Download Interpreting the Standard Deviation

Interpreting the Standard
Deviation
1. Chebyshev’s Rule
2. 68 – 95 – 99.7 Rule
3. Measures of Relative Standing
1. z - score
2. Percentile
Chebyshev’s Rule
• In any set of data, regardless of the shape
of the frequency distribution of the data, at
least
1 – 1/k2
of the values must lie within ±k standard
deviation of the mean.
Chebyshev’s Rule
(Rephrasing)
•
In any set of data, regardless of the shape
of the frequency distribution of the data, at
most
1/k2
of the values must lie outside ± k standard
deviation of the mean.
Histogram of Runtime
(Example 2.11 pg 74)
9
8
7
Freq
6
5
4
3
2
1
0
1
2
3
4
5
Runtime
6
7
8
9
10
Chebyshev’s Rule
Example 2.11, pg 74
x: Rat's time to move through a maze (minutes)
Raw
Sort
x
x
1.97 6.06 0.60 3.65
Class
1.74 5.63 1.06 3.77
3.77 4.25 1.15 3.81
0.5
0.60 4.44 1.65 4.02
1.5
2.75 5.21 1.71 4.25
2.5
5.36 1.93 1.74 4.44
3.5
4.02 2.02 1.93 4.55
4.5
3.81 4.55 1.97 5.15
5.5
1.06 5.15 2.02 5.21
6.5
3.20 3.37 2.06 5.36
7.5
9.70 7.60 2.47 5.63
8.5
1.71 2.06 2.75 6.06
9.5
1.15 3.65 3.16 7.60
8.29 3.16 3.20 8.29
xbar = 3.74
2.47 1.65 3.37 9.70
xbar - s = 1.54
xbar - 2s = -0.66
xbar - 3s = -2.86
Freq
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
3
8
4
6
4
2
0
2
0
1
s = 2.2
xbar + s = 5.94
xbar + 2s = 8.14
xbar + 3s = 10.34
Count %
23 77
28 93
30 100
68 – 95 – 99.7 Rule
(Also called Empirical Rule)
• If a set of data has a mound shaped and
symmetric frequency distribution, then
approximately
• 68% of the values will fall within ± 1
standard deviation of the mean.
• 95% of the values will fall within ± 2
standard deviation of the mean.
• 99.7% of the values will fall within ± 3
standard deviation of the mean.
68 – 95 – 99.7 Rule (Rephrased)
(Also called Empirical Rule)
• If a set of data has a mound shaped and
symmetric frequency distribution, then
approximately
• 32 % of the values will fall outside ± 1
standard deviation of the mean.
• 5% of the values will fall outside ± 2
standard deviation of the mean.
• 0.3% of the values will fall outside ± 3
standard deviation of the mean
Symmetric and mound shaped frequency
distribution
• µ = 60, σ = 10 (like in example 2.13, pg 76)
Symmetric and Mound Shaped frequency
distribution
• Normal Model
3. Measures of Relative Standing
z-score
• Represents the distance between a given
measurement and the mean, expressed in
standard deviations.
• Sample: z = (x – xbar)/s
• Population: z = (x - µ)/σ
3. Measures of Relative Standing
z-score
• Example (2.13 pg 76), µ = 60, σ = 10.
• Assume the frequency distribution is
symmetric and mound shaped, then the z-score
of x = 40 is
•
z = (40 – 60)/10 = -2
• With what frequency will we find
measurements less than 40?
• Since 40 is 2 standard deviation of the mean,
that frequency will be ½ of 5% or 2.5%
Measures of Relative Standing
pth percentile
• p% of the measurements fall below and
(100–p)% fall above
• Formula:
• pth percentile = measurement in p(n+1)/100
order of magnitude
• Example 2.14 pg 82: n=51 (data from expl 2.2)
• The 25 percentile is the measurement in
25(52)/100 = 13 order of magnitude, that is to
say x (13) = 7.9 (from the sorted data base)
Relation between z-scores and
percentiles, after the 68-95-99.7 rule
z-score
-3
-2
-1
0
1
2
3
Percentile
0.15
2.5
16
50
84
97.5
99.85