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

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.

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

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

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

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

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

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