Normal distribution

+
Normal distribution
Slides edited by Valerio Di Fonzo for www.globalpolis.org
Based on the work of Mine Çetinkaya-Rundel of OpenIntro
The slides may be copied, edited, and/or shared via the CC BY-SA license
Some images may be included under fair use guidelines (educational purposes)

Obtaining Good Samples
● Unimodal and symmetric, bell shaped curve
● Many variables are nearly normal, but none are exactly
normal
● Normal Distribution has two parameters and it is denoted as
N(μ, σ) → Normal with mean μ and standard deviation σ

Heights of males
“The male heights on OkCupid very
nearly follow the expected normal
distribution -- except the whole thing is
shifted to the right of where it should be.
Almost universally guys like to add a
couple inches.”
“You can also see a more subtle vanity at
work: starting at roughly 5' 8", the top of
the dotted curve tilts even further
rightward. This means that guys as they
get closer to six feet round up a bit more
than usual, stretching for that coveted
psychological benchmark.”
http://blog.okcupid.com/index.php/the
-biggest-lies-in-online-dating

Heights of females
“When we looked into the
data for women, we were
surprised to see height
exaggeration was just as
widespread, though without
the lurch towards a
benchmark height.”
http://blog.okcupid.com/index.php/the
-biggest-lies-in-online-dating

Normal distributions
with different parameters

SAT scores are distributed nearly normally with mean 1500 and
standard deviation 300. ACT scores are distributed nearly
normally with mean 21 and standard deviation 5. A college
admissions officer wants to determine which of the two
applicants scored better on their standardized test with respect
to the other test takers: Pam, who earned an 1800 on her SAT,
or Jim, who scored a 24 on his ACT?

Standardizing with Z scores
Since we cannot just compare these two raw scores, we instead compare
how many standard deviations beyond the mean each observation is.
● Pam's score is (1800 - 1500) / 300 = 1 standard deviation above the
mean.
● Jim's score is (24 - 21) / 5 = 0.6 standard deviations above the mean.
As we can see, Pam
performed better than Jim

Standardizing with Z scores (cont.)
These are called standardized scores, or Z scores.
● Z score of an observation is the number of standard
deviations it falls above or below the mean.
● Z score of mean = 0
● Z scores are defined for distributions of any shape, but
only when the distribution is normal can we use Z scores
to calculate percentiles.
● Observations that are more than 2 SD away from the
mean (|Z| > 2) are usually considered unusual.

Percentiles
● Percentile is the percentage of observations that fall below a
given data point.
● Graphically, percentile is the area below the probability
distribution curve to the left of that observation.

Calculating percentiles --
using computation
There are many ways to compute percentiles/areas under the
curve. R:
Applet: www.socr.ucla.edu/htmls/SOCR_Distributions.html

Computing percentiles using applet
http://bitly.com/dist_calc

Calculating percentiles using tables

Practice
Z = 1.28 [1.2 (vertical table) + 0.08 (horizontal table)]

68-95-99.7 Rule
For nearly normally distributed data,
● about 68% falls within 1 SD of the mean,
● about 95% falls within 2 SD of the mean,
● about 99.7% falls within 3 SD of the mean.
It is possible for observations to fall 4, 5, or more standard deviations away from the
mean, but these occurrences are very rare if the data are nearly normal.

Describing variability using the
68-95-99.7 Rule
SAT scores are distributed nearly normally with mean 1500 and standard deviation
300.
● ~68% of students score between 1200 and 1800 on the SAT.
● ~95% of students score between 900 and 2100 on the SAT.
● ~$99.7% of students score between 600 and 2400 on the SAT.

Number of hours of sleep
on school nights
Mean = 6.88 hours, SD = 0.92 hrs

on school nights
72% of the data are within 1 SD of the mean: 6.88 ± 0.93

on school nights
92% of the data are within 1 SD of the mean: 6.88 ± 2 x 0.93

on school nights

Evaluating normal distribution

NBA players have more variable heights than other people.
As we can see, the plot shows a left skew.

Six sigma
The term six sigma process comes from the notion that if one
has six standard deviations between the process mean and
the nearest specification limit, as shown in the graph,
practically no items will fail to meet specifications.
http://en.wikipedia.org/wiki/Six_Sigma

Normal distribution

More Related Content

What's hot (20)

Similar to Normal distribution (20)

More from Global Polis (11)

Recently uploaded (20)

Normal distribution