Chapter 4
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Chapter 4. The Description of Data: Measures of Variation and Dispersion. Measures of Variation. We have looked at measures of the center, or location, of data. We also need a measure of the dispersion of data. Range. The range is the distance spanned by the data .

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Chapter 4

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Chapter 4

Chapter 4

The Description of Data:Measures of

Variation and Dispersion


Measures of variation

Measures of Variation

  • We have looked at measures of the center, or location, of data.

  • We also need a measure of the dispersion of data.


Range

Range

  • The range is the distance spanned by the data.

  • The range is calculated by subtracting the smallest data value from the largest.

  • The range is sensitive to outliers.

  • The range does not provide any information regarding the data between the minimum and maximum.


Interquartile range

Interquartile Range

  • The interquartile range is the distance spanned by the middle 50% of the data.

  • The interquartile range is calculated by subtracting Q1 from Q3.

  • The interquartile range is not sensitive to outliers, but still gives insight into the dispersion of the data.


Mean absolute deviation

Mean Absolute Deviation

  • The mean absolute deviation is the mean distance to the mean. In other words, it’s the average distance from the data to µ.


Variance and standard deviation

Variance andStandard Deviation

  • The variance is the average squared distance to the mean.

  • The standard deviation is the square root of the variance.


Variance and standard deviation1

Variance andStandard Deviation

  • For samples, we divide by n-1 to avoid bias.

  • The standard deviations of populations and samples are available from your calculator. Variance can be calculated as the square of the standard deviation.


Chebyshev s theorem

Chebyshev’s Theorem

  • The minimum proportion of data that can be found within k standard deviations from the mean is:


Chebyshev s theorem1

Chebyshev’s Theorem

  • Chebyshev’s Theorem works for any distribution, but it does not work very well.

  • This theorem gives the minimum proportion of data that will be found in a given interval, but in reality, the actual amount is usually much higher than Chebyshev predicts.


The empirical rule

The Empirical Rule

  • If the distribution of data is normal (bell shaped), then:

    • 68% of the data will be found within one standard deviation of the mean.

    • 95% of the data will be found within two standard deviations of the mean.

    • 99.7% of the data will be found within three standard deviations of the mean.


The empirical rule1

The Empirical Rule

  • The empirical rule only works for distributions that are normal (bell shaped).

  • The empirical rule is much more accurate than Chebyshev’s Theorem.


Coefficient of variation

Coefficient of Variation

  • The coefficient of variation measures the relative variation of a distribution.

  • Since this is a relative measure, there are no units, making it easier to compare the variation of two different populations.


Skewness

Skewness

  • Distributions with a long right tail are positively skewed.

  • Distributions with a long left tail are negatively skewed.

  • Distributions that are not skewed are symmetric.


Pearson s coefficient of skewness

Pearson’s Coefficient of Skewness

  • Pearson’s coefficient of skewness gives a numeric measurement of the skewness of a distribution.

  • Distributions with an SK of 0 are symmetric.

  • Distributions with a positive SK are positively skewed, while distributions with a negative SK are negatively skewed.


Try it

Try it!

  • The median price of a home selling in San Diego during 1991 was $195,000. The first and third quartile prices were $170,500 and $232,000 respectively. What was the semi-interquartile range for the cost of a home in San Diego in 1991?

  • $30,750


Try it1

Try it!

  • A sample of 6 prices quoted for a particular television set are $326, $299, $345, $295, $310, and $345.

    • Find the range of this sample.

    • $50


Try it2

Try it!

  • A sample of 6 prices quoted for a particular television set are $326, $299, $345, $295, $310, and $345.

    • Find the variance for the quoted price of the TV.

    • $490.40


Try it3

Try it!

  • A sample of 6 prices quoted for a particular television set are $326, $299, $345, $295, $310, and $345.

    • Find the standard deviation for the quoted price of the TV.

    • $22.14


Try it4

Try it!

  • Given a set of data with a mean of 220.8 and a standard deviation of 17.0, find the k, or z, value of:

    • 200

    • k = -1.2235


Try it5

Try it!

  • Given a set of data with a mean of 220.8 and a standard deviation of 17.0, find the k, or z, value of:

    • 238.4

    • k = 1.0353


Try it6

Try it!

  • Given a set of data with a mean of 220.8 and a standard deviation of 17.0, find the k, or z, value of:

    • 229

    • k = .4824


Try it7

Try it!

  • Given a set of data with a mean of 220.8 and a standard deviation of 17.0, find the k, or z, value of:

    • 198.1

    • k = -1.3353


Try it8

Try It!

  • Exercise 4.12

  • SK = -.5430


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