Transcript Normal curves - Greer Middle College

Get out your Density Curve WS!
Today’s Objectives:
You will be able to describe a Normal curve.
You will be able to find percentages and
standard deviations based on a Normal curve.
Normal Curves
The Normal curves are one of the most
common types of density curve.
All Normal curves are...
•
•
•
Symmetric
Single-peaked
Bell-shaped
Normal Curves
Normal curves describe Normal distributions.
Normal distributions are anything but “normal.”
They play a huge role in statistics.
Capitalize the “n” in normal when referring to a
Normal distribution or curve.
Why Important?
Normal distributions are important in
statistics because…
1. Normal distributions are good descriptions
for some distributions of real data.
Ex. Scores on tests, repeated careful measures of the same
quantity, characteristics of biological populations…
Why Important?
Normal distributions are important in
statistics because…
2. Normal distributions are good
approximations to the results of many kinds
of chance outcomes.
Ex. tossing a coin many times, rolling a die…
Why Important?
Normal distributions are important in
statistics because…
2. Most importantly, Normal distributions are
the basis for many statistical inference
procedures.
Inference—the process of arriving at some conclusion that
possesses some degree of probability relative to the premises
Normal Curves
• Symmetric, single-peaked, and bell-shaped.
• Tails fall off quickly, so do not expect outliers.
• Mean, Median, and Mode are all located at
the peak in the center of the curve.
Mean Median Mode
Normal Curves
The mean fixes the center of the curve and
the standard deviation determines its
shape.
The standard deviation fixes the spread of
a Normal curve.
Remember, spread tells us how much a data sample is
spread out or scattered.
Normal Curves
The mean and the standard deviation
completely specifies the curve.
Changing the mean changes its location on the axis.
Changing the standard deviation changes the shape of
a Normal curve.
The Empirical Rule
Also known as the 68—95—99.7 Rule
because these values describe the
distribution.
The empirical rule applies only to NORMAL
DISTRIBUTIONS!!!!!
The Empirical Rule
The Empirical Rule states that
•
Approximately 68% of the data values fall within one
standard deviation of the mean
•
Approximately 95% of the data values fall within 2
standard deviations of the mean
•
Approximately 99.8% of the data values fall within 3
standard deviations of the mean
The Empirical Rule
68% of the observations fall within ±1
standard deviation around the mean.
68% of data
−1 𝑆
𝑥
1𝑆
The Empirical Rule
95% of the observations fall within ±2
standard deviation around the mean.
95% of data
−2 𝑆
−1 𝑆
𝑥
1𝑆
2𝑆
The Empirical Rule
99.8% of the observations fall within ±3
standard deviation around the mean.
99.8% of data
−3 𝑆
−2 𝑆
−1 𝑆
𝑥
1𝑆
2𝑆
3𝑆
The Empirical Rule
99.8% of data
95% of data
68% of data
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
𝑥 + 3𝑆
Examples
Assume you have a normal distribution of test
scores with a mean of 82 and a standard
deviation of 6.
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
.1%
𝑥 + 3𝑆
Examples
1. What percent of the data scores
were above a 76?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
𝑥 + 3𝑆
64
70
76
82
88
94
100
Examples
2. 68% of the data fall between what
two scores?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
𝑥 + 3𝑆
64
70
76
82
88
94
100
Examples
3. What percent of the data scores fall
between 70 and 100?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
𝑥 + 3𝑆
64
70
76
82
88
94
100
Examples
4. How many standard deviations away
from the mean is 88, and in which
direction?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
𝑥 + 3𝑆
64
70
76
82
88
94
100
Examples
5. How many standard deviations away
from the mean is 64, and in which
direction?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
𝑥 + 3𝑆
64
70
76
82
88
94
100
Examples
A charity puts on a relay race to raise money.
The times of the finishes are normally
distributed with a mean of 53 minutes and a
standard deviation of 9.5 minutes.
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
𝑥
𝑥 + 1𝑆
𝑥 + 2𝑆
.1%
𝑥 + 3𝑆
Examples
1. What percent of the data times were
between 34 and 81.5 minutes?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
24.5 34 43.5
𝑥
53
𝑥 + 1𝑆
𝑥 + 2𝑆
62.5 72
.1%
𝑥 + 3𝑆
81.5
Examples
2. 95% of the data fall between what
two times?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
24.5 34 43.5
𝑥
53
𝑥 + 1𝑆
𝑥 + 2𝑆
62.5 72
.1%
𝑥 + 3𝑆
81.5
Examples
3. What percent of the data times
were below 72 minutes?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
24.5 34 43.5
𝑥
53
𝑥 + 1𝑆
𝑥 + 2𝑆
62.5 72
.1%
𝑥 + 3𝑆
81.5
Examples
4. How many standard deviations away
from the mean is 34 and in which
direction?
13.5%
34%
34%
13.5%
2.4%
2.4%
.1%
𝑥 − 3𝑆
𝑥 − 2𝑆
𝑥 − 1𝑆
24.5 34 43.5
𝑥
53
𝑥 + 1𝑆
𝑥 + 2𝑆
62.5 72
.1%
𝑥 + 3𝑆
81.5
Homework
Normal Curve Worksheet
Due Monday