Chapter 3 Location and Spread

1. Chapter 3 Location and Spread Location and Spread Central Tendency Variation Adding and Multiplying a Constant Overall Shape Distribution Empirical Rule

2. Location and Spread A data consists of test scores of 33 students What score did the students get? An answer would be somewhere around 55 points. A better answer would be about 55 points give or take 10 points.

3. Location and Spread • Central Tendency (Location) • a measure of where most of the data is located • a measure of where the data is centered • Variation (Spread) • by how much do the observations vary • by how much will observations miss the center In the phrase ‘the scores are about 55 give or take 10 points’, 55 would be a measure of location and 10 a measure of spread.

4. Central Tendency • Non-resistant : affected by extreme values • Mean or average : = xi / n • Midrange: MR = ( Min + Max )/2 Mean = (39 + 41 + . . . + 73 + 75 + 75)/33 = 1873/33 = 56.76 MR = ( 39 + 75 )/2 = 114/2 = 57

5. Central Tendency • Resistant : not affected by extreme values • Median : value of the middle observations when the data is sorted • Midhinge : (Q1 + Q3)/2 • Mode : most frequent occurring observation Median = 55 Midhinge = ( 50.5 + 62.5 ) / 2 = 56.5

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7. Variation / Spread Variance : average squared deviations Standard Deviation (SD) : average distance from the mean SD

8. Variation/Spread Example for Variance and SD Var = [ (39 – 56.76)2 + (41-56.76)2 + . . . + (75-56.76)2 ] / (33-1) = 91.06 SD = Squareroot (91.06) = 9.54

9. Variation/Spread Range: r = max – min Interquartile range: IQR = Q3 – Q1 Coefficient of Variation : SD/ • Range: r = 75 – 39 = 36 • Interquartile range: IQR = 62.5 – 50.5 = 12 • Coefficient of Variation : CV = (9.54 / 56.76)*100% • = 16.81%

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11. Summary • Non-Resistant • Location • Mean, Midrange • Spread • Standard Deviation, Variance, Range and CV • Resistant • Location • Median, Mode, Midhinge • Spread • Interquartile range

12. Adding and Multiplying a Constant c • Data : 25 30 28 36 • Mean = 29.75 • SD = 4.65 What happens if we add or multiply a constant number to all the observations?

13. Adding and Multiplying a Constant c

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16. Overall Distribution Shape • Symmetric • Balanced : “Midrange = Midhinge = Median” • Uniform : Rectangular shape • Bell-shaped • Skewed • Right-skewed : presence of extremely high values • Left-skewed : presence of extremely low values

17. Uniform : Rectangular shape

18. Bell-shaped Short-tailed Long-tailed

19. Skewed Distribution • Right-Skewed : presence of extremely high values in the data • Midrange > Median • Left-Skewed : presence of extremely low values in the data • Midrange < Median

20. Distribution of the Middle 50% • Symmetric • Midhinge = Median • Left-Skewed • Midhinge < Median • Right-Skewed • Midhinge > Median

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22. Empirical Rule • 68% of the observations are within 1 SD from the Mean • ± 1SD • 95% of the observations are within 2 SD from the Mean • ± 2SD • 99.7% of the observations are within 3 SD from the Mean • ± 3SD

23. Empirical Rule • Example: A light bulb manufacturer tested the lifetimes of their product. The average lifetime is 300 hours with SD = 20 hours computed from a sample of 50 light bulbs. • 68 % of the light bulbs have a lifetime within 300 ± 20 hours (280, 320). • 95% of the light bulbs have a lifetime within 300 ± 40 hours (260, 340). • 99.7% of the light bulbs have a lifetime within 300 ± 60 hours (240, 360).

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