Statistics Chapter 3 Averages and Variations

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1 Statistics Chapter 3 Averages and Variations Measures of Central Tendency Average a measure of the center value or central tendency of a distribution of values. Three types of average: Mode Median Mean Mode The mode is the most frequently occurring value in a data set. Example - Sixteen students are asked how many college math classes they have completed. {0, 3, 2, 2, 1, 1, 0, 5, 1,1, 0, 2, 2, 7, 1, 3} What is the mode? Note: not every data set has a mode i.e. data: has no mode. Example Count the number of letters in each word of this sentence and give the mode. What number appears the most in this list? Median - Median (M) is a number so that half the observations are smaller than it and half are larger. - The median can be a value that is not a data point. Finding the median: 1). Order the data from smallest to largest. 2). For an odd number of data values: Median = Middle data value 3). For an even number of data values: Sum of middle two values Median 2 The median is resistant (or robust) to an outlier. 1 P a g e

2 Example - Find the median of the following data set. { 4, 6, 6, 7, 9, 12, 18, 19} Mean - Let X 1, X 2,, X n be a set of n observations. The sample mean is X Sample mean x 1 x 2 xn... 1 n n n i 1 x i Population mean x N Example - Find the mean of the following data set. {3, 8, 5, 4, 8, 4, 10} Example Find the mean of the given 5 exam scores 72, 93, 79, 86, 81 Comment: The sample mean is affected by outliers. Suppose the last person did not take the exam and received a 0. X The mean is not resistant (or robust) to an outlier. Comment: (1.) If the distribution of data values is approximately symmetric, the mean and median are approximately the same. (2.) If the distribution is skewed right then (3.) If the distribution is skewed left then (4.) Often it is appropriate to stat both the mean and the median. 2 P a g e

3 Trimmed Mean - A trimmed mean is the mean of the data values left after trimming a specified percentage of the smallest and largest values from the data set. This allows us to eliminate the influence of extreme data values while still getting the benefit of the mean which takes into account the sample size as opposed to the median. Order the data and remove k% of the data values from the bottom and top. 5% and 10% trimmed means are common. Then compute the mean with the remaining data values. Example - Average Class size of 19 schools in California for an introductory course. Find the 5% trimmed mean Resistant Measures of Central Tendency A resistant measure will not be affected by extreme values in the data set. The mean is not resistant to extreme values. The median is resistant to extreme values. A trimmed mean is also resistant. Critical Thinking Four levels of data nominal, ordinal, interval, ratio (Chapter 1) Mode can be used with all four levels. Median may be used with ordinal, interval, of ratio level. Mean may be used with interval or ratio level. Mound-shaped data values of mean, median and mode are nearly equal. 3 P a g e

4 Mound-shaped symmetrical mean = median = mode. Skewed-left data mean < median < mode. Skewed-right data mean > median > mode. Weighted Average - Sometimes we wish to average numbers, but assign more importance to some values than others. This is called a weight. At times, we may need to assign more importance (weight) to some of the data values. xw Weighted Average w x is a data value. w is the weight assigned to that value. 4 P a g e

5 Example Grades: Given that the exam average is 85, the final exam score was 87, the homework average was 98. Find the weighted average if the tests are worth 40% and the final was worth 40% and the homework was 20%. Measures of Variation Three measures of variation: range variance standard deviation Range = Largest value smallest value Only two data values are used in the computation, so much of the information in the data is lost. Sample Variance and Standard Deviation Sample Variance Can be thought of as a kind of average of the values divide by (n-1) instead of n for technical reasons. values. However we Sample Standard Deviation Can be thought of as a measure of variability or risk. Larger values of s imply greater variability in the data. Why do we take the square root? Because the units before the square root are so we take the square root to get back to the original units. 5 P a g e

6 Example - Find the standard deviation of the data set. {2,4,6} Example set 1: 6, 7, 8, 9, 10 Set 1: X 8 set 2: 4, 6, 8, 10, 12 Set 2: X 8 Set 2 is more spread out than set 1. We will measure spread in a data set by looking at the deviations x x, i 1,..., n. We will square these deviations and add them up and then average them by dividing by (n-1) to get a measure of spread. n 1 xi x s n 1 The (sample) variance 2 2 i 1 i The standard deviation is the square root of the variance. s s 2 = standard deviation s has the same dimensions as the original x s Example Show mathematically which set is has more spread. Set 1 or Set2 from above. 6 P a g e

7 Properties of Standard Deviation 1.) Use the standard deviation to measure spread when it is appropriate to use the mean to measure the center of the distribution. 2.) If all the observations have the same value then s 2 s 0. 3.) The standard deviations and variance are not resistant to outliers. Rule of Thumb: For symmetric distributions with no outliers the mean and standard deviations are good measures of center and spread of a distribution. For skewed distributions or distributions with outliers the 5 number summary is good. We will talk about the 5 number summary a bit later. Population Variance and Standard Deviation How to With your Calculator: Press STAT EDIT This is where you enter your data To Analyze your Data: Press STAT CALC 1-Var Stats Example Big Blossom Greenhouse was commissioned to develop an extra large rose for the Rose Bowl Parade. A random sample of blossoms from Hybrid A bushes yielded the following diameters (in inches) for mature peak blooms. Find some descriptive statistics for the given data. 2, 3, 3, 8, 10, 10 7 P a g e

8 Coefficient of Variation A disadvantage of the standard deviation as a comparative measure of variation is that it depends on the units of measurements. This means that it is difficult to use the standard deviation to compare measurements from different populations. For this reason we have the coefficient of variation which expresses the standard deviation as a percentage of the sample or population mean. For Samples For Populations s CV 100 CV 100 x Example Find the CV of Set 1 and Set 2 from earlier example Chebyshev s Theorem 8 P a g e

9 Critical Thinking Standard deviation or variance, along with the mean, gives a better picture of the data distribution than the mean alone. Chebyshev s theorem works for all kinds of data distribution. Data values beyond 2.5 standard deviations from the mean may be considered as outliers. Example Students who care is a student volunteer program in which college students donate work time to various community projects. For a random sample of students in the program, the mean number of hours was hours each semester with a sample standard deviation of hours each semester. Find an interval to A to B for the number of hours volunteered into which at least 75% of the students in this program would fall. Solution: Chevbyshev s Theorem states that 75 % of the data must fall within 2 standard deviations of the mean. The mean and the standard deviation the interval is: 25.7 to 32.5 At least 75% of the students would fall into the group that volunteered from 25.7 to 32.5 hours each semester. Percentiles and Quartiles For whole numbers P, 1 P 99, the P th percentile of a distribution is a value such that P% of the data fall below it, and (100-P)% of the data fall at or above it. Q 1 = 25 th Percentile Q 2 = 50 th Percentile = The Median Q 3 = 75 th Percentile Quartiles and Interquartile Range (IQR) : 9 P a g e

10 Quantiles divide the data into four parts To find quantiles (1.) Arrange the data from smallest to largest and find the median. (2.) First quantile (Q 1 ) is the median of the observations whose position in the ordered list is to the left of the median. (3.) Third quantile (Q 3 ) is the median of the observations whose position in the ordered list is to the right of the median. A measure of spread based on Quantiles is the Interquantile range = IQR IQR =Q 3 Q 1 The IQR gives the spread of the middle 50% of the data. Computing Quartiles Example Find the mean ( ) and and the interquartile range (IQR) of the following data P a g e

11 Five Number Summary A listing of the following statistics: Minimum data value, Q 1, Median = Q 2, Q 3, Maximum data value Box-and-Whisker plot represents the five-number summary graphically. Box-and-Whisker Plot Construction Critical Thinking Box-and-whisker plots display the spread of data about the median. If the median is centered and the whiskers are about the same length, then the data distribution is symmetric around the median. Fences may be placed on either side of the box. Values lie beyond the fences are outliers. Using the Interquantile Range to look for possible outliers Any data point more than 1.5*IQR above the third Quantile or below the first Quantile is a possible outlier. Example - Which of the following box-and-whiskers plots suggests a symmetric data distribution? a. b. c. d. 11 P a g e

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