Transcript standard deviations underneath the curve

Get out your Normal Curve WS!
Today’s Objectives:
You will be able to standardize a Normal
distribution.
Warm Up
Take a 3x5 Card from the front table and clear
your desk of everything but a pencil.
From memory, draw the Normal curve including all %’s inside
the curve and the standard deviations underneath the curve.
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
Homework Check
Homework Check
Measurement Activity
The big idea of this activity is that repeated
measurements of the same quantity tend
to follow a Normal distribution.
You will produce your own data that will be
displayed as a dot plot.
Measurement Activity
1. Describe the shape, center, and spread of the
distribution of measurements. Are there any outliers?
2. Calculate the average textbook length measurement
for your class. Compare with the value provided by
your teacher.
Class Average: _________
Diener Value: _________
Z-Scores
In Lesson 1 we learned the equation to find the
z-score of a particular observation…
𝑥 − 𝑚𝑒𝑎𝑛
𝑧=
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
The usual notation for the mean of a density
curve is 𝜇 (the Greek letter “mu”) and the
standard deviation of a density curve as 𝜎 (the
Greek letter “sigma”)
Z-Score
𝜇 and 𝜎 are the same symbols we use to
represent the mean and standard deviation
of a population distribution.
Z-Score
Any time you are asked to standardize a
Normal curve you must use the new
equation using the new symbols,
𝑥−𝜇
𝑧=
𝜎
New symbols—Same meaning
Standard Normal Distribution
If the variable we standardize has a Normal
distribution, then so does the new variable
𝑧. This new distribution is called the
standard Normal distribution.
Standard Normal Distribution
The standard Normal distribution is the
Normal distribution with mean 0 and
standard deviation 1.
Analyze
The 68-95-99.8 rule tells us that about 68% of
the observations fall within ±1 standard
deviation of the mean.
What if I wanted to find the percent of
observations that fall between ±1.25?
Does the rule help?
Standard Normal Table
All Normal distributions are the same when
we standardize.
Therefore we can find areas under any
Normal curve from a table of values.
Turn to the back of your book—Table A.
Standard Normal Table
Table A is a table of areas under the
standard Normal curve.
𝑍
Example 1
Find the proportion of observations from
the standard Normal distributions that are
(a) Less than -1.25
Example 2
Find the proportion of observations from
the standard Normal distributions that are
(b) Greater than 0.81
CAUTION!
A common mistake is to look up a z-value in
Table A and report the entry corresponding to
that z-value, regardless of whether the
problem asks for the area to the left or to the
right of that z-value!
The table give you the area to the LEFT
Example 3
Find the proportion of observations from
the standard Normal distribution that are
between -1.25 and 0.81.
How to Solve
From Table A, the area to the left of 0.81 is
0.7910 and the area to the left of -1.25 is
0.1056.
So just subtract the values to find the answer.
0.81
-1.25
-1.25
0.81
Practice Questions
Use Table A to answer the following
questions.
1. 𝑧 is less than -0.37
2. 𝑧 is greater than -0.37
3. 𝑧 is less than 2.15
4. 𝑧 is greater than 2.15
You Try!
Use Table A to answer the following
questions.
1. 𝑧 is less than -1.58
2. 𝑧 is greater than -1.58
3. 𝑧 is greater than -0.46
4. 𝑧 is less than 0.93
Practice Questions
Use Table A to answer the following
questions.
1. 𝑧 is between -1.33 and 1.65
2. 𝑧 is between 0.50 and 1.79
You Try!
Use Table A to answer the following
questions.
1. 𝑧 is between -2.05 and 0.78
2. 𝑧 is between -1.11 and -0.32
Practice Questions-Backwards
Use Table A to answer the following questions. Use
the values in Table A that come closest to satisfying
the condition.
In each case sketch a standard Normal curve with
your value of 𝒛 marked on the axis.
1. The 20th percentile of the standard normal
distribution.
2. 45% of all observations are greater than 𝑧.
You Try!
Use Table A to answer the following questions. Use
the values in Table A that come closest to satisfying
the condition.
In each case sketch a standard Normal curve with
your value of 𝒛 marked on the axis.
1. The 63rd percentile of the standard Normal
distribution.
2. 75% of all observations are greater than 𝑧.
Homework
Normal Distribution Worksheet
Due Wednesday