Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Student Academic Learning Services Page 1 of 6 Statistics: The Normal Distribution A Student Academic Learning Services Guide www.durhamcollege.ca/sals Student Services Building, SSB 204 905.721.2000 ext. 2491 This document last updated: 7/21/2011 Student Academic Learning Services Page 2 of 6 The Normal Distribution: things to remember A normal distribution is a continuous probability distribution… that is bell shaped o that is symmetrical o the right and left halves are identical (mirror images of each other) that has tails (ends) that approach the bottom axis (but never touch it) o it looks like a bell with a single peak in the middle “asymptotic” where the mean, median, and mode are equal o represented by the symbol μ www.durhamcollege.ca/sals Student Services Building, SSB 204 905.721.2000 ext. 2491 This document last updated: 7/21/2011 Student Academic Learning Services Page 3 of 6 Points to Ponder Each point on the curve represents the measurement (x) of an individual in the population The peak in the middle is the average/mean of the population, μ To the left of the peak are measurements less than μ To the right of the peak are measurements greater than μ Because the curve is symmetrical, ½ the curve (or 50%) is to the left of the peak, and ½ the curve (or 50%) is to the right of the peak. When choosing a point on the curve randomly, there is a 50% chance of selecting a point to the left of the peak (a measurement less than μ) , and a 50% chance of selecting a point to the right of the peak (a measurement greater than μ). www.durhamcollege.ca/sals Student Services Building, SSB 204 905.721.2000 ext. 2491 This document last updated: 7/21/2011 Student Academic Learning Services Page 4 of 6 Any point (not just the peak) on the curve can divide the distribution into different parts. We can say that the proportion of the curve to the left of any point is equal to the probability of randomly selecting an individual with less than that measurement. We can also say that the proportion of the curve to the right of any point is equal to the probability of randomly selecting an individual with more than that measurement. So how do we determine the relative proportions (left of/right of) created by selecting a random point (x) on the curve? Fortunately, there is a characteristic of the normal curve that can help us (and someone else has already done the really hard math). The normal distribution follows the empirical rule, so we know that about 68% of the measurements are between one standard deviation (σ) to the left of the mean and one standard deviation to the right of the mean, 95% are between two standard deviations, and 99.7% are between three standard deviations. www.durhamcollege.ca/sals Student Services Building, SSB 204 905.721.2000 ext. 2491 This document last updated: 7/21/2011 Student Academic Learning Services Page 5 of 6 The Standard Normal Distribution If we convert a measurement (x) to the number of standard deviations it is away from the mean, we can look up the probability that goes with that number. The number of standard deviations a measurement is away from the mean is called the z-score. The formula used to calculate a z-score is: The numerator of this formula (x – μ) determines how far away a measurement is away from the mean and in which direction (positive when x is larger than μ, negative when x is less than µ) When we divide the result by the standard deviation (σ), the result is how many standard deviations that measurement is from the mean. This is the z-score. For example, let us say that a population has a mean of 100 with a standard deviation of 15. What is the z-score for a measurement of 125? Calculating the numerator (x – μ) we have: 125-100=25. We then divide the result (25) by the standard deviation (15) and have: 25÷15=1.67 The measurement of 125 has a z-score of 1.67. This means the measurement is 1.67 standard deviations to the right of the mean (we know it’s to the right because it is a positive number) www.durhamcollege.ca/sals z= x µ Student Services Building, SSB 204 905.721.2000 ext. 2491 This document last updated: 7/21/2011 Student Academic Learning Services Page 6 of 6 There are a number of ways to look up the probabilities that are associated with a z-score of 1.67. For example, your textbook may have an appendix located in the back, or you may be using computer software (Microsoft Excel). Whatever method is used, a z-score of +1.67 divides the distribution, with 95.22% of the distribution to the left of the score and 4.78% to the right of the score. We can now say that (when chosen randomly) there is a 95.22% chance of choosing an individual with a measurement less than 125, or a 4.78% chance of choosing an individual with a measurement greater than 125. www.durhamcollege.ca/sals Student Services Building, SSB 204 905.721.2000 ext. 2491 This document last updated: 7/21/2011