Lecture 23
The Normal Distribution
Summer 2021
Announcements
Weekly Goals
Variance &
Standard Deviation
How Far from the Mean (Average)?
R ← M ← S ← D ← A ← D
Salient Feature of Standard Deviation
No matter what the shape of the distribution,�
the bulk of the data are in the range ��Mean ± A Few SDs
Standard Units
Standard Units
(Demo)
Discussion Question
Find whole numbers that are close to:
Average in standard units ≅ 0
≅ 27
Average + 1SD in standard units ≅ 1
27 + SD ≅ 33
SD ≅ 33-27 ≅ 6
(Demo)
Chebyshev's Inequality
How Big are Most of the Values?
Chebyshev’s Inequality
No matter what the shape of the distribution,
the proportion of values (i.e., fraction of the population) in the range “Mean ± z SDs” is
is at least 1 - 1/z²
Chebyshev’s Bounds
Range | Proportion |
Mean ± 2 SDs | at least 1 - 1/4 (75%) |
Mean ± 3 SDs | at least 1 - 1/9 = 8/9 (88.89%) |
Mean ± 4 SDs | at least 1 - 1/16 = 15/16 (93.75%) |
Mean ± 5 SDs | at least 1 - 1/25 = 24/25 (96%) |
No matter what the distribution looks like!
The SD and the Histogram
The SD and Bell-Shaped Curves
If a histogram is bell-shaped, then
(Demo: Maternal Heights)
Point of Inflection
The Normal Distribution
(AKA, Bell-Shaped Curve)
(AKA, Gaussian Distribution)
The Standard Normal Curve
A beautiful formula that we won’t use at all:
Bell Curve
Normal Proportions
How Big are Most of the Values?
No matter what the shape of the distribution,
the bulk of the data are in the range
“Mean ± A Few SDs”
If a histogram is bell-shaped, then
“Mean ± 3 SDs”
Bounds and Normal Approximations
NOTE: If our random distribution is Normal, then we don’t need to bootstrap.
Our 95% confidence interval is simply: Mean ± 2 SDs.
Percent in Range | All Distributions (Chebyshev’s) ≥ 1-1/z2 | Normal Distribution |
Mean ± 1 SD | ≥ 1-1/1 = 0 (0%) | ≈ 68% |
Mean ± 2 SDs | ≥ 1-1/22 = 3/4 (75%) | ≈ 95% |
Mean ± 3 SDs | ≥ 1-1/32 = 8/9 (88.89%) | ≈ 99.73% |
A “Central” Area
Source: Statistics How To
Central Limit Theorem
Sample Means (Averages)
Central Limit Theorem
If the sample is
Then, regardless of the distribution of the population,
the probability distribution of the sample average
is roughly normal
(Demo)