Z-Scores The Normal Distribution

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Presentation transcript:

Z-Scores The Normal Distribution Algebra 1 Notes Z-Scores The Normal Distribution

Z-scores z-scores allow for comparison of data that are measured in different units. The comparison comes from standardizing the data in terms of the standard deviation and the mean.

z-scores Formula:

Questions about z-scores Can a z-score be negative? Can a z-score be zero?

Interpeting Z-scores = -1 The heights of 16 year-old males are normally distributed with mean 68 inches and standard deviation 2 inches. Determine the z score for a male who is 66 inches tall. Facts: x = 66 = 68 = 2 = -1 This height is 1 standard deviation below the mean.

Interpeting Z-scores = 1.26 Jack scored 27 on the mathematics portion of the ACT. If the math scores on the ACT are normally distributed with mean of 20.7 and standard deviation of 5.0, determine Jack’s z- score. Facts: x = 27 = 20.7 = 5.0 = 1.26 This score is 1.26 standard deviations above the mean.

Interpeting Z-scores = 1.42 Jill scored 680 on the mathematics portion of the SAT. If the math scores on the SAT are normally distributed with mean of 518 and standard deviation of 114, determine Jill’s z score. Facts: x = 680 = 518 = 114 = 1.42 This score is 1.42 standard deviations above the mean.

Interpreting z-scores Assume the SAT and the ACT measure the same kind of ability. Who did better…Jack or Jill? Since z = 1.42 is greater than z = 1.26, Jill’s score of 680 on the SAT is relatively “higher” than Jack’s score of 27 on the ACT. Jill did better.

Interpreting z-scores The mean time it takes runners to complete a cross-country race is 85 minutes. The standard deviation of running times is 5 minutes. A runner has a z-score of -2. What does this mean?

Interpreting z-scores The runner ran 2 minutes faster than the overall mean. The runner ran 2 minutes faster than his personal mean. The runner ran 2 standard deviations faster than the overall mean. The runner ran 2 standard deviations faster than his personal mean.

Consider the data shown below Consider the data shown below. This histogram shows the heights of 1000 eighteen year old girls at a college campus and the frequency with which those heights occurred. What do you notice about the heights of the girls?

If we represented the data with a line plot, by connecting the center of each bar of the histogram, our plot would look like this: Notice the general shape of this curve. Curves that have this shape are often referred to as bell-shaped.

This curve can be smoothed even further: This continuous smoothed curve is known as a normal curve. This normal curve is part of a family of curves known as the normal distribution.

The 68-95-99.7 Rule for Normal Distributions Approximately 68% of the observations fall within 1 standard deviation of the mean

Approximately 95% of the observations fall within 2 standard deviations of the mean

Approximately 99.7% of the observations fall within 3 standard deviations of the mean

Another way of looking at it!