Theorems About Variability Section 9.4 Theorems About Variability

Objectives: 1. To describe variability in an arbitrary set of data. 2. To describe variability in mound- shaped data.

Theorem 9.3: Chebyshev’s Theorem Given any n measurements, at least 1 – 1/k2 of the measurements must lie within k standard deviations of the mean for k > 1. (At most 1/k2 of the data can lie outside the interval [x – ks, x + ks].)

EXAMPLE 1 What portion of scores must always lie within 3 standard deviations of the mean? 2 - k 9 1 - = 9 8 =

EXAMPLE 2 Check the theorem for the following set of data when k = 3. 1, 6, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12

Definition Mound-shaped data Mound-shaped data is a set of data in which the graph of frequencies is roughly symmetrical, and the mode is in the middle. The term “bell-shaped” is also used.

5 4 3 2 1 x1 x2 x3 x4 x5 x6 x7 x8 x9

Theorem 9.4: Empirical Rule Given a set of data that is mound-shaped, approximately 68% of the data will fall within 1 st. dev. of the mean, 95% will fall within 2 st. dev. of the mean, and 99.7% will fall within 3 st. dev. of the mean.

-3 -2 -1 0 1 2 3 68%

-3 -2 -1 0 1 2 3 95%

-3 -2 -1 0 1 2 3 99.7%

EXAMPLE 3 Using the data shown, check that it is roughly mound-shaped and show that the empirical rule applies. 8, 9, 10, 8, 7, 5, 11, 6, 8, 7, 9, 7, 8, 10, 9, 8, 3, 9, 6, 4

8, 9, 10, 8, 7, 5, 11, 6, 8, 7, 9, 7, 8, 10, 9, 8, 3, 9, 6, 4. 3 - 1 4 - 1 5 - 1 6 - 2 7 - 3 8 - 5 9 - 4 10 - 2 11 - 1 1 2 3 4 5 6 7 8 9 10 11

Find the mean and st. dev. 3 + 4 + 5 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 8 + 8 + 8 + 9 + 9 + 9 + 9 + 10 + 10 + 11 x = 152 / 20 = 7.6 s = 2.0

Find the intervals of 1, 2, and 3 st. dev. around the mean. [x - s, x + s] = [5.6, 9.6] contains 14 of 20 or 70% [x - 2s, x + 2s] = [3.6, 11.6] contains 19 of 20 or 95% [x - 3s, x + 3s] = [1.6, 13.6] contains 20 of 20 or 100%

Theorem 9.5: Central Limit Theorem If you repeatedly choose random samples each of size n (n  30) from the sample population and compute the mean for each sample, the graph of the frequency distribution of the population of all sample means will be mound-shaped.

Homework: pp. 464-465

►A. Exercises What percentage of any set of data must always fall within 1. 1 standard deviation of the mean?

►A. Exercises What percentage of any set of data must always fall within 5. How many full standard deviations are necessary to guarantee that 97% of the data is included?

►A. Exercises If a set of data is mound-shaped with x = 110 and s = 3, 6. What percentage of data must lie in [104, 116]?

►A. Exercises If a set of data is mound-shaped with x = 110 and s = 3, 7. Give the interval containing 68% of the data.

►A. Exercises If a set of data is mound-shaped with x = 110 and s = 3, 8. How many full standard deviations are necessary to guarantee that 97% of the data is included?

►A. Exercises If a set of data is mound-shaped with x = 110 and s = 3, 9. Give the interval containing at least 97% of this data.

►A. Exercises If a set of data is mound-shaped with x = 110 and s = 3, 10. If the data were not mound-shaped, how spread out could 97% of the data be? Give the interval.

►B. Exercises Verify the empirical rule on the following data. 41, 43, 46, 43, 39, 40, 43, 45, 41, 44, 47, 43, 44, 42 16. Show that the data is mound shaped.

41, 43, 46, 43, 39, 40, 43, 45, 41, 44, 47, 43, 44, 42 39 - 1 40 - 1 41 - 2 42 - 1 43 - 4 44 - 2 45 - 1 46 - 1 47 - 1 39 40 41 42 43 44 45 46 47

►B. Exercises Verify the empirical rule on the following data. 41, 43, 46, 43, 39, 40, 43, 45, 41, 44, 47, 43, 44, 42 17. For 1 standard deviation

►B. Exercises Verify the empirical rule on the following data. 41, 43, 46, 43, 39, 40, 43, 45, 41, 44, 47, 43, 44, 42 18. For 2 standard deviations

►B. Exercises Verify the empirical rule on the following data. 41, 43, 46, 43, 39, 40, 43, 45, 41, 44, 47, 43, 44, 42 19. For 3 standard deviations

►B. Exercises 20. Since test scores for national tests such as the SAT are typically mound-shaped, what percentage of students score 700 or above?

■ Cumulative Review: 25. Give the three smallest positive and three largest negative angles coterminal with 42°.

■ Cumulative Review: 26. Give the exact values of the sine, cosine, and tangent of 210° and of 135°.

■ Cumulative Review: 27. g(x) = 4 + ex Identify the horizontal asymptotes of each function. 27. g(x) = 4 + ex

■ Cumulative Review: 28. h(x) = Tan-1 x Identify the horizontal asymptotes of each function. 28. h(x) = Tan-1 x

■ Cumulative Review: 29. f(x) = Identify the horizontal asymptotes of each function. 29. f(x) = 8x – 5 9x + 10