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This document summarizes key concepts from Chapter 3 of a statistics textbook. It discusses different measures used to describe the central tendency and variation in data, including the mean, median, mode, range, and standard deviation. It provides definitions and formulas for calculating these descriptive statistics, and explains how to interpret and apply them to analyze quantitative data sets.

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Chapter 3 Statistics for Describing, Exploring, and Comparing Data Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Created by Tom Wegleitner, Centreville, Virginia Section 3-1 Review and Preview
Chapter 1 Distinguish between population and sample, parameter and statistic Good sampling methods: simple random sample , collect in appropriate ways Chapter 2 Frequency distribution: summarizing data Graphs designed to help understand data Center, variation, distribution, outliers, changing characteristics over time Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Review
Important Statistics Mean, median, standard deviation, variance Understanding and Interpreting these important statistics Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Preview
Descriptive Statistics In this chapter we’ll learn to summarize or describe the important characteristics of a known set of data Inferential Statistics In later chapters we’ll learn to use sample data to make inferences or generalizations about a population Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Preview
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 3-2 Measures of Center
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept Characteristics of center. Measures of center, including mean and median, as tools for analyzing data. Not only determine the value of each measure of center, but also interpret those values.
Basics Concepts of Measures of Center Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 1
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Measure of Center Measure of Center the value at the center or middle of a data set
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Arithmetic Mean Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values What most people call an average .
Notation  denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of data values in a sample . N represents the number of data values in a population . Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
µ is pronounced ‘mu’ and denotes the mean of all values in a population Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x N µ =  x
Advantages Is relatively reliable, means of samples drawn from the same population don’t vary as much as other measures of center Takes every data value into account Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center
Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Median is not affected by an extreme value - is a resistant measure of the center often denoted by x (pronounced ‘x-tilde’) ~
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Median 1. If the number of data values is odd, the median is the number located in the exact middle of the list. 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers. First sort the values (arrange them in order), the follow one of these
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 5.40 1.10 0.42 0.73 0.48 1.10 0.66 0.42 0.48 0.66 0.73 1.10 1.10 5.40 (in order - odd number of values) exact middle MEDIAN is 0.73 5.40 1.10 0.42 0.73 0.48 1.10 0.42 0.48 0.73 1.10 1.10 5.40 0.73 + 1.10 2 (in order - even number of values – no exact middle shared by two numbers) MEDIAN is 0.915
Mode the value that occurs with the greatest frequency Data set can have one, more than one, or no mode Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mode Mode is the only measure of central tendency that can be used with nominal data Bimodal two data values occur with the same greatest frequency Multimodal more than two data values occur with the same greatest frequency No Mode no data value is repeated
Mode is 1.10 Bimodal - 27 & 55 No Mode Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 Mode - Examples
Midrange the value midway between the maximum and minimum values in the original data set Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definition Midrange = maximum value + minimum value 2
Sensitive to extremes because it uses only the maximum and minimum values, so rarely used Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Midrange Redeeming Features (1) very easy to compute (2) reinforces that there are several ways to define the center (3) Avoids confusion with median
Carry one more decimal place than is present in the original set of values. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-off Rule for Measures of Center
Think about the method used to collect the sample data. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Critical Thinking Think about whether the results are reasonable.
Beyond the Basics of Measures of Center Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 2
Assume that all sample values in each class are equal to the class midpoint. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean from a Frequency Distribution
use class midpoint of classes for variable x Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean from a Frequency Distribution
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Weighted Mean When data values are assigned different weights, we can compute a weighted mean . x = w  ( w • x ) 
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Best Measure of Center
Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewed and Symmetric
Skewed to the left (also called negatively skewed) have a longer left tail, mean and median are to the left of the mode Skewed to the right (also called positively skewed) have a longer right tail, mean and median are to the right of the mode Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewed Left or Right
The mean and median cannot always be used to identify the shape of the distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Shape of the Distribution
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewness
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed: Types of measures of center Mean Median Mode Mean from a frequency distribution Weighted means Best measures of center Skewness
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Section 3-3 Measures of Variation
Key Concept Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Discuss characteristics of variation, in particular, measures of variation, such as standard deviation, for analyzing data. Make understanding and interpreting the standard deviation a priority.
Basics Concepts of Measures of Variation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Part 1
Definition The range of a set of data values is the difference between the maximum data value and the minimum data value. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range = (maximum value) – (minimum value) It is very sensitive to extreme values; therefore not as useful as other measures of variation.
Round-Off Rule for Measures of Variation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation.
Definition Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The standard deviation of a set of sample values, denoted by s , is a measure of variation of values about the mean.
Sample Standard Deviation Formula Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education  ( x – x ) 2 n – 1 s =
Sample Standard Deviation (Shortcut Formula) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education n ( n – 1 ) s = n  x 2 ) – (  x ) 2
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Standard Deviation - Important Properties The standard deviation is a measure of variation of all values from the mean . The value of the standard deviation s is usually positive. The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others). The units of the standard deviation s are the same as the units of the original data values.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Comparing Variation in Different Samples It’s a good practice to compare two sample standard deviations only when the sample means are approximately the same. When comparing variation in samples with very different means, it is better to use the coefficient of variation, which is defined later in this section.
Population Standard Deviation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education 2  ( x – µ ) N  = This formula is similar to the previous formula, but instead, the population mean and population size are used.
Variance Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Population variance:  2 - Square of the population standard deviation  The variance of a set of values is a measure of variation equal to the square of the standard deviation. Sample variance: s 2 - Square of the sample standard deviation s
Unbiased Estimator Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The sample variance s 2 is an unbiased estimator of the population variance  2 , which means values of s 2 tend to target the value of  2 instead of systematically tending to overestimate or underestimate  2 .
Variance - Notation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education s = sample standard deviation s 2 = sample variance  = population standard deviation  2 = population variance
Beyond the Basics of Measures of Variation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Part 2
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range Rule of Thumb is based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within two standard deviations of the mean.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range Rule of Thumb for Interpreting a Known Value of the Standard Deviation Informally define usual values in a data set to be those that are typical and not too extreme. Find rough estimates of the minimum and maximum “usual” sample values as follows: Minimum “usual” value (mean) – 2  (standard deviation) = Maximum “usual” value (mean) + 2  (standard deviation) =
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range Rule of Thumb for Estimating a Value of the Standard Deviation s To roughly estimate the standard deviation from a collection of known sample data use where range = (maximum value) – (minimum value) range 4 s 
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Properties of the Standard Deviation Measures the variation among data values Values close together have a small standard deviation, but values with much more variation have a larger standard deviation Has the same units of measurement as the original data
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Properties of the Standard Deviation For many data sets, a value is unusual if it differs from the mean by more than two standard deviations Compare standard deviations of two different data sets only if the they use the same scale and units, and they have means that are approximately the same
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Empirical (or 68-95-99.7) Rule For data sets having a distribution that is approximately bell shaped , the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The Empirical Rule
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The Empirical Rule
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The Empirical Rule
Chebyshev’s Theorem Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1–1/ K 2 , where K is any positive number greater than 1. For K = 2, at least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. For K = 3, at least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Rationale for using n – 1 versus n There are only n – 1 independent values. With a given mean, only n – 1 values can be freely assigned any number before the last value is determined. Dividing by n – 1 yields better results than dividing by n . It causes s 2 to target  2 whereas division by n causes s 2 to underestimate  2 .
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Coefficient of Variation The coefficient of variation (or CV ) for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean. Sample Population s x CV =  100%  CV =   100%
Recap Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education In this section we have looked at: Range Standard deviation of a sample and population Variance of a sample and population Coefficient of variation (CV) Range rule of thumb Empirical distribution Chebyshev’s theorem
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 3-4 Measures of Relative Standing and Boxplots
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept This section introduces measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set. They can be used to compare values from different data sets, or to compare values within the same data set. The most important concept is the z score . We will also discuss percentiles and quartiles, as well as a new statistical graph called the boxplot.
Basics of z Scores, Percentiles, Quartiles, and Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 1
z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Z score
Sample Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Population Round z scores to 2 decimal places Measures of Position z Score x – µ z =  z = x – x s
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score < –2 or z score > 2
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Percentiles are measures of location. There are 99 percentiles denoted P 1 , P 2 , . . . P 99 , which divide a set of data into 100 groups with about 1% of the values in each group.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Percentile of a Data Value Percentile of value x = • 100 number of values less than x total number of values
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. n total number of values in the data set k percentile being used L locator that gives the position of a value P k k th percentile Notation Converting from the k th Percentile to the Corresponding Data Value L = • n k 100
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Converting from the k th Percentile to the Corresponding Data Value
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Quartiles Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. Are measures of location, denoted Q 1 , Q 2 , and Q 3 , which divide a set of data into four groups with about 25% of the values in each group.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Q 1 , Q 2 , Q 3 divide ranked scores into four equal parts Quartiles 25% 25% 25% 25% Q 3 Q 2 Q 1 (minimum) (maximum) (median)
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interquartile Range (or IQR): Q 3 – Q 1 10 - 90 Percentile Range: P 90 – P 10 Some Other Statistics Semi-interquartile Range: 2 Q 3 – Q 1 Midquartile: 2 Q 3 + Q 1
For a set of data, the 5-number summary consists of the minimum value; the first quartile Q 1 ; the median (or second quartile Q 2 ); the third quartile, Q 3 ; and the maximum value. 5-Number Summary Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
A boxplot (or box-and-whisker-diagram ) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q 1 ; the median; and the third quartile, Q 3. Boxplot Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplot of Movie Budget Amounts
Boxplots - Normal Distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Normal Distribution: Heights from a Simple Random Sample of Women
Boxplots - Skewed Distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches
Outliers and Modified Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 2
Outliers An outlier is a value that lies very far away from the vast majority of the other values in a data set. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Important Principles An outlier can have a dramatic effect on the mean. An outlier can have a dramatic effect on the standard deviation. An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Outliers for Modified Boxplots For purposes of constructing modified boxplots , we can consider outliers to be data values meeting specific criteria. In modified boxplots, a data value is an outlier if it is . . . above Q 3 by an amount greater than 1.5  IQR below Q 1 by an amount greater than 1.5  IQR or
Modified Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplots described earlier are called skeletal (or regular ) boxplots. Some statistical packages provide modified boxplots which represent outliers as special points.
Modified Boxplot Construction Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. A special symbol (such as an asterisk) is used to identify outliers. The solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier. A modified boxplot is constructed with these specifications:
Modified Boxplots - Example Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Pulse rates of females listed in Data Set 1 in Appendix B.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed: z Scores z Scores and unusual values Quartiles Percentiles Converting a percentile to corresponding data values Other statistics Effects of outliers 5-number summary Boxplots and modified boxplots
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Putting It All Together Always consider certain key factors: Context of the data Source of the data Measures of Center Sampling Method Measures of Variation Outliers Practical Implications Changing patterns over time Conclusions Distribution

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Triola 11 chapter 3

  • 1. Chapter 3 Statistics for Describing, Exploring, and Comparing Data Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots
  • 2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Created by Tom Wegleitner, Centreville, Virginia Section 3-1 Review and Preview
  • 3. Chapter 1 Distinguish between population and sample, parameter and statistic Good sampling methods: simple random sample , collect in appropriate ways Chapter 2 Frequency distribution: summarizing data Graphs designed to help understand data Center, variation, distribution, outliers, changing characteristics over time Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Review
  • 4. Important Statistics Mean, median, standard deviation, variance Understanding and Interpreting these important statistics Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Preview
  • 5. Descriptive Statistics In this chapter we’ll learn to summarize or describe the important characteristics of a known set of data Inferential Statistics In later chapters we’ll learn to use sample data to make inferences or generalizations about a population Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Preview
  • 6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 3-2 Measures of Center
  • 7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept Characteristics of center. Measures of center, including mean and median, as tools for analyzing data. Not only determine the value of each measure of center, but also interpret those values.
  • 8. Basics Concepts of Measures of Center Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 1
  • 9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Measure of Center Measure of Center the value at the center or middle of a data set
  • 10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Arithmetic Mean Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values What most people call an average .
  • 11. Notation  denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of data values in a sample . N represents the number of data values in a population . Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
  • 12. µ is pronounced ‘mu’ and denotes the mean of all values in a population Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x N µ =  x
  • 13. Advantages Is relatively reliable, means of samples drawn from the same population don’t vary as much as other measures of center Takes every data value into account Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center
  • 14. Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Median is not affected by an extreme value - is a resistant measure of the center often denoted by x (pronounced ‘x-tilde’) ~
  • 15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Median 1. If the number of data values is odd, the median is the number located in the exact middle of the list. 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers. First sort the values (arrange them in order), the follow one of these
  • 16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 5.40 1.10 0.42 0.73 0.48 1.10 0.66 0.42 0.48 0.66 0.73 1.10 1.10 5.40 (in order - odd number of values) exact middle MEDIAN is 0.73 5.40 1.10 0.42 0.73 0.48 1.10 0.42 0.48 0.73 1.10 1.10 5.40 0.73 + 1.10 2 (in order - even number of values – no exact middle shared by two numbers) MEDIAN is 0.915
  • 17. Mode the value that occurs with the greatest frequency Data set can have one, more than one, or no mode Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mode Mode is the only measure of central tendency that can be used with nominal data Bimodal two data values occur with the same greatest frequency Multimodal more than two data values occur with the same greatest frequency No Mode no data value is repeated
  • 18. Mode is 1.10 Bimodal - 27 & 55 No Mode Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 Mode - Examples
  • 19. Midrange the value midway between the maximum and minimum values in the original data set Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definition Midrange = maximum value + minimum value 2
  • 20. Sensitive to extremes because it uses only the maximum and minimum values, so rarely used Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Midrange Redeeming Features (1) very easy to compute (2) reinforces that there are several ways to define the center (3) Avoids confusion with median
  • 21. Carry one more decimal place than is present in the original set of values. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-off Rule for Measures of Center
  • 22. Think about the method used to collect the sample data. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Critical Thinking Think about whether the results are reasonable.
  • 23. Beyond the Basics of Measures of Center Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 2
  • 24. Assume that all sample values in each class are equal to the class midpoint. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean from a Frequency Distribution
  • 25. use class midpoint of classes for variable x Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean from a Frequency Distribution
  • 26. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Weighted Mean When data values are assigned different weights, we can compute a weighted mean . x = w  ( w • x ) 
  • 27. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Best Measure of Center
  • 28. Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewed and Symmetric
  • 29. Skewed to the left (also called negatively skewed) have a longer left tail, mean and median are to the left of the mode Skewed to the right (also called positively skewed) have a longer right tail, mean and median are to the right of the mode Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewed Left or Right
  • 30. The mean and median cannot always be used to identify the shape of the distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Shape of the Distribution
  • 31. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewness
  • 32. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed: Types of measures of center Mean Median Mode Mean from a frequency distribution Weighted means Best measures of center Skewness
  • 33. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Section 3-3 Measures of Variation
  • 34. Key Concept Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Discuss characteristics of variation, in particular, measures of variation, such as standard deviation, for analyzing data. Make understanding and interpreting the standard deviation a priority.
  • 35. Basics Concepts of Measures of Variation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Part 1
  • 36. Definition The range of a set of data values is the difference between the maximum data value and the minimum data value. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range = (maximum value) – (minimum value) It is very sensitive to extreme values; therefore not as useful as other measures of variation.
  • 37. Round-Off Rule for Measures of Variation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation.
  • 38. Definition Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The standard deviation of a set of sample values, denoted by s , is a measure of variation of values about the mean.
  • 39. Sample Standard Deviation Formula Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education  ( x – x ) 2 n – 1 s =
  • 40. Sample Standard Deviation (Shortcut Formula) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education n ( n – 1 ) s = n  x 2 ) – (  x ) 2
  • 41. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Standard Deviation - Important Properties The standard deviation is a measure of variation of all values from the mean . The value of the standard deviation s is usually positive. The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others). The units of the standard deviation s are the same as the units of the original data values.
  • 42. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Comparing Variation in Different Samples It’s a good practice to compare two sample standard deviations only when the sample means are approximately the same. When comparing variation in samples with very different means, it is better to use the coefficient of variation, which is defined later in this section.
  • 43. Population Standard Deviation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education 2  ( x – µ ) N  = This formula is similar to the previous formula, but instead, the population mean and population size are used.
  • 44. Variance Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Population variance:  2 - Square of the population standard deviation  The variance of a set of values is a measure of variation equal to the square of the standard deviation. Sample variance: s 2 - Square of the sample standard deviation s
  • 45. Unbiased Estimator Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The sample variance s 2 is an unbiased estimator of the population variance  2 , which means values of s 2 tend to target the value of  2 instead of systematically tending to overestimate or underestimate  2 .
  • 46. Variance - Notation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education s = sample standard deviation s 2 = sample variance  = population standard deviation  2 = population variance
  • 47. Beyond the Basics of Measures of Variation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Part 2
  • 48. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range Rule of Thumb is based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within two standard deviations of the mean.
  • 49. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range Rule of Thumb for Interpreting a Known Value of the Standard Deviation Informally define usual values in a data set to be those that are typical and not too extreme. Find rough estimates of the minimum and maximum “usual” sample values as follows: Minimum “usual” value (mean) – 2  (standard deviation) = Maximum “usual” value (mean) + 2  (standard deviation) =
  • 50. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Range Rule of Thumb for Estimating a Value of the Standard Deviation s To roughly estimate the standard deviation from a collection of known sample data use where range = (maximum value) – (minimum value) range 4 s 
  • 51. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Properties of the Standard Deviation Measures the variation among data values Values close together have a small standard deviation, but values with much more variation have a larger standard deviation Has the same units of measurement as the original data
  • 52. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Properties of the Standard Deviation For many data sets, a value is unusual if it differs from the mean by more than two standard deviations Compare standard deviations of two different data sets only if the they use the same scale and units, and they have means that are approximately the same
  • 53. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Empirical (or 68-95-99.7) Rule For data sets having a distribution that is approximately bell shaped , the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean.
  • 54. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The Empirical Rule
  • 55. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The Empirical Rule
  • 56. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The Empirical Rule
  • 57. Chebyshev’s Theorem Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education The proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1–1/ K 2 , where K is any positive number greater than 1. For K = 2, at least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. For K = 3, at least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean.
  • 58. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Rationale for using n – 1 versus n There are only n – 1 independent values. With a given mean, only n – 1 values can be freely assigned any number before the last value is determined. Dividing by n – 1 yields better results than dividing by n . It causes s 2 to target  2 whereas division by n causes s 2 to underestimate  2 .
  • 59. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education Coefficient of Variation The coefficient of variation (or CV ) for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean. Sample Population s x CV =  100%  CV =   100%
  • 60. Recap Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010 Pearson Education In this section we have looked at: Range Standard deviation of a sample and population Variance of a sample and population Coefficient of variation (CV) Range rule of thumb Empirical distribution Chebyshev’s theorem
  • 61. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 3-4 Measures of Relative Standing and Boxplots
  • 62. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept This section introduces measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set. They can be used to compare values from different data sets, or to compare values within the same data set. The most important concept is the z score . We will also discuss percentiles and quartiles, as well as a new statistical graph called the boxplot.
  • 63. Basics of z Scores, Percentiles, Quartiles, and Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 1
  • 64. z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Z score
  • 65. Sample Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Population Round z scores to 2 decimal places Measures of Position z Score x – µ z =  z = x – x s
  • 66. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score < –2 or z score > 2
  • 67. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Percentiles are measures of location. There are 99 percentiles denoted P 1 , P 2 , . . . P 99 , which divide a set of data into 100 groups with about 1% of the values in each group.
  • 68. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Percentile of a Data Value Percentile of value x = • 100 number of values less than x total number of values
  • 69. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. n total number of values in the data set k percentile being used L locator that gives the position of a value P k k th percentile Notation Converting from the k th Percentile to the Corresponding Data Value L = • n k 100
  • 70. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Converting from the k th Percentile to the Corresponding Data Value
  • 71. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Quartiles Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. Are measures of location, denoted Q 1 , Q 2 , and Q 3 , which divide a set of data into four groups with about 25% of the values in each group.
  • 72. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Q 1 , Q 2 , Q 3 divide ranked scores into four equal parts Quartiles 25% 25% 25% 25% Q 3 Q 2 Q 1 (minimum) (maximum) (median)
  • 73. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interquartile Range (or IQR): Q 3 – Q 1 10 - 90 Percentile Range: P 90 – P 10 Some Other Statistics Semi-interquartile Range: 2 Q 3 – Q 1 Midquartile: 2 Q 3 + Q 1
  • 74. For a set of data, the 5-number summary consists of the minimum value; the first quartile Q 1 ; the median (or second quartile Q 2 ); the third quartile, Q 3 ; and the maximum value. 5-Number Summary Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
  • 75. A boxplot (or box-and-whisker-diagram ) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q 1 ; the median; and the third quartile, Q 3. Boxplot Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
  • 76. Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplot of Movie Budget Amounts
  • 77. Boxplots - Normal Distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Normal Distribution: Heights from a Simple Random Sample of Women
  • 78. Boxplots - Skewed Distribution Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches
  • 79. Outliers and Modified Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Part 2
  • 80. Outliers An outlier is a value that lies very far away from the vast majority of the other values in a data set. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
  • 81. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Important Principles An outlier can have a dramatic effect on the mean. An outlier can have a dramatic effect on the standard deviation. An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.
  • 82. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Outliers for Modified Boxplots For purposes of constructing modified boxplots , we can consider outliers to be data values meeting specific criteria. In modified boxplots, a data value is an outlier if it is . . . above Q 3 by an amount greater than 1.5  IQR below Q 1 by an amount greater than 1.5  IQR or
  • 83. Modified Boxplots Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplots described earlier are called skeletal (or regular ) boxplots. Some statistical packages provide modified boxplots which represent outliers as special points.
  • 84. Modified Boxplot Construction Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. A special symbol (such as an asterisk) is used to identify outliers. The solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier. A modified boxplot is constructed with these specifications:
  • 85. Modified Boxplots - Example Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Pulse rates of females listed in Data Set 1 in Appendix B.
  • 86. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed: z Scores z Scores and unusual values Quartiles Percentiles Converting a percentile to corresponding data values Other statistics Effects of outliers 5-number summary Boxplots and modified boxplots
  • 87. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Putting It All Together Always consider certain key factors: Context of the data Source of the data Measures of Center Sampling Method Measures of Variation Outliers Practical Implications Changing patterns over time Conclusions Distribution