Two types of deviation: • Standard Deviation. • Quartile Deviation. 2. Aspirin
Example. You buy 10 bottles of Bare Aspirin (50 count) and 10 bottles of MM
Brand.
Statistics Lecture Notes
1. Deviation Two types of deviation: • Standard Deviation • Quartile Deviation
2. Aspirin Example You buy 10 bottles of Bare Aspirin (50 count) and 10 bottles of MM Brand Aspirin (50 count) and record the exact number of aspirin in each bottle. The results are:
3. Bare Aspirin Distribution Frequency Distribution Bare Aspirin Number Freq Freq×Num 47 1 47 48 1 48 49 2 98 50 3 150 51 1 51 52 1 52 54 1 54 Total 10 500 1
2
4. MM Aspirin Distribution MM Aspirin Number Freq Freq×Num 20 1 20 31 1 31 39 1 39 45 1 45 50 2 100 57 1 57 64 1 64 66 1 66 78 1 78 Total 10 500
5. Comparison Average mean median mode
Bare 50 50 50
MM 50 50 50
6. Bare Aspirin Deviation Bare Aspirin X Freq X − µ (X−µ)2 Freq×(X−µ)2 47 1 -3 9 9 48 1 -2 4 4 49 2 -1 1 2 50 3 0 0 0 51 1 1 1 1 52 1 2 4 4 54 1 4 16 16 Total 10 36
Standard Deviation =
r
Total = 10
r
36 = 1.90 10
3
7. Sigma The Greek letter sigma σ is often used to represent the standard deviation of a set of data.
8. MM Aspirin Deviation MM Aspirin X Freq X − µ (X−µ)2 Freq×(X−µ)2 20 1 -30 900 900 31 1 -19 361 361 39 1 -11 121 121 45 1 -5 25 25 50 2 0 0 0 57 1 7 49 49 64 1 14 196 196 66 1 16 256 256 78 1 28 784 784 Total 10 2692
9. Comparison For the MM Aspirin Standard Deviation =
r
Total = 10
r
2692 = 16.4 10
Expected number of aspirin per bottle • Bare Brand: 50 ± 1.9 • MM Brand: 50 ± 16.4
10. Quartile Deviation • Given a data set {X1 , X2 , · · · .Xn } listed in order from smallest to largest (repetitions allowed). • To compute quartile deviation: – Step 1. Find the median of the data set and call this median Q2 – Q2 is also called the second quartile values. – Step 2. Use Q2 to divide the original data set into two parts.
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– The lower 50% consists of those values ≤ Q2 ; – the upper 50% consists of values ≥ Q2 . – Note: when n is odd, include Q2 in both the lower and upper data sets.
11. Quartile Deviation Cont’d • Step 3. Q1 = the median of the lower data set – Q3 = the median of the upper data set – Q1 is called the first quartile position. – Q3 is called the third quartile position. Q3 − Q 1 • Step 4. is called the quartile deviation. 2
12. Example 1. • Test Scores: {49, 54, 59, 60, 62, 65, 65, 68, 77, 83, 84, 89, 90} • The number of data points is n = 13 Q2 = the median of the data set • = the 7th data point = 65.
13. Example 1. Cont’d • {49, 54, 59, 60, 62, 65, 65, 68, 77, 83, 84, 89, 90} • Lower data set: {49, 54, 59, 60, 62, 65, 65} Q1 = the median of the lower data set • = the 4th data point = 60. • Upper data set: {65, 68, 77, 83, 84, 89, 90} Q3 = the median of the upper data set • = the 4th data point = 83. Q3 − Q 1 83 − 60 • Quartile deviation = = = 11.5. 2 2
14. Example 2. • Test Scores: {49, 54, 59, 60, 62, 65, 68, 77, 83, 84, 89, 90} • The number of data points is n = 12
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Q2 = the median of the data set = the average of X6 and X7 • = (65 + 68)/2 = 66.5.
15. Example 2. Cont’d • {49, 54, 59, 60, 62, 65, 68, 77, 83, 84, 89, 90} • Lower data set: {49, 54, 59, 60, 62, 65} Q1 = the median of the lower data set • = (59 + 60)/2 = 59.5. • Upper data set: {68, 77, 83, 84, 89, 90} Q3 = the median of the upper data set • = (83 + 84)/2 = 83.5. 83.5 − 59.5 Q3 − Q 1 = = 12. • Quartile deviation = 2 2
16. Exercise Puzzler • Suppose you jog for one mile at a speed of 5 mph • For the one mile return trip, you walk at a speed of 3 mph • What’s your average speed for the 2 miles? – (a) 3.75 mph – (b) 4 mph – (c) 4.25 mph
17. Distance–Time–Rate Formula • The important formula is • distance = rate × time • Equivalently distance • time = rate • For example, the time required to drive 120 miles if your speed is 60 mph is 120 • t= = 2 hours 60
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18. Jogging One Mile • d = 1 mile • r = 5 mph 1 mile • t= 5mile/hr 60 min 1 = hr × 5 1 hr = 12 min
19. Walking One Mile • d = 1 mile • r = 3 mph 1 mile • t= 3mile/hr 1 60 min = hr × 3 1 hr = 20 min
20. Puzzler Solved Jogging one Mile and Walking One Mile: • d = 2 mile • t = 12 + 20 = 32 min d • r= t =
2 miles 60 min × 32 min 1 hr
=
60 miles 16 hr
=
15 miles = 3.75 mph 4 hr
21. Find the Missing Data Point • The mean of four numbers is 75 • Three of the numbers are 79, 62, and 71 • What is the fourth number?
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• Solution: – Call the fourth number X. 79 + 62 + 71 + X – Then µ = = 75 4 – Multiply by 4: – 79 + 62 + 71 + X = 4 × 75 = 300 – 212 + X = 300 – X = 300 − 212 = 88
