is a whole number, what we have to do is get the 35 th value and the 36 th value and compute their average.

If L is a whole number, you must get that number and the number that comes after it, then compute their average. If L is a decimal number, round up and with that number you will find P k. Remember, you will have to order the data in increasing order first. Example: Find the 3 rd quartile of the data Q 3 =P 75 =63

Graphs using Percentiles. Boxplot: Consists of a 5 number summary that is made up of the minimum, Q 1, Q 2, Q 3, and the maximum. minimum Q1Q1 Q2Q2 Q3Q3 maximum Statistic defined by using Quartiles. Interquartile Range (IQR): Q 3 ─ Q 1

Minitab Printout: Descriptive Statistics: Final Percentage Variable N Mean Median TrMean StDev SE Mean Final % Variable Minimum Maximum Q1 Q3 Final %

Example: As an incentive to attract additional customers, a Caribbean hotel recently installed a toll-free 800 phone number. During the first three weeks of its operation, the hotel received the following number of requests for additional information on a daily basis: Construct a boxplot with a 5-number summary to represent this data.

Example: In “Age of Oscar winning Best Actors and Actresses” by Richard Brown, the author compares the ages of actors and actresses at the time that they won their Oscar. The results for winners from both categories are listed bellow. Use a boxplot to compare their ages. Male: Female: