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Chapter 4

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Chapter 4

The Description of Data:Measures of

Variation and Dispersion

- We have looked at measures of the center, or location, of data.
- We also need a measure of the dispersion of data.

- 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.

- 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.

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

- The variance is the average squared distance to the mean.
- The standard deviation is the square root of the variance.

- 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.

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

- 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.

- 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 rule only works for distributions that are normal (bell shaped).
- The empirical rule is much more accurate than Chebyshev’s Theorem.

- 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.

- 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 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.

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- Exercise 4.12
- SK = -.5430