Download Normal distribution

Warm-Up: Determine whether a linear model is
appropriate. Find the line of best fit.
Water use (thousand of
litres)
Power use (kWh)
21
38
22
42
24
64
26
72
31
87
35
101
25
54
24
55
23
49
25
52
y=4.5x-53.92
Normal
Distribution
Normal Distributions
• Approximately normal distribution
Data sets that can be described as having bar
graphs that roughly fit a bell-shaped pattern.
• Normal distribution
A distribution of data that has a perfect bell shape.
• Normal curves
Perfect bell-shaped curves.
3
Normal Distributions
Data which is normally distributed has the majority of the
data centered at the mean.
• Symmetry
Every normal curve has a vertical axis of symmetry,
splitting the bell-shaped region outlined by the
curve into two identical halves. We can refer to it
as the line of symmetry.
4
Normal Distributions
• Median/mean.
We call the point of
intersection of the horizontal
axis and the line of symmetry
of the curve the center of the
distribution. The center is both
the median and the mean
(average) of the data. We use
the Greek letter µ (mu) to
denote this value.
5
Normal Distributions
The standard deviation of a
normal distribution is the
horizontal distance between
the line of symmetry of the
curve and one of the two
points of inflection (P or P' )
6
Special Characteristics
·68% of the data is within 1 standard deviation (1σ)
of the mean (µ).
·95% is within 2σ of µ
·99.7% (or practically 100%) is within 3σ of µ
Example
The distribution of head circumference among males is
normally distributed with µ=22.8 in and σ=1.1 in.
·What percent have a head circumference greater than 23.9 in?
.16 or 16%
·What percent have a head circumference between 21.7 and 25 in?
.815 or 81.5%
·What percent have a head circumference less than 20.6 in?
.025 or 2.5%
What if the values we want to know about
are not σ, 2σ, or 3σ away?
• Standardizing
To standardize a data value x, we measure how far x
has strayed from the mean  using the standard
deviation  as the unit of measurement.
• Z-value
A standardized data value.
·We have to find the standardized value and use the zchart
Example:
µ=22.8 in and σ=1.1 in.
We want to know the percent of men with
head circumference less than 24.3 inches?
Reading
your Z-Chart
Example:
µ=22.8 in and σ=1.1 in.
We want to know the percent of men with
head circumference less than 24.3 inches?
0.9131 or
91.31%
Example:
µ=22.8 in and σ=1.1 in.
We want to know the percent of men
with head circumference greater than
24.3 inches?
We know that 91.31% is greater than
24.3
Subtract from 100% to find the
percentage that’s less.
.0869 or
8.69%
Example:
µ=22.8 in and σ=1.1 in.
We want to know the percent of men with head
circumference between 20.9 and 24.3 inches?
Find percent less than 24.3 (We already did that)
91.31%
Find percent less than 20.9
z=-1.72
.0427 or 4.27%
Subtract percent less than 20.9 from percent less
than 24.3
91.31– 4.27
= 87.04%
You try it!
The distribution of weights for 6-month-old
boys is approximately normal with mean
µ=17.25 lbs and σ=2 lbs.
What is the probability a child will weigh greater than
20.5 lb?
.9484
What is the probability a child will weigh between 16.5
and 19.2 lbs?
.4845