Z Scores and Normal Distribution (12-6) Objective: Draw and label normal distributions, compute z-scores, and interpret and analyze data using z-scores.

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Presentation transcript:

Z Scores and Normal Distribution (12-6) Objective: Draw and label normal distributions, compute z-scores, and interpret and analyze data using z-scores.

Distribution When there are a large number of values in a data set, the frequency distribution tends to cluster around the mean of the set in a distribution (or shape) called a normal distribution. The graph of a normal distribution is called a normal curve. Since the shape of the graph resembles a bell, the graph is also called a bell curve.

Distribution

Example 1 The mean height of 15-year-old boys in the city where Isaac lives is 67 inches, with a standard deviation of 2.8 inches. Use normal distribution to represent these data. 1.Use the mean value and the standard deviation to find your minimum and maximum values.  Xmin = 67 – 3(2.8) =  Xmax = (2.8) =  Ymax = 1 ÷ (2 x 2.8) =

Example 1 The mean height of 15-year-old boys in the city where Isaac lives is 67 inches, with a standard deviation of 2.8 inches. Use normal distribution to represent these data. 2.Go to WINDOW. Set the Xmin, Xmax, and Ymax using the values from Step 1. The Xscl should be the standard deviation. The Ymin should be 0. The Yscl should be 1.

Example 1 The mean height of 15-year-old boys in the city where Isaac lives is 67 inches, with a standard deviation of 2.8 inches. Use normal distribution to represent these data. 3.By entering the mean and standard deviation into the calculator, we can graph the corresponding normal curve. Enter the values using the following keystrokes: Y= 2 nd VARS ENTER X,T,Θ,n, 67, 2.8 ) GRAPH.

Example 1 The mean height of 15-year-old boys in the city where Isaac lives is 67 inches, with a standard deviation of 2.8 inches. Use normal distribution to represent these data. μ 58.6 in 75.4 in

Z-Scores The distance between a data value and the mean value is called the z-score. A z-score is the number of standard deviations a data value is from the mean. The unit of measure for a z-score is a standard deviation. The z-score for a data value x is given by z =, where μ is the mean and σ is the standard deviation.

Example 2 Find the z-score for a height of 74 inches using the following data set. Heights: 70, 66, 72, 74, 70, 73, 68, 70, 67, 70, 71, 69, 68, 66, 69, 69, 64 μ = 69.2 σ = 2.5 The z-score for 74 inches is 1.92.

Example 3 Find the z-score of someone who is 73 inches tall.

Example 4 Find the z-score of someone who is 66 inches tall.

Z-Scores  A negative z-score is the number of standard deviations to the left or less than the mean.  A positive z-score is the number of standard deviations to the right or more than the mean.

Example 5 Z-scores can be used to compare scores from two different data sets. In the year 2000, Barry Bonds hit 42 home runs. In the year 1950, Joe Dimaggio hit 38 home runs. Which player was the better homerun hitter for their corresponding years? In other words, which one broke farther from the pack?  Z bonds =  Z dimaggio = Since Dimaggio has a higher z-score, he was the better homerun hitter for his time. YearMean Total Home RunsStandard Deviation = 1.83 = 2