The bell shape curve

1. The bell shape curve Normal distributionFETP India

2. Competency to be gained from this lecture Use the properties of normal distributions to estimate the proportion of a population between selected values

3. Key issues • Normal distribution • Properties of the normal distribution • Z score

4. Frequency distribution • For a continuous variable, the values taken by the variable may be listed • One can examine how commonly the variable will take specific values • The relative frequency with which the variable is taking selected values is called a frequency distribution Normal distribution

5. Distribution • We observe the frequency distribution of values • If we smoothen the distribution, we obtain a curve • If the curve can correspond to a mathematical formula, we can apply formula that allow predicting a number of parameters Normal distribution

6. The normal curve presented as an histogram Normal distribution

7. Observation • Many naturally occurring events follow a rough pattern with: • Many observations clustered around the mean • Few observations with values away from the mean • This bell-shaped curve was named Normal distribution by a mathematician called Gauss Normal distribution

8. The normal distribution • Normal distribution • The symmetrical clustering of values around a central location • Normal curve • The bell-shaped curve that results when a normal distribution is graphed Normal distribution

9. Properties of the normal distribution of a continuous variable • Symmetric about its mean • The median = the mode = the mean • The entire distribution is known if two parameters are known: • The mean • The standard deviation Properties

10. Additional properties of the normal distribution • 68% of the values lie between: • Mean + one standard deviation • Mean - one standard deviation • 95% of the values lie between: • Mean + two standard deviations • Mean - two standard deviations • > 99% of the values lie between • Mean + three standard deviations • Mean - three standard deviations Properties

11. Distribution of the values according to the standard deviation

12. Characterizing a normal distribution • The mean specifies the location • The standard deviation specifies the spread • Hence: • For different values of mean or standard deviation or both, • We get different normal distributions. Properties

13. Usefulness of the normal distribution • Many statistical tests are based on the assumption that the variable is normally distributed in the population • Using the standard deviation it is possible to: • Describe the “normal” range between x-standard deviations • Compare the degree of variability in the distribution of a factor: • Between two populations • Between two different variables in the same population Properties

14. Distributions that are approximately normal • For distributions that are approximately normal: • Unimodal (One mode) • Symmetrical • Having a bell shaped curve • The standard deviation and the mean together provide sufficient information to describe the distribution totally Properties

15. z-score • Every normal distribution can be standardized in terms of a quantity called the normal deviate (z) • The z score is an index of the distance from the mean in units of standard deviations Z-score

16. Standardizing a normal distribution • Z is defined as: • Observation - Mean • Z =----------------------------------------- • Standard deviation • The probabilities associated with normal distribution are obtained from the knowledge of z Z-score

17. Representing a normal curve on a standard deviation scale Mean One standard deviation Minus one standard deviation The x-axis expresses the data values in a standardized format

18. Knowing what proportion of the values lies between two values Between the mean and + 1 standard deviation, there is 68% / 2 = 34% of the values Z-score

19. Area under the curve and Z-score • What proportion of the population is between between 0 and 1.96? Z-score

20. First example of use of the normal distribution: Heights • We are examining a population of persons with heights that are normally distributed • Consider the normal distribution of heights: • Mean height = X = 65" • Standard deviation = SD = 2" Z-score

21. What is the proportion of persons whose height exceeds 68”? • Normal deviate • Z = (x-x)/SD = (68-65)/2 = 1.5 • The area under the curve from Z = 1.5 to : • 0.0668 • 6.68% • 6.68% of persons have a height that exceeds 68" Z-score

22. What is the proportion of persons whose height is less than 60”? • Normal deviate : • Z = (x-x)/SD = (60 - 65 ) / 2 = - 2.5 • The area under the curve from z = -  to z = -2.5 is equal to the one from Z = 2.5 to z = +  • The area under the curve from Z = 2.5 to  : • 0.0062 • 0.62% • 0.62% of persons have a height below 60” Z-score

23. What is the proportion of persons whose height is between 64 " and 67 ” (1/2)? } • Normal deviate for x=64”: • Z1 =(65-64)/2 = - 0.5 • Area under the curve: • From Z1 = -  to - 0.5 = • From Z1 = 0.5 to  • Proportion of the population with height less than 64”: • 0.3085 • 30.8 Z-score

24. What is the proportion of persons whose height is between 64 " and 67 ” (2/2)? • Normal deviate for X=67" = Z2 = (67-65)/2 = • Area under the curve from Z2 = 1 to  • 0.1587 • 15.8% of the population has a height exceeding 67" • Heights between 64" and 67’’ • 1 - 0.3085 - 0.1587 = 0.5328 = 53.28% Z-score

25. Second example of use of the normal distribution: Cholesterol level • We are examining a population of persons with cholesterol levels that are normally distributed • Consider the normal distribution of cholesterol levels: • Mean cholesterol = 242 mg % • Standard deviation = 45 mg % Z-score

26. Example 2: What is the cholesterol level exceeded by 10% of men? • What is the Z corresponding to an area of 10% (0.1) on the right? • The Z value from the table is: • 1.282 • Z = (x-242)/45 =1.3 • X - 242 = 1.3 x 45 = 58.5 • X= 58.5 + 242 = 300.5 mg% Z-score

27. What is the cholesterol level that is exceeded by 2.5% of men ? • What is the Z corresponding to 2.5% of the area (0.025) on the left? • From the table, • Z corresponding to an upper area of 0.025 = 1.96 • By symmetry, the lower value of Z is -1.96 • (x-x) / SD = z • (x-242) /45 = -1.96 • (x-242) /45 = -1.96 • X – 242 = -1.96 x 45 = - 88.2 • X = 242 - 88.2 = 153.8 mg% Z-score

28. How does one know the distribution is normal? • Is the distribution symmetrical? • A normal distribution is symmetrical • Is the distribution skewed? • A normal distribution is not skewed • What is the kurtosis of the distribution? • The normal distribution is neither too sharp nor too shallow • Computer programmes are available to test the normality of the distribution Z-score

29. Scores and normal curve Z-score

30. Key messages • The symmetrical clustering of values around a central location is called a normal distribution • Normal distributions are symmetric, have a common value for the mean, the median and the mode and are solely characterized by their mean and their standard deviation • Z-scores allow estimating the proportion of the population lying between selected values