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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

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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