Chapter 3

Averages and Variation

Understanding Basic Statistics Fifth Edition

By Brase and Brase Prepared by Jon Booze

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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

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ModeThe 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.

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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:Sum of middle two valuesMedian

2

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Median

Find the median of the following data set.{ 4, 6, 6, 7, 9, 12, 18, 19}

a). 6 b). 7 c). 8 d). 9

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Median

Find the median of the following data set.{4, 6, 6, 7, 9, 12, 18, 19}

a). 6 b). 7 c). 8 d). 9

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Sample mean Population mean

Mean

xx

n x

N

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Sample mean Population mean

Mean

xx

n x

N

Find the mean of the following data set.{3, 8, 5, 4, 8, 4, 10}

a). 8 b). 6.5 c). 6 d). 7

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Sample mean Population mean

Mean

xx

n x

N

Find the mean of the following data set.{3, 8, 5, 4, 8, 4, 10}

a). 8 b). 6.5 c). 6 d). 7

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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.

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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.

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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.

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Critical Thinking

• Mound-shaped data – values of mean, median and mode are nearly equal.

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Critical Thinking

• Skewed-left data – mean < median < mode.

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Critical Thinking

• Skewed-right data – mean > median > mode.

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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.

Weighted Averagexw

w

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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: rangevariancestandard deviation

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Sample Variance and Standard Deviation

Sample Variance Sample Standard Deviation

2( )12

1

nx xi

isn

2s s

Find the standard deviation of the data set.{2,4,6}

a). 2 b). 3 c). 4 d). 3.67

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Sample Variance and Standard Deviation

Sample Variance Sample Standard Deviation

2( )12

1

nx xi

isn

2s s

Find the standard deviation of the data set.{2,4,6}

a). 2 b). 3 c). 4 d). 3.67

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Population Variance Population Standard Deviation

Population Varianceand Standard Deviation

N

xN

i

i

1

2

2

)( 2

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The Coefficient of Variation

100x

sCV 100

CV

For Samples For Populations

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Chebyshev’s Theorem

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Chebyshev’s Theorem

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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.

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Percentiles and Quartiles

• For whole numbers P, 1 ≤ P ≤ 99, the Pth 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.

• Q1 = 25th Percentile• Q2 = 50th Percentile = The Median• Q3 = 75th Percentile

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Quartiles and Interquartile Range (IQR)

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Computing Quartiles

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Five Number Summary

• A listing of the following statistics:

– Minimum, Q1, Median, Q3, Maximum

• Box-and-Whisder plot – represents the five-number summary graphically.

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Box-and-Whisker Plot Construction

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• 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

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Which of the following box-and-whiskers plots suggests a symmetric data distribution?

(a) (b) (c) (d)

Critical Thinking

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Which of the following box-and-whiskers plots suggests a symmetric data distribution?

(a) (b) (c) (d)

Critical Thinking