# A.9 and AII.11 and *AII.11 - Normal Curve

A.9  The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores.

AII.11    The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

AII.11  The student will

a)  identify and describe properties of a normal distribution;

b)  interpret and compare z-scores for normally distributed data; and

c)  apply properties of normal distributions to determine probabilities associated with areas under the standard normal curve.

Bloom's Level:  Apply

### 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

• Descriptive statistics may include measures of center and dispersion.
• Variance, standard deviation, and mean absolute deviation  measure the dispersion of the data.
• The sum of the deviations of data points from the mean of a data set is 0.
• Standard deviation is expressed in the original units of measurement of the data.
• Standard deviation is calculated by taking the square root of the variance.
• The greater the value of the standard deviation, the further the data tend to be dispersed from the mean.
• For a data distribution with outliers, the mean absolute deviation may be a better measure of dispersion than the standard deviation or variance.
• A z-score (standard score) is a measure of position derived from the mean and standard deviation of data.
• A z-score derived from a particular data value tells how many standard deviations that data value is 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 normal distribution curve is a symmetrical, bell-shaped curve defined by the mean and the standard deviation of a data set.  The mean is located on the line of symmetry of the curve.
• Areas under the curve represent probabilities associated with continuous distributions.
• 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.
• The mean of the data in a standard normal distribution is 0 and the standard deviation is 1.
• The standard normal curve allows for the comparison of data from different normal distributions.
• A z-score is a measure of position derived from the mean and standard deviation of data.
• A z-score expresses, in standard deviation units, how far an element falls from the mean of the data set.
• A z-score is a derived score from a given normal distribution.
• A standard normal distribution is the set of all z-scores.

·  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.

·  Graphing utilities can be used to represent a normally distributed data set and explore relationships between the data set and its descriptive statistics.

### ESSENTIALS

Standard Deviation

A.93  Given data, including data in a real-world context, calculate variance and standard deviation of a data set and interpret the standard deviation.

Z-scores

A.94  Given data, including data in a real-world context, calculate and interpret z-score for a data set.

Comparing Z-scores

A.94  Given data, including data in a real-world context, calculate and interpret z-score for a data set.

·  AII.11b1  Solve problems involving the relationship of the mean, standard deviation, and z-score of a normally distributed data set.

Properties of Normal Curve

All.111  Identify the properties of a normal probability distribution.

·  AII.11a1  Identify the properties of a normal distribution.

·  AII.111  Use a graphing utility to investigate, represent, and determine relationships between a normally distributed data set and its descriptive statistics.

All.112  Describe how the standard deviation and mean affect the graph of the normal distribution.

·  AII.11a2  Describe how the standard deviation and the mean affect the graph of the normal distribution.

All.114  Represent probability as area under the curve of a standard normal probability distribution.

·  AII.11c1  Represent probability as area under the curve of a standard normal distribution.

Probability / Percentages

All.113  Compare two sets of normally distributed data using a standard normal distribution and z-scores.

·  AII.11b2  Compare two sets of normally distributed data using a standard normal distribution and z-scores, given the mean and standard deviation.

All.114  Represent probability as area under the curve of a standard normal probability distribution.

·  AII.11c1  Represent probability as area under the curve of a standard normal distribution.

All.115  Use the graphing calculator or a standard normal probability table to determine probabilities or percentiles based on z-scores

·  AII.11c2  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: Oct 27, 2017