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The normal distribution and standard deviations

The normal distribution and standard deviations. z -scores. When a set of data values are normally distributed, we can standardize each score by converting it into a z -score . . z -scores make it easier to compare data values measured on different scales. z -scores.

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The normal distribution and standard deviations

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  1. The normal distribution and standard deviations

  2. z-scores When a set of data values are normally distributed, we can standardize each score by converting it into a z-score. z-scores make it easier to compare data values measured on different scales.

  3. z-scores A z-score reflects how many standard deviations above or below the mean a raw score is. The z-score is positive if the data value lies above the mean and negative if the data value lies below the mean.

  4. Analyzing the data Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, how many standard deviations away from the mean is her score?

  5. z-score formula Where x represents an element of the data set, the mean is represented by and standard deviation by .

  6. Analyzing the data Her z-score would be 2 which means her score is two standard deviations above the mean.

  7. Comparing variables with very different observed units of measure • Comparing an SAT score to an ACT score • Mary’s ACT score is 26. Jason’s SAT score is 900. Who did better? • The mean SAT score is 1000 with a standard deviation of 100 SAT points. The mean ACT score is 22 with a standard deviation of 2 ACT points. • Jason’s score is 1 standard deviation below the mean SAT score and Mary’s score is 2 standard deviations above the mean ACT score. Therefore, Mary’s score is relatively better.

  8. Analyzing the data • A set of math test scores has a mean of 70 and a standard deviation of 8. • A set of English test scores has a mean of 74 and a standard deviation of 16. • For which test would a score of 78 have a higher standing?

  9. Analyzing the data To solve: Find the z-score for each test. The math score would have the highest standing since it is 1 standard deviation above the mean while the English score is only .25 standard deviation above the mean.

  10. Analyzing the data A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative? The z-score must be positive since the element of the data set is above the mean.

  11. The normal distribution and standard deviations

  12. The normal distribution and standard deviations In a normal distribution: Approximately 68% of scores will fall within one standard deviation of the mean

  13. The normal distribution and standard deviations In a normal distribution: Approximately 95% of scores will fall within two standard deviations of the mean

  14. The normal distribution and standard deviations In a normal distribution: Approximately 99.7% of scores will fall within three standard deviations of the mean

  15. Empirical Rule 68-95-99.7%

  16. Z-score Distribution • Mean of zero • Zero distance from the mean • The z-score has two parts: • The number • The sign • Negative z-scores aren’t bad • Z-score distribution always has same shape

  17. Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and standard deviation of 10: 80 90 100 110 120 -2 sd -1 sd 1 sd 2 sd What score is one sd below the mean? 90 What score is two sd above the mean? 120

  18. Using standard deviation units to describe individual scores 80 90 100 110 120 -2 sd -1 sd 1 sd 2 sd How many standard deviations below the mean is a score of 90? 1 How many standard deviations above the mean is a score of 120? 2

  19. Example • Suppose a student is applying to various law schools and wishes to gain an idea of what his GPA and LSAT scores will need to be in order to be admitted. • Assume the scores are normally distributed • The mean GPA is a 3.0 with a standard deviation of .2 • The mean LSAT score is a 155 with a standard deviation of 7

  20. GPA SD SD SD SD SD SD SD 2.4 2.6 2.8 3.0 3.2 3.4 3.6 68% 95% 99%

  21. LSAT Scores SD SD SD SD SD SD SD 134 141 148 155 162 169 176 68% 95% 99%

  22. Fun facts about z scores the mean of a distribution has a z score of ____? zero positive z scores represent raw scores that are __________ (above or below) the mean? above negative z scores represent raw scores that are __________ (above or below) the mean? below

  23. Conclusions • Z-score is defined as the number of standard deviations from the mean. • Z-score is useful in comparing variables with very different observed units of measure. • Z-score allows for precise predictions to be made of how many of a population’s scores fall within a score range in a normal distribution.

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