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Describing Spread: The Quartiles

Describing Spread: The Quartiles. Section 5.5. Chapter 5: Exploring Data: Distributions Describing Spread: The Quartiles.

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Describing Spread: The Quartiles

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  1. Describing Spread: The Quartiles Section 5.5

  2. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • The mean and median provide two different measures of the center of a distribution. But a measure of center alone can be misleading. Two neighborhoods with median house price $193,000 are very different if one has both mansions and modest homes and the other has little variation among houses.

  3. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • We are interested in the spread or variability of house prices as well as their centers. • The simplest useful numerical description consists of both a measure of center and a measure of spread.

  4. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • The simplest way to measure spread is with range. • The rangeis a measure of spread of a set of observations. It is obtained by subtracting the smallest observation from the largest observation.

  5. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • Turn to page 152 and find the range of the Percent of Adult Population of Hispanic Origin. • Range = 42.1% – 0.7% = 41.4%

  6. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • Turn to page 157 and find the range in the city mileage and the range in the highway mileage. • Range of city mileage = 48 – 15 = 33 mpg • Range of hwy. mileage = 45 – 23 = 22 mpg

  7. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • Find the range in the city mileage and the range in the highway mileage without the Toyota Prius. • Range of city mileage = 27 – 15 = 12 mpg • Range of hwy. mileage = 33 – 23 = 10 mpg • As you can see range can be influenced by outliers.

  8. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • We can improve our description of spread by also looking at the spread of the middle half of the data. • The first and third quartiles separate out the middle half.

  9. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • The first quartile (Q1) of a distribution or dataset is the point that exceeds one-quarter (or 25%) of the values. • Q1 is also the 25th percentile.

  10. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • The third quartile is the point that exceeds three quarters (or 75%) of the values. • The second quartile exceeds two quarters (or 50%) of the values, and so is equivalent to the median.

  11. Calculating the Quartiles Q1 and Q3

  12. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • To calculate the quartiles: • Use the median to split the ordered data set into two halves-an upper half and a lower half. • The first quartile, Q1 is the median of the lower half. • The third quartile, Q3 is the median of the upper half.

  13. Example 1

  14. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • The city mileages of the 12 gasoline-powered midsized cars, after sorting: • 15 16 18 19 20 20 21 21 21 22 24 27 • The first quartile is the median of the 6 observations in the lower half. • 15 16 18 19 20 20 • Q1 = (18 + 19)/2 = 18.5

  15. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • The city mileages of the 12 gasoline-powered midsized cars, after sorting: • 15 16 18 19 20 20 21 21 21 22 24 27 • The third quartile is the median of the 6 observations in the upper half. • 21 21 21 22 24 27 • Q3 = (21 + 22)/2 = 21.5

  16. Example 2

  17. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • Look on page 157 at the highway mileage of the 13 midsized cars. • Sort the mileage from smallest to largest. • 23 24 27 27 28 29 29 30 30 31 33 33 45 • Find the median of the entire data set. • Exclude it from the data set so two equal-sized groups can be formed.

  18. Chapter 5: Exploring Data: DistributionsDescribing Spread: The Quartiles • 23 24 27 27 28 29 30 30 31 33 33 45 • Find Q1 and Q3. • Q1 = (27 + 27) / 2 = 27mpg • Q3 = (31 + 33)/2 = 32 mpg

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