Georgia Tech, MATH 3070
Excerpt: ... MATH 3070 Introduction to Probability and Statistics Lecture notes Measures of Position Objectives: 1. Understand and use the Empirical Rule The Empirical Rule (68-95-99.7) If the distribution of the data is approximately symmetrical with a single peak we usually refer to this as a normal distribution. The distribution is centered around the mean and has noticeable changes in slope at various points related to the standard deviation. We can say this about the data: 1. 68% of the observations fall within 1 standard deviation of the mean. 2. 95% of the observations fall within 2 standard deviations of the mean. 3. 99.7% of the observations fall within 3 standard deviations of the mean. How can we know this? It depends upon the curve and the characteristics of a symmetrical curve, or density curve. The area underneath the curve is 1 and always positive. The curve bends at each multiple of the standard deviation. These bending points are called inflection points. ...
Cornell, PAM 210
Excerpt: ... Book Problem Set #1 Circle all of your final answers. From the book, do: 1.62 (page 49, refers to page 24) Note that the stems are whole percents and the leaves are tenths of a percent so the first number is 5.7. For part b, explicitly list all of th ...
Western New England University, BIS 220
Excerpt: ... 1. The Standard Normal distribution is a special case of a normal distribution It has a mean and standard deviation of: a. mean=1, standard deviation=0 b. mean=0, standard deviation=1 c. mean=1, standard deviation=1 d. mean=0, standard deviation=0 2. ...
UCLA, STATS 11
Excerpt: ... mple size is "large" (your textbook believes that samples greater than or equal to 15 are large), the distribution of the possible sample means will be close to the normal distribution. It is a very powerful theorem and it is the reason why the normal distribution is so well studied. C. Summary Take a simple random sample from a population with mean and standard deviation . Let x be the average of the samples taken from the population. If either the original population is normally distributed OR the sample size n is sufficiently large, then x will be normally distributed with expected value and standard deviation . n If the histogram for the population follows a normal curve, or if the sample size is large enough each time, then the histogram for the possible values for x-bar will follow a normal curve that has a mean of and a standard deviation of . n Thus, about 68% of the x-bars will be within one standard deviation, about 95% of the x-bars will be within two standard deviations , and 99.7% of th ...
Mercyhurst, M 109
Excerpt: ... Section 2.4: Today we will study How to nd the range of a data set. Measures of Variation How to nd the variance and standard deviation of a population and a sample. How to interpret the standard deviation. Range DEFINITION The range of a data ...
Arizona, PSYC 230
Excerpt: ... Compare and contrast descriptive statistics and inferential statistics Compare and contrast quantitative data and qualitative data Define and contrast continuous versus discrete variables Be able to identify the independent variable (IV) and the depe ...
UCLA, ECON ECON 41
Excerpt: ... UCLA Econ 41 Statistics for Economists Second Quiz, July 23, 20007 Answers Version A: I. Multiple Choice: (5 points each) 1._d_ 2._d_ 3._b_ 4._a_ 5._d_ 6._b_ 7._d_ 8._a_ II. Essay Question (20 points): 1. For distribution A, we have 180 = + (1.0) ...
Cornell, PAM 2100
Excerpt: ... PAM 2100 Lecture 1 Instructor: Salam Abdus Problem Set #1 Posted: 9th September, 2008. Due: 16th September, Tuesday, 2008, in class, at 1:25 pm Circle all of your final answers. Put your name and your section number at the top. #1) Conservationists have despaired over destruction of tropical rainforest by logging, clearing, and burning. These words begin a report on a statistical study of the effects of logging in Borneo. Researchers compared forest plots that had never been logged (Group 1) with similar plots nearby that had been logged 1 year earlier (Group 2) and 8 years earlier (Group 3). All plots were 0.1 hectare in area. Here are the counts of trees for plots in each group: Group 1 Group 2 Group 3 27 12 18 22 12 4 29 15 22 21 9 15 19 20 18 33 18 19 16 17 22 20 14 12 24 14 12 27 2 28 17 19 19 a) Draw a separate stemplot for each group. b) Calculate the means and standard deviations for Group 1. (Round the mean to two decimal places. Use this number to calculate the standard deviation. ...
Fayetteville State University, MGF 1107
Excerpt: ... the data set, approximately how many are between 440 and 460? 9) In what percentile is a data value that is 0.75 standard deviations above the mean? 10) In what percentile is a data value that is 0.75 standard deviations below the mean? 11) Jane scored 405. What is the percentile for her score? -12) Suppose that in a company with 8000 employees, the annual salaries are normally distributed with a mean salary of $25,000 and a standard deviation of $3800. Approximately how many of the employees earn more than $30,000? 13) The resting heart rates for a group of 1500 individuals is normally distributed with a mean of 75 and a standard deviation of 12. Approximately how many of the heart rates were between 70 and 80? Answers: 1) 84.13% 2) 97.72% 3) 91.08% 4) 473.4 (473 ok) 5) 445.5 (445 or 446 ok) 6) 203.55 (204 ok) 7) 750 8) 626.4 (626 ok) 9) 77th 10) 23rd 11) 0.62nd 12) 774.4 (774 ok) 13) 466.2 (466 ok) 1 ...
San Jacinto, MATH 150
Excerpt: ... .608178 83.66151 260.7148 21 23 14 5 494.6517333 105.9880889 1171.261156 1303.574222 514.5 724.5 539 227.5 ( xi s i 1 x) 2 fi 75 5191.386667 8.37579094 29.35333 n 1 Interpreting the standard deviation 1. The more variation in a data set, the greater the standard deviation. 2. The larger the standard deviation, the more "spread" in the shape of the histogram representing the data. 3. Standard deviation is used for quality control in business and industry. If there is too much variation in the manufacturing of a certain product, the process is out of control and adjustments to the machinery must be made to insure more uniformity in the production process. Three standard deviations rule " Almost all" the data will lie within 3 standard deviations of the mean Mathematically, nearly 100% of the data will fall in the _ _ interval determined by ( x 3s, x 3s) Empirical Rule If a data set is "mound shaped" or "bell-shaped", then: 1. approximat ...
N.C. State, BUS 350
Excerpt: ... es It is often useful to know how many standard deviations a data point x is from the mean. We define z-scores, or standardized scores, as the distance x is from xbar (the mean) in units of the standard deviation. We calculate the z-score for x as follows: z = (x - xbar)/s A z-score of +2 means that an observation is 2 standard deviations above the mean. A z-score of -1.5 means that an observation is 1.5 standard deviations below the mean. End of z-score explanation. Question: Three landmarks of baseball achievement (besides the achievements of Babe Ruth and Henry Aaron) are Ty Cobb's batting average of .420 in 1911, Ted William's .406 in 1941 and George Brett's .390 in 1980. These batting averages cannot be compared directly because the distribution of major league batting averages has changed over the decades. The distributions are quite symmetric and (except for outliers such as Cobb, Williams, and Brett) reasonably mound shaped. Although the mean batting average has been held roughly constant by rules ...
CSU Northridge, MR 31841
Excerpt: ... Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has the following pr ...
UCLA, ECON 41
Excerpt: ... Econ 41 Ekaterini Kyriazidou UCLA Statistics for Economists Class 8: Population and Sampling Distributions In the last lectures we discussed probability distributions of discrete and continuous random variables. The distributions we've seen come ...
CSU Fullerton, C 210
Excerpt: ... e from being zero (owing to negative and positive values), we take the average of the squares of the differences and define what is called the standard deviation of a series of measurements Xi: 1 N (X i X )2 N 1 i =1 (note that one divides by N-1 rather than N in order to indicate that the standard deviation is undefined if only one measurement is made). If the probability distribution of our random measurements is normally distributed (e.g., has a Gaussian distribution), the probability of our measurement landing within 1 standard deviation is 68% and within 2 standard deviations is 95.4 %, that is X 68% X 2 95.4% Another important measure of uncertainty is the standard deviation of the mean, X . The standard deviation is the uncertainty in any given measurement, whereas the = standard deviation of the mean is the uncertainty in our best estimate of the true answer using N measurements, that is, the mean X . The standard deviation of the mean is given by X = / N Thus, if our ...
Duke, STA 102
Excerpt: ... Note: Data sets for this chapter are not available; only selected summary measures (means and standard deviations ) were provided in the referenced publications ...
Duke, STA 102
Excerpt: ... Note: Some data sets for this chapter are not available; only selected summary measures (means and standard deviations ) were provided in the referenced publications ...
North Shore, MAT 143
Excerpt: ... Jack-inthe-Box data? Intro to Statistics 3 Measures of Variation Empirical (or 68-95-99.7) Rule for Data with a Bell Shaped Distribution For data sets having a distribution that is approximately bell-shaped, About 68% of all values fall within 1 standard deviation of the mean; About 95% of all values fall within 2 standard deviations of the mean; and About 99.7% of all values fall within 3 standard deviations of the mean. Draw a picture illustrating the empirical rule: Problem(p. 84, #24, 25): Heights of women have a mean of 63.6 in. and a standard deviation of 2.5 in. (based on data from the National Health Survey). Use the range rule of thumb to estimate the minimum and maximum "usual" heights of women. In this context, is it unusual for a woman to be 6 ft tall? Suppose that the heights of women has a bell-shaped distribution. Using the empirical rule, what is the approximate percentage of women whose height is between a. 61.6 in. and 66.1 in.? b. 56.1 in. and 71.1 in.? Intro to ...
Allegheny, ECON 203
Excerpt: ... Economic Statistics Class Notes 1/28/09 Recall Continuous Probability Distribution Example The Normal Distribution The Normal Distribution is a distribution described by its mean and standard deviation such that (plus or minus one standard de ...