#### Math - 2018-19

# AII.11 - Normal Curve

AII.11The student willa)

identifyanddescribeproperties of a normal distribution;b)

interpretandcomparez-scores for normally distributed data; andc)

applyproperties of normal distributions todetermineprobabilities associated with areas under the standard normal curve.

*Adopted: 2016*

### BIG IDEAS

- I can predict election
winners from polling data, decide the official weight
of a Hershey bar by weighing a fraction of the candy bars coming off a
production line, and determine theriskassociated with price-fluctuations of
Verizon stock.

- I will understand that deviations are measures used to quantify the amount of variation or scattering in a set of data values.

### UNDERSTANDING THE STANDARD

·
The focus of
this standard is on the interpretation of descriptive statistics, *z*-scores, probabilities, and their
relationship to the normal curve in the context of a data set.

· Descriptive statistics include measures of center (mean, median, mode) and dispersion or spread (variance and standard deviation).

·
Variance (*σ*^{ 2}) and standard deviation (*σ*) measure the spread of data about the
mean in a data set.

· Standard deviation is expressed in the original units of measurement of the data.

· The greater the value of the standard deviation, the further the data tends to be dispersed from the mean.

· In order to develop an understanding of standard deviation as a measure of dispersion (spread), students should have experiences analyzing the formulas for and the relationship between variance and standard deviation.

·
A normal
distribution curve is the family of symmetrical, bell-shaped curves defined
by the mean and the standard deviation of a data set. The arithmetic mean (*μ*) is located on the line of symmetry of the curve and is approximately equivalent
to the median and mode of the data set.

· The normal curve is a probability distribution and the total area under the curve is 1.

· For a normal distribution, approximately 68 percent of the data fall within one standard deviation of the mean, approximately 95 percent of the data fall within two standard deviations of the mean, and approximately 99.7 percent of the data fall within three standard deviations of the mean. This is often referred to as the Empirical Rule or the 68-95-99.7 rule.

NOTE: This chart illustrates percentages that correspond to subdivisions in one standard deviation increments. Percentages for other subdivisions require the table of Standard Normal Probabilities or a graphing utility.

The mean and standard deviation of a normal distribution affect the location and shape of the curve. The vertical line of symmetry of the normal distribution falls at the mean. The greater the standard· deviation, the wider (“flatter” or “less peaked”) the distribution of the data.

·
A *z*-score derived from a
particular data value tells how many standard deviations that data value falls
above or below the mean of the data set. It is positive if the data value lies
above the mean and negative if the data value lies below the mean.

·
A standard normal distribution is the set of all *z*-scores. The mean of the data in a standard
normal distribution is 0 and the standard deviation is 1. This allows for the comparison of
unlike normal data.

· The table of Standard Normal Probabilities and graphing utilities may be used to determine normal distribution probabilities.

·
Given a *z*-score (*z*), the table of Standard Normal
Probabilities (*z*-table) shows the
area under the curve to the left of *z*.
This area represents the proportion of observations with a *z*-score less than the one specified. Table rows show the *z*-score’s whole number and tenths place.
Table columns show the hundredths place.

### ESSENTIALS

**The student will use
problem solving, mathematical communication, mathematical reasoning,
connections, and representations to**

· AII.11b_{1} **Solve**
problems involving the relationship of the mean, standard deviation, and
z-score of a normally distributed data set.

· AII.11a_{1} **Identify**
the properties of a normal distribution.

· AII.11_{1} **Use**
a graphing utility to **investigate**, **represent**, and **determine** relationships between a normally distributed data set and
its descriptive statistics.

· AII.11a_{2} **Describe**
how the standard deviation and the mean affect the graph of the normal distribution.

· AII.11c_{1} **Represent**
probability as area under the curve of a standard normal distribution.

· AII.11b_{2} **Compare**
two sets of normally distributed data using a standard normal distribution and *z*-scores, given the mean and standard
deviation.

· AII.11c_{1} **Represent**
probability as area under the curve of a standard normal distribution.

· AII.11c_{2} **Use**
the graphing utility or a table of Standard Normal Probabilities to **determine** probabilities associated with
areas under the standard normal curve.

### KEY VOCABULARY

normal
distribution, probabilities, standard normal curve, area under normal curve,
properties, standard deviation, normal probability distribution, mean, standard
normal distribution, standard normal probability distribution, z-score,
percentile

*Updated: Aug 23, 2018*