Objective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode.

Transcription

1 Chapter 3 Numerically Summarizing Data 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 N number of observations in the population). x Sample mean: x i where n observations in the sample). x i x is each data value and N is the population size (the i x is each data value and n in the sample size (the number of i 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) μ = x i N N = 8 = = = (b) Sample mean: From a lottery method, were selected. x = x i n n = 4 = = 70 5 = 17.5 (c) Does the sample mean equal to the population mean? No. 1

2 (d) Does the population mean or sample mean stay the same? Explain. μ stays the same. x varies from sample to sample. A. Median The median, M, is the value that lies in the middle of the data when arranged in ascending order. Example 1: Find the median of the data given below Reorder: n = 9 = odd The location of the median is at = n = = 5 th position. The median is 18. Example : Find the median of the data given below. $35.34 $4.09 $38.7 $43.8 $39.45 $49.36 $30.15 $40.88 Reorder: $30.15 $35.34 $38.7 $39.45 $40.88 $4.09 $43.8 $49.36 N = 8 = even The location of median is between n n and 1 which is between 8 8 and= 1 4 th and 5 th position. The median = = =

3 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 : 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 Example 3: The following data represent the G.P.A. of 1 students Find the mean, median, and mode G.P.A. Reorder: (a) mean x = x i n n = 1 = = (b) median Reorder: median = ( )/ =.655 (c) mode Reorder: mode = None 3

4 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. Definition: A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially. Mean or Median versus Skewness 4

5 Chapter 3. Measures of Dispersion 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 = R = largest data value - smallest data value The range is not resistant because it is affected by extreme values in the data set. A. Variance and Standard Deviation Variance 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 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, N. ( x ) i Definition Formula N x i x i N N Computational Formula The sample variance, s, of a variable is the sum of the squared deviations about the sample mean, x, divided by the number of observations in the sample minus 1, n 1. i ( x x) s n 1 s x i x i n n 1 Definition Formula Computational Formula 5

6 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 n 1.We call n 1 the degree of freedom because the first n 1 observations have freedom to be whatever value they wish, but the nth value has no freedom in order to force ( x x) to be zero. The population standard deviation,, is the square root of the population variance or σ Population Variance The sample standard deviation, s, is the square root of the sample variance or s Sample Variance i 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. Example 1: Use the definition formula to find the population variance and standard deviation. Population: 4, 10, 1, 13, 1 Definition formula δ = (x i μ) N μ = where μ = x i N = 60 5 = 1 x i xi ( x i ) = -8 (-8)² = = - (-)² = = 0 0² = = 1 1² = = 9 9² = 81 ( ) 150 Population variance: σ = = 30 Population standard deviation: σ = σ = x i 6

7 Example : Use the definition formula to find the sample variance and standard deviation. Sample: 83, 65, 91, 84 Definition formula s = (x i x ) n 1 Sample mean: x = = 33 4 where x = x i n = xi x i ( x i ) =.5 (.5)² = = (-15.75)² = = = 3.5 Sample variance: s = (10.5) ( (3.5) x i x) Sample standard deviation: s = Example 3: Use StatCrunch to find the sample variance and standard deviation. Sample: 83, 65, 91, 84 (same data set as Example ) 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 Step : 7

8 Click var1 under Select column(s): Under Statistics:, choose Variance and Std. dev. (click them while holding Ctrl key on the keyboard) Click Compute! Variance and standard deviation are computed. s 1.9 s 11.1 For more detailed instructions, please download Q3..0 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. 8

9 Objective C : Empirical Rule The figure below illustrates the Empirical Rule 9

10 Example 1: SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114. (Source: College Board, 007) (a) What percentage of SAT scores is between 401 and 69? 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 69. (b) What percentage of SAT scores is between 87 and 743? (114) (114) 743 According to the Empirical Rule, approximately 95% of the data will lie within standard deviations of the mean. 95% of SAT scores is between 87 and

11 (c) What percentage of SAT scores is less than 401 or greater than 69? 100 % 68% = % (d) What percentage of SAT scores is between 515 and 743? 95% = = % 11

12 (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. (μ 3σ, μ + 3σ) = (515-3(114), (114)) = (173, 857) Chapter 3.4 Measures of Position and Outliers Measures of position determine the relative position of a certain data value within the entire set of data. Objective A : z -scores The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. x Population z -score: z Sample z -score: z x x s Example 1: The average 0- to 9-year-old man is 69.6 inches tall, with a standard deviation of 3.0 inches, while the average 0- to 9-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 6-inch woman? Man: µ = 69.6 inches σ = 3.0 inches x = 67 inches Z = x μ = = σ standard deviation below the mean. Woman: µ = 64.1 inches σ = 3.8 inches x = 67 inches Z = x μ = = σ standard deviation below the mean Therefore, the 6- inch woman is relatively taller than the 67- inch man. 1

13 Objective B:Percentiles and Quartiles B1. Percentiles The k th percentile, P k, of a set of data is a value such that k 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 1.0 kg. 5% of 36-month-old males weigh 1.0 kg or less. 95% of 36-month-old males weigh more than 1.0 kg. The most common percentiles are quartiles. The first quartile, Q 1, is equivalent to P. 5 The second quartile, Q, is equivalent to P. 50 The third quartile, Q 3, is equivalent to P

14 Example : Determine the quartiles of the following data Ascending order: M = 58 Q = 58 Lower half of the data: Q 1 = 45 Upper half of the data: B. Interquartile Q 3 = 71 The interquartile range, IQR, is the measure of dispersion that is based on quartiles. The range and standard deviation are affected by extreme values. The IQR is resistant to extreme values. 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 007 semester in a College Algebra class at Joliet Junior College. (a) Provide an interpretation of Q 1. Q 1 4 : Q 51.5: Q 3 7.5: Q1 4 Q 51.5 Q % of the students spend 4 minutes or less on homework for each section, and 75% of the students spend more than 4 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 7.5 minutes or less on homework for each section, and 5% of the students spend more than 7.5 minutes. 14

15 (b) Determine and interpret the interquartile range. IQR = Q 3 - Q 1 = = 30.5 minutes 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 Q and Q1 is less than the difference between Q3 and Q. 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. 15

16 Chapter 3.5 The Five-Number Summary and Boxplots Objective A :The Five-Number Summary Example 1: The number of chocolate chips in a randomly selected 1 name-brand cookies were recorded. The results are shown below Find the five-number Summary. M 5 Ascending order: Lower half of the data: Q 1 = +3 =.5 Upper half of the data: Q 3 = 8+9 = 8.5 Five-number summary: Minimum = 19, =.5, = 5, Q = 8.5, Maximum = 33 Q1 Q 3 16

17 Objective B: Boxplot The five-number summary can be used to construct a graph called the boxplot. Objective C:Using a Boxplot to describe the shape of a distribution 17

18 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 x? 15 (b) To the nearest integer, what is the first quartile of variable y? (c) Which variable has more dispersion? Why? The y variable has more dispersion because the IQR on y is wider than the IQR on the x variable. (d) Does the variable x have any outliers? If so, what is the value of the outlier? Yes, there is an asterisk on the right side of the boxplot. Outlier 30 (e) Describe the shape of the variable y. Support your position. Since there is a longer whisker on the left and Q Q 1 is bigger than Q3 Q, the shape of the distribution is skewed to the left. 18