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Thinking Mathematically

Learn about the 68-95-99.7 Rule in normal distribution, z-scores, percentiles, finding data item percentages, and margin of error in surveys.

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Thinking Mathematically

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  1. Thinking Mathematically The Normal Distribution

  2. The 68-95-99.7 Rule for the Normal Distribution • Approximately 68% of the measurements will fall within 1 standard deviation of the mean. • Approximately 95% of the measurements will fall within 2 standard deviations of the mean. • Approximately 99.7% (essentially all) the measurements will fall within 3 standard deviations of the mean.

  3. The 68-95-99.7 Rule for the Normal Distribution 99.7% 95% 68% - 3 - 2 -1 1 2 3

  4. Computing z-Scores A z-score describes how many standard deviations a data item in a normal distribution lies above or below the mean. The z-score can be obtained using z-score = data item – mean standard deviation Data items above the mean have positive z - scores. Data items below the mean have negative z-scores. The z-score for the mean is 0.

  5. Percentiles If n% of the items in a distribution are less than a particular data item, we say that the data item is in the nth percentile of the distribution. For example, if a student scored in the 93rd percentile on the SAT, the student did better than about 93% of all those who took the exam.

  6. Finding the Percentage of Data Items between Two Given Items in a Normal Distribution • Convert each given data item to a z-score: z = data item - mean standard deviation • Use the table to find the percentile corresponding to each z-score in step 1. • Subtract the lesser percentile from the greater percentile and attach a % sign.

  7. Margin of Error in a Survey If a statistic is obtained from a random sample of size n, there is a 95% probability that it lies within 1/n of the true population statistic, where ± 1/ n is called the margin of error.

  8. Thinking Mathematically The Normal Distribution

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