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Understanding the Normal Curve: Probability, Percentiles, and Data Distribution

In this lecture, we explore the Normal Curve and its importance in probability and statistics. We discuss how data can often be approximated using this model and the significance of standard units. Learn how to convert values to standard units and find areas under the Normal Curve, including calculations for areas between specified points. Additionally, we delve into the concept of percentiles, explaining how to summarize data distributions, particularly for histograms that do not fit the normality assumption. Understand the interquartile range and its implications.

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Understanding the Normal Curve: Probability, Percentiles, and Data Distribution

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  1. MA-250 Probability and Statistics Nazar Khan PUCIT Lecture 4

  2. The Normal Curve • An approximation to data distribution that is normally quite accurate • Normally data follows such a distribution

  3. The Normal Curve

  4. Standard Units • Express the data in terms of standard deviation • Converting a value X to standard units • (X-average)/SD

  5. The Normal Approximation to Data

  6. The Normal Approximation to Data

  7. Finding Areas under Normal Curve

  8. Find the area between 0 and 2 under the normal curve. • Find the area between -2 and 1 under the normal curve.

  9. Find the area to the right of 1 under the normal curve. • Find the area to the left of 2 under the normal curve.

  10. Find the area between 1 and 2 under the normal curve

  11. Key point: Notice how the original data and its histogram can now be discarded. This is the advantage of the normal approximation!

  12. Normal approximation is not always accurate Verify this.

  13. Percentiles • Some histograms can not be accurately summarised by the normal curve. • Can they be summarised some other way? • Yes, using percentiles. • If 10% of the data lies below x, then x is the 10th percentile.

  14. Percentiles • If 10% of the data lies below x, then x is the 10th percentile. • If k% of the data lies below x, then x is the kth percentile. • A percentile is not a percentage! • It is a number that lies on the horizontal axis.

  15. Percentiles • The 50th percentile is …. • Is there a technical term for the 50th percentile? • Think about what the 50th percentile represents.

  16. Percentiles • How can we summarize spread of a histogram using percentiles? • Subtract the 25th percentile from the 75th percentile. • This is called the interquartile range. • Is the interquartile range a percentage?

  17. Percentiles • Percentiles can summarize any histogram, whether it follows the normal curve or not. • Question: • Assuming that a set of exam scores follows the normal curvewith mean 63 and SD 10, what is the 95th percentile?

  18. A note on reading Normal Tables • The table in the book is not a standard table. • Gives area under the normal curve from –z to z • Standard tables give area from –inf to z • So always note what a normal table represents before using area values from it. • Is there a relationship between this z-score and percentiles?

  19. Summary

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