# Numerically Summarizing Data

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Numerically Summarizing Data
Chapter 3 Numerically Summarizing Data Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In order to use this PowerPoint presentation, the required textbook for the class is the Fundamentals of Statistics, Informed Decisions Using Data, Michael Sullivan, III, fourth edition. Los Angeles Mission College Prepared by DW

Chapter 3.1 Measures of Central Tendency
Objective A : Mean, Median, and Mode Objective B : Relation Between the Mean, Median, and Distribution Shape Los Angeles Mission College Prepared by DW

Chapter 3.1 Measures of Central Tendency
Objective A : Mean, Median, and Mode Three measures of central of tendency: the mean, the median, and the mode. A1. Mean The mean of a variable is the sum of all data values divided by the number of observations. Population mean: where is each data value and is the population size (the number of observations in the population). Sample mean: where is each data value and in the sample size (the number of observations in the sample). Los Angeles Mission College Prepared by DW

Example 1: Population : Compute the population mean and sample mean from a simple random sample of size 4. Does the sample mean equal to the population mean? Does the population mean or sample mean stay the same? Explain. (a) Population mean : (Round the mean to one more decimal place than that in the raw data) Los Angeles Mission College Prepared by DW

From a lottery method, 23 16 14 17 were selected.
(b) Sample mean: From a lottery method, were selected. (c) Does the sample mean equal to the population mean? No. (d) Does the population mean or sample mean stay the same? Explain. stays the same. varies from sample to sample. Los Angeles Mission College Prepared by DW

A2. Median The median, , is the value that lies in the middle of the data when arranged in ascending order. If is odd, the median is the data value in the middle of the data set; the location of the median is the position. If is even, the median is the mean of the two middle observations in the data set that lie in the and position respectively. Los Angeles Mission College Prepared by DW

Example 1: Find the median of the data given below.
Reorder: The location of median is at = = = 5th position The median is 18. Los Angeles Mission College Prepared by DW

Example 2: Find the median of the data given below.
\$ \$ \$ \$ \$ \$ \$ \$40.88 Reorder: \$ \$ \$ \$ \$ \$ \$ \$49.36 The location of median is between and which is between and = 4th and 5th position. The median Los Angeles Mission College Prepared by DW

A3. Mode Mode is the most frequent observation in the data set.
Example 1: Find the mode of the data given below. Reorder: Mode = 60 and 80 Example 2: Find the mode of the data given below. A C D C B C A B B F B W F D B W D A D C D Reorder: A A A B B B B B C C C C D D D D D F F W W Mode = B and D Los Angeles Mission College Prepared by DW

Example 3: The following data represent the G.P.A. of 12 students.
Find the mean, median, and mode G.P.A. Reorder: (a) mean Los Angeles Mission College Prepared by DW

(b) median The location of median is between and which is between and = 6th and 7th position. Reorder: 6th 7th The median (c) mode None. Los Angeles Mission College Prepared by DW

Chapter 3.1 Measures of Central Tendency
Objective A : Mean, Median, and Mode Objective B : Relation Between the Mean, Median, and Distribution Shape Los Angeles Mission College Prepared by DW

Objective B : Relation Between the Mean, Median, and Distribution Shape
The mean is sensitive to extreme data. For continuous data, if the distribution shape is a bell-shaped curve, the mean is a better measure of central tendency because it includes all data values in a data set. The median is resistant to extreme data. For continuous data, if the distribution shape is skewed to the right or left, the median is a better measure of central tendency. The mode is used to represent the measure of central tendency for qualitative data. Los Angeles Mission College Prepared by DW

Mean or Median versus Skewness
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Chapter 3.2 Measures of Dispersion
Objective A : Range, Variance, and Standard Deviation Objective B : Empirical Rule Objective C : Chebyshev’s Inequality Los Angeles Mission College Prepared by DW

Chapter 3.2 Measures of Dispersion (Part I)
Measurement of dispersion is a numerical measure that can quantify the spread of data. In this section, the three numerical measures of dispersion that we will discuss are the range, variance, and standard deviation. In the later section, we will discuss another measure of dispersion called interquartile range (IQR). Objective A : Range, Variance, and Standard Deviation A1. Range Range = = largest data value – smallest data value The range is not resistant because it is affected by extreme values in the data set. Los Angeles Mission College Prepared by DW

A2. Variance and Standard Deviation
Standard Deviation is based on the deviation about the mean. Since the sum of deviation about the mean is zero, we cannot use the average deviation about the mean as a measure of spread. We use the average squared deviation (variance) instead. The population variance, , of a variable is the sum of the squared deviations about the population mean, , divided by the number of observations in the population, Definition Formula Computational Formula Los Angeles Mission College Prepared by DW

Computational Formula
The sample variance, , of a variable is the sum of the squared deviations about the sample mean, , divided by the number of observations in the sample minus 1, Definition Formula Computational Formula Los Angeles Mission College Prepared by DW

In order to use the sample variance to obtain an unbiased estimate of the population variance, we divide the sum of the squared deviations about the sample mean by We call the degree of freedom because the first observations have freedom to be whatever value they wish, but the th value has no freedom in order to force to be zero. The population standard deviation, , is the square root of the population variance or The sample standard deviation, , is the square root of the sample variance or To avoid round-off error, never use the rounded value of the variance to compute the standard deviation. Keep a few more decimal places for an intermediate step calculation. Los Angeles Mission College Prepared by DW

Population standard deviation:
Example 1: Use the definition formula to find the population variance and standard deviation. Population: 4, 10, 12, 13, 21 Definition formula where Population variance: Population standard deviation: Los Angeles Mission College Prepared by DW

Sample standard deviation:
Example 2: Use the definition formula to find the sample variance and standard deviation. Sample: 83, 65, 91, 84 Definition formula where Sample mean: Sample variance: Sample standard deviation: Los Angeles Mission College Prepared by DW

Sample: 83, 65, 91, 84 (same data set as Example 2)
Example 3: Use the computational formula to find the sample variance and standard deviation. Sample: 83, 65, 91, (same data set as Example 2) Computational Formula (Sample variance) Sample standard deviation: Los Angeles Mission College Prepared by DW

Sample: 83, 65, 91, 84 (same data set as Example 2)
Example 4: Use StatCrunch to find the sample variance and standard deviation. Sample: 83, 65, 91, (same data set as Example 2) Step 1: Click StatCrunch navigation button under the Course Home page  Click StatCrunch website  Click Open StatCrunch  Input the raw data in Var 1 column  Click Stat  Click Summary Stats  Columns Los Angeles Mission College Prepared by DW

Step 2: Click var1 under Select column(s):  Under Statistics:, choose Variance and Std. dev. (click them while holding Ctrl key on the keyboard)  Click Compute! Los Angeles Mission College Prepared by DW

Variance and standard deviation are computed.
For more detailed instructions, please download “Q “ by clicking the StatCrunch Handout navigation button of the course homepage. Note : For a small data set, students are expected to calculate the standard deviation by hand. Los Angeles Mission College Prepared by DW

Chapter 3.2 Measures of Dispersion
Objective A : Range, Variance, and Standard Deviation Objective B : Empirical Rule Objective C : Chebyshev’s Inequality Los Angeles Mission College Prepared by DW

Objective B : Empirical Rule
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The figure below illustrates the Empirical Rule.
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(a) What percentage of SAT scores is between 401 and 629?
Example 1: SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114. (Source: College Board, 2007) (a) What percentage of SAT scores is between 401 and ? According to the Empirical Rule, approximately 68% of the data will lie within 1 standard deviation of the mean. 68% of SAT scores is between 401 and 629. Los Angeles Mission College Prepared by DW

Example 1: (b) What percentage of SAT scores is between 287 and 743?
According to the Empirical Rule, approximately 95% of the data will lie within 2 standard deviations of the mean. 95% of SAT scores is between 287 and 743. Los Angeles Mission College Prepared by DW

Example 1: (c) What percentage of SAT scores is less than 401 or greater than 629?
= Los Angeles Mission College Prepared by DW

= Example 1: (d) What percentage of SAT scores is between 515 and 743?
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Example 1: (e) About 99.7% of SAT scores will be between what scores?
According to the Empirical Rule, approximately 99.7% of the data will lie within 3 standard deviations of the mean. Los Angeles Mission College Prepared by DW

Chapter 3.2 Measures of Dispersion
Objective A : Range, Variance, and Standard Deviation Objective B : Empirical Rule Objective C : Chebyshev’s Inequality Los Angeles Mission College Prepared by DW

Objective C : Chebyshev’s Inequality
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What minimum percentage of commuters in Boston has a
Example 1: According to the U.S. Census Bureau, the mean of the commute time to for a resident to Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following: What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean? Standard deviation → According to the Chebyshev’s Inequality, at least will lie within 2 standard deviations of the mean. Los Angeles Mission College Prepared by DW

(i) According to the Chebyshev’s Inequality, at least
Example 1: (b) (i) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? (ii) What are the commute times within standard deviations of the mean? (i) According to the Chebyshev’s Inequality, at least of the data will lie within standard deviations of the mean. Since , 55.6% of commuters in Boston has a commute time. (ii) At least 55.6% of commuters in Boston has a commute time between minutes and minutes Los Angeles Mission College Prepared by DW

Chapter 3.3 Measures of Central Tendency and Dispersion from Grouped Data
This section we are going to learn how to calculate the mean, , and the weighted mean, , from data that have already been summarized in frequency distributions (group data). Since raw data cannot be retrieved from a frequency table, the class midpoint is used to represent the mean of the data values within each class. Midpoint = (Adding consecutive lower class limits) ÷ 2 Los Angeles Mission College Prepared by DW

Chapter 3.3 Measures of Central Tendency and Dispersion
from Grouped Data Objective A : Approximate the sample mean of a variable from grouped data. Objective B : The weighted Mean, Los Angeles Mission College Prepared by DW

Objective A : Approximate the sample mean of a variable
from grouped data. Sample Mean: where is the midpoint of the th class is the frequency of the th class is the number of classes Los Angeles Mission College Prepared by DW

Example 1: The following frequency distribution represents the second test scores of my Math 227 from last semester Approximate the mean of the score. Los Angeles Mission College Prepared by DW

From the previous slide,
The mean of the score : Los Angeles Mission College Prepared by DW

Chapter 3.3 Measures of Central Tendency and Dispersion from Grouped Data
Objective A : Approximate the sample mean of a variable from grouped data. Objective B : The weighted Mean, Los Angeles Mission College Prepared by DW

Objective B : The weighted Mean,
We compute the weighted mean when data values are not weighted equally. where is the weight of the th observation is the value of the th observation Los Angeles Mission College Prepared by DW

The price per pound of the mix :
Example 1: Michael and Kevin want to buy nuts. They can't agree on whether they want peanuts, cashews, or almonds They agree to create a mix. They bought 2.5 pounds of peanuts for \$1.30 per pound, 4 pounds of cashews for \$4.50 per pounds, and 2 pounds of almonds for \$ per pound. Determine the price per pound of the mix. The price per pound of the mix : Los Angeles Mission College Prepared by DW

Marissa’s course average :
Example 2: In Marissa's calculus course, attendance counts for 5% of the grade, quizzes count for 10% of the grade, exams count for 60% of the grade, and the final exam counts for % of the grade. Marissa had a 100% average for attendance, 93% for quizzes, 86% for exams, and 85% on the final. Determine Marissa's course average. Marissa’s course average : Los Angeles Mission College Prepared by DW

Ch 3.4 Measures of Positions and Outliers
Objective A : -scores Objective B : Percentiles and Quartiles Objective C : Outliers Los Angeles Mission College Prepared by DW

Ch3.4 Measures of Positions and Outliers
Measures of position determine the relative position of a certain data value within the entire set of data. Objective A : -scores The -score represents the distance that a data value is from the mean in terms of the number of standard deviations. Population -score: Sample -score: Los Angeles Mission College Prepared by DW

0.87 standard deviation below the mean.
Example 1: The average 20- to 29-year-old man is 69.6 inches tall, with a standard deviation of 3.0 inches, while the average 20- to 29-year-old woman is 64.1 inches tall, with a standard deviation of 3.8 inches. Who is relatively taller, a 67-inch man or 62-inch woman? Man : 0.87 standard deviation below the mean. Woman : 0.55 standard deviation below the mean Therefore, the 62-inch woman is relatively taller than the 67-inch man. Los Angeles Mission College Prepared by DW

Ch 3.4 Measures of Positions and Outliers
Objective A : -scores Objective B : Percentiles and Quartiles Objective C : Outliers Los Angeles Mission College Prepared by DW

Objective B : Percentiles and Quartiles
The th percentile, , of a set of data is a value such that percent of the observations are less than or equal to the value. Example 1: Explain the meaning of the 5th percentile of the weight of males 36 months of age is 12.0 kg. 5% of 36-month-old males weighs 12.0 kg or less. 95% of 36-month-old males weighs more than 12.0 kg. Los Angeles Mission College Prepared by DW

The most common percentiles are quartiles.
The first quartile, , is equivalent to The second quartile, , is equivalent to The third quartile, , is equivalent to Los Angeles Mission College Prepared by DW

Example 2: Determine the quartiles of the following data
Ascending order : Lower half of the data : Upper half of the data : Los Angeles Mission College Prepared by DW

B2. Interquartile The interquartile range, IQR, is the measure of dispersion that is based on quartiles. The range and standard deviation are effected by extreme values. The IQR is resistant to extreme values. Los Angeles Mission College Prepared by DW

(a) Provide an interpretation of these results.
Example 1: One variable that is measured by online homework systems is the amount of time a student spends on homework for each section of the text. The following is a summary of the number of minutes a student spends for each section of the text for the fall 2007 semester in a College Algebra class at Joliet Junior College. (a) Provide an interpretation of these results. 25% of the students spend 42 minutes or less on homework for each section, and 75% of the students spend more than 42 minutes. 50% of the students spend 51.5 minutes or less on homework for each section, and 50% of the students spend more than 51.5 minutes. 75% of the students spend 72.5 minutes or less on homework for each section, and 25% of the students spend more than 72.5 minutes. Los Angeles Mission College Prepared by DW

(b) Determine and interpret the interquartile range.
The middle of 50% of all students has a range of 30.5 minutes of time spent on homework. (c) Do you believe that the distribution of time spent doing homework is skewed or symmetric? Why? Skewed right. The difference between and is less than the difference between and Los Angeles Mission College Prepared by DW

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Ch 3.4 Measures of Positions and Outliers
Objective A : -scores Objective B : Percentiles and Quartiles Objective C : Outliers Los Angeles Mission College Prepared by DW

Objective C : Outliers Extreme observations are called outliers; they may occur by error in the measurement or during data entry or from errors in sampling. Los Angeles Mission College Prepared by DW

(a) Determine the quartiles.
Example 1: The following data represent the hemoglobin ( in g/dL ) for 20 randomly selected cats. (Source: Joliet Junior College Veterinarian Technology Program) (a) Determine the quartiles. Ascending order : Los Angeles Mission College Prepared by DW

(b) Compute and interpret the interquartile range, IQR.
Lower half of the data: Upper half of the data: Los Angeles Mission College Prepared by DW

Ascending order of the original data :
(c) Determine the lower and upper fences. Are there any outliers, according to this criterion? Ascending order of the original data : All data falls within 6.23 to except 5.7. 5.7 is the outlier. Prepared by DW Los Angeles Mission College

Ch 3.5 The Five-Number Summary and Boxplots
Objective A : The Five-Number Summary Objective B : Boxplots Objective C : Using a Boxplot to describe the shape of a distribution Los Angeles Mission College Prepared by DW

Ch 3.5 The Five-Number Summary and Boxplots
Objective A : The Five-Number Summary Los Angeles Mission College Prepared by DW

Find the Five-Number Summary.
Example 1: The number of chocolate chips in a randomly selected name-brand cookies were recorded. The results are shown Find the Five-Number Summary. Ascending order : Lower half of the data: Upper half of the data: Five-number summary: Minimum = 19, = 22.5, = 25, = 28.5, Maximum = 33 Los Angeles Mission College Prepared by DW

Ch 3.5 The Five-Number Summary and Boxplots
Objective A : The Five-Number Summary Objective B : Boxplots Objective C : Using a Boxplot to describe the shape of a distribution Los Angeles Mission College Prepared by DW

Objective B : Boxplots The five-number summary can be used to construct a graph called the boxplot. Los Angeles Mission College Prepared by DW

Example 1: A stockbroker recorded the number of clients she saw each day over an 11-day period. The data are shown Draw a boxplot. Ascending order : Since all data fall between the lower fence, 12, and upper fence, 60. There is no outlier. 20 25 30 35 40 50 45 55 Los Angeles Mission College Prepared by DW

Ch 3.5 The Five-Number Summary and Boxplots
Objective A : The Five-Number Summary Objective B : Boxplots Objective C : Using a Boxplot to describe the shape of a distribution Los Angeles Mission College Prepared by DW

Objective C : Using a Boxplot to describe the shape of a distribution
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Example 1: Use the side-by-side boxplots shown to answer the
questions that follow. (a) To the nearest integer, what is the median of variable ? (b) To the nearest integer, what is the first quartile of variable ? Los Angeles Mission College Prepared by DW

(c) Which variable has more dispersion? Why?
The variable has more dispersion because the IQR on is wider than the IQR on the variable. (d) Does the variable have any outliers? If so, what is the value of the outlier? Yes, there is an asterisk on the right side of the boxplot. Outliers (e) Describe the shape of the variable Support your position. Since there is a longer whisker on the left and is bigger than , the shape of the distribution is skewed to the left. Los Angeles Mission College Prepared by DW

Example 2: The following data represent the carbon dioxide emissions per capita (total carbon dioxide emissions, in tons, divided by total population) for the countries of Western Europe in 2004. Los Angeles Mission College Prepared by DW

(a) Find the five-number summary.
Ascending order: Minimum = 1.01, = 1.61, = 2.165, = 2.68, Maximum = 6.81 (b) Determine the lower and upper fences. Los Angeles Mission College Prepared by DW

(c) Construct a boxplot.
1.00 2.00 3.00 4.00 5.00 7.00 6.00 8.00 5.22 is a mild outlier which is represented by an asterisk. 6.81 is an extreme outlier because it is larger than An extreme outlier is presented by an open circle. (d) Use the boxplot and quartiles to describe the shape of the distribution. Since there are two extreme large outliers, the shape of the distribution is skewed to the right. Los Angeles Mission College Prepared by DW

Note: Part (a) and (c) can be easily done by using StatCrunch
Note: Part (a) and (c) can be easily done by using StatCrunch. For the instructions, please refer to the StatCrunch handout. Los Angeles Mission College Prepared by DW