Process Distributions A process distribution can be characterized by its location, spread, and shape. Location is measured by the mean of the distribution and spread is measured by the range or standard deviation. The shape of process distributions can be characterized as either symmetric or skewed. A symmetric distribution has the same number of observations above and below the mean. A skewed distribution has a greater number of observations either above or below the mean. © 2007 Pearson Education
View full slide show




Standard Deviation = SD • Standard Deviation uses all the data points, not just some like Range and Interquartile Range • Standard Deviation does not have squared units (like Variance) and is thus easier to interpret • Standard deviation has the same units as the data!! • The sample standard deviation is a point estimator of the population standard deviation • Interpretation of Standard Deviation: • • • • • • • A Numerical Measure that says how much variability there is in the data points Standard Deviation Is Like An Average Of The Deviations Standard Deviation tells us how fairly the mean represents its data points Standard Deviation tells us how clustered the data points are around the mean For financial assets standard deviation is a measure of risk or fluctuation in asset value Use STDEV.P function for population data Use STDEV.S for sample data. 56
View full slide show




 The sampling distribution of    When    we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts. MEAN AND STANDARD DEVIATION OF A SAMPLE MEAN Suppose that is the mean of an SRS of size drawn from a large population with mean and standard deviation . Then the sampling distribution of has mean and standard deviation . We say the statistic is an unbiased estimator of the parameter . Because it’s standard deviation is , the averages are less variable than individual observations, and the results of large samples are less variable than the results of small samples. SAMPLING DISTRIBUTION OF A SAMPLE MEAN If individual observations have the distribution, then the sample mean of an SRS of size has the distribution.
View full slide show




Choosing Measures of Center and Spread We now have a choice between two descriptions for center and spread:  Mean and Standard Deviation  Median and Interquartile Range Choosing Measures of Center and Spread •The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. •Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. •NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA! 15
View full slide show




Important Properties of the Student t Distribution 1. The Student t distribution is different for different sample sizes (see Figure 7-5, following, for the cases n = 3 and n = 12). 2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a = 1). 5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution. Slide 72 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.
View full slide show




Z-score: Number of Standard Deviations • Formula for z-score = Deviation/SD = (Xi - Xbar)/SD • Excel Function: STANDARDIZE(X,Mean,SD) • z Score = How Many Standard Deviation is a particular value ways from the mean? • • • • z < 0, value below mean z > 0, value above mean z = 0, value is equal to mean Z score measures the relative location of a particular x in the data set (as compared to the mean), in units of standard deviation. • Relative Location in terms of "Number of Standard Deviations • z Score = Standardized Value • Observations in 2 different data sets that have the same z-score are said to have the same relative location or the same number of standard deviations away from the mean. 59
View full slide show




Describing Distributions A distribution is symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right
View full slide show




Describing Distributions A distribution is symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right
View full slide show




Describing Distributions 9  A distribution is symmetric if the right and left sides of the graph are approximately mirror images of each other.  A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side.  It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right
View full slide show




Does it look like a Normal distribution? • The variance of a measured speed distribution normally should be less than the variance of a random distribution (estimated by the sample mean) • The standard deviation should be approximately one-sixth of the total range since the mean plus and minus three standard deviations encompasses 99.73 percent of the observations of a normal distribution • The standard deviation should be approximately one-half of the 15 to 85 percentile range since the mean plus and minus one standard deviation encompasses 68.27 percent of the observation of a normal distribution, • The 10-mph pace should be spread equally on each side of the mean • The normal distribution has little skewness and the coefficient of skewness should be approximately zero. mean - mode mean - median coef skew = = 3( ) s s Example on pages 102,103
View full slide show