Normal Distribution Central Limit TheoremLecture4.ppt
Download as PPT, PDF0 likes82 views
Lecture on Normal Distribution
More Related Content
Similar to Normal Distribution Central Limit TheoremLecture4.ppt (20)
More from muhammad shahid (20)
Recently uploaded (20)
Normal Distribution Central Limit TheoremLecture4.ppt
- 1. 悋惘忰 悋惘忰 悋 惡愕 悋惘忰 悋惘忰 悋 惡愕
- 2. Introduction to Introduction to Biostatistics Biostatistics Dr. Mubashir Ahmed Research Coordinator & Head of Research Department Kharadar General Hospital, Karachi
- 3. Normal Distribution In a normal distribution, data is symmetrically distributed with no skew. When plotted on a graph, the data follows a bell shape, with most values clustering around a central region and tapering off as they go further away from the center. Normal distributions are also called Gaussian distributions or bell curves because of their shape.
- 4. Bell Shaped Bell Shaped Symmetrical Symmetrical Mean, Median and Mode Mean, Median and Mode are Equal are Equal Location is determined by the Location is determined by the mean, mean, 亮 亮 Spread is determined by the Spread is determined by the standard deviation, standard deviation, The random variable has an The random variable has an infinite theoretical range: infinite theoretical range: + + to to Mean = Median = Mode X f(X) 亮 The Normal Distribution
- 5. Why Normal Distribution All kinds of variables in natural and social sciences are normally or approximately normally distributed. Height, birth weight, reading ability, job satisfaction, or SAT scores are few examples of such variables. Because normally distributed variables are so common, many statistical tests are designed for normally distributed populations. Understanding the properties of normal distributions means you can use inferential statistics to compare different groups and make estimates about populations using samples.
- 10. Characteristics of Normal Distribution The mean, median and mode are exactly the same. The distribution is symmetric about the mean half the values fall below the mean and half above the mean. The distribution can be described by two values: the mean and the standard deviation.
- 11. Characteristics of Normal Distribution: Empirical rule
- 12. Normal Distribution and % of area under the curve
- 16. Kurtosis
- 17. The Standardized Normal Distribution Any normal distribution (with any mean and Any normal distribution (with any mean and standard deviation combination) can be standard deviation combination) can be transformed into the standardized normal transformed into the standardized normal distribution distribution (Z) (Z) To compute normal probabilities need to To compute normal probabilities need to transform transform X X units into units into Z Z units units The standardized normal distribution The standardized normal distribution (Z) (Z) has has a mean of a mean of 0 0 and a standard deviation of and a standard deviation of 1 1
- 18. Translation to the Standardized Normal Distribution (Z Score Calculation) Translate from X to the standardized normal Translate from X to the standardized normal (the Z distribution) by (the Z distribution) by subtracting the subtracting the mean mean of X and of X and dividing by its standard dividing by its standard deviation deviation: : 亮 X Z The Z distribution always has mean = 0 and standard deviation = 1
- 19. The Standardized Normal Distribution Also known as the Z distribution Also known as the Z distribution Mean is 0 Mean is 0 Standard Deviation is 1 Standard Deviation is 1 Z f(Z) 0 1 Values above the mean have positive Z- values. Values below the mean have negative Z-values.
- 20. Example If X is distributed normally with mean of If X is distributed normally with mean of $100 and standard deviation of $50, the Z $100 and standard deviation of $50, the Z value for X = $200 is value for X = $200 is This says that X = $200 is two standard This says that X = $200 is two standard deviations (2 increments of $50 units) above deviations (2 increments of $50 units) above the mean of $100. the mean of $100. 2.0 $50 100 $ $200 亮 X Z
- 21. Comparing X and Z units Note that the shape of the distribution is the same, only the scale has changed. We can express the problem in the original units (X in dollars) or in standardized units (Z) Z $100 2.0 0 $200 $X(亮 = $100, = $50) (亮 = 0, = 1)
- 22. The Standardized Normal Table The Cumulative Standardized Normal table in The Cumulative Standardized Normal table in the textbook the textbook (Appendix table E.2) (Appendix table E.2) gives the gives the probability probability less than less than a desired value of Z (i.e., a desired value of Z (i.e., from negative infinity to Z) from negative infinity to Z) Z 0 2.00 0.9772 Example: P(Z < 2.00) = 0.9772
- 23. The Standardized Normal Table The value within the The value within the table gives the table gives the probability probability from Z = from Z = up to the desired Z value up to the desired Z value .9772 2.0 P(Z < 2.00) = 0.9772 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point 2.0 . . . (continued) Z 0.00 0.01 0.02 0.0 0.1