Chapter 3 Averages and Variation Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

3 | 2 Copyright © Cengage Learning. All rights reserved. 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

3 | 3 Copyright © Cengage Learning. All rights reserved. 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} The mode is 1.

3 | 4 Copyright © Cengage Learning. All rights reserved. Median 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:

3 | 5 Copyright © Cengage Learning. All rights reserved. Median Find the median of the following data set. { 4, 6, 6, 7, 9, 12, 18, 19} a). 6b). 7c). 8d). 9

3 | 6 Copyright © Cengage Learning. All rights reserved. Median Find the median of the following data set. {4, 6, 6, 7, 9, 12, 18, 19} a). 6b). 7c). 8d). 9

3 | 7 Copyright © Cengage Learning. All rights reserved. Sample meanPopulation mean Mean

3 | 8 Copyright © Cengage Learning. All rights reserved. Sample meanPopulation mean Mean Find the mean of the following data set. {3, 8, 5, 4, 8, 4, 10} a). 8b). 6.5c). 6d). 7

3 | 9 Copyright © Cengage Learning. All rights reserved. Sample meanPopulation mean Mean Find the mean of the following data set. {3, 8, 5, 4, 8, 4, 10} a). 8b). 6.5c). 6d). 7

3 | 10 Copyright © Cengage Learning. All rights reserved. Trimmed Mean 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.

3 | 11 Copyright © Cengage Learning. All rights reserved. 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.

3 | 12 Copyright © Cengage Learning. All rights reserved. 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.

3 | 13 Copyright © Cengage Learning. All rights reserved. Critical Thinking Mound-shaped data – values of mean, median and mode are nearly equal.

3 | 14 Copyright © Cengage Learning. All rights reserved. Critical Thinking Skewed-left data – mean < median < mode.

3 | 15 Copyright © Cengage Learning. All rights reserved. Critical Thinking Skewed-right data – mean > median > mode.

3 | 16 Copyright © Cengage Learning. All rights reserved. Weighted Average At times, we may need to assign more importance (weight) to some of the data values. x is a data value. w is the weight assigned to that value.

3 | 17 Copyright © Cengage Learning. All rights reserved. Measures of Variation Range = Largest value – smallest value Only two data values are used in the computation, so much of the information in the data is lost. Three measures of variation: range variance standard deviation

3 | 18 Copyright © Cengage Learning. All rights reserved. Sample Variance and Standard Deviation Sample Variance Sample Standard Deviation Find the standard deviation of the data set. {2,4,6} a). 2b). 3c). 4d). 3.67

3 | 19 Copyright © Cengage Learning. All rights reserved. Sample Variance and Standard Deviation Sample Variance Sample Standard Deviation Find the standard deviation of the data set. {2,4,6} a). 2b). 3c). 4d). 3.67

3 | 20 Copyright © Cengage Learning. All rights reserved. Population Variance Population Standard Deviation Population Variance and Standard Deviation

3 | 21 Copyright © Cengage Learning. All rights reserved. The Coefficient of Variation For Samples For Populations

3 | 22 Copyright © Cengage Learning. All rights reserved. Chebyshev’s Theorem

3 | 23 Copyright © Cengage Learning. All rights reserved. Chebyshev’s Theorem

3 | 24 Copyright © Cengage Learning. All rights reserved. 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.

3 | 25 Copyright © Cengage Learning. All rights reserved. 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

3 | 26 Copyright © Cengage Learning. All rights reserved. Quartiles and Interquartile Range (IQR)

3 | 27 Copyright © Cengage Learning. All rights reserved. Computing Quartiles

3 | 28 Copyright © Cengage Learning. All rights reserved. Five Number Summary A listing of the following statistics: –Minimum, Q 1, Median, Q 3, Maximum Box-and-Whisder plot – represents the five- number summary graphically.

3 | 29 Copyright © Cengage Learning. All rights reserved. Box-and-Whisker Plot Construction

3 | 30 Copyright © Cengage Learning. All rights reserved. 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. (See problem 10) Critical Thinking

3 | 31 Copyright © Cengage Learning. All rights reserved. Which of the following box-and-whiskers plots suggests a symmetric data distribution? (a) (b) (c) (d) Critical Thinking

3 | 32 Copyright © Cengage Learning. All rights reserved. Which of the following box-and-whiskers plots suggests a symmetric data distribution? (a) (b) (c) (d) Critical Thinking