NUMERICAL SUMMARIES (lecture 3)Summaries for symmetric distribution  2 numeric measures: the mean and standard deviation o The mean describes the center or the balancing point of the distribution. We can Also use the mean as the typical value for symmetric mound shaped distributions .  The mean can be used as a balancing point. In symmetric distribution mean is at the center. For a right skewed distribution the mean is to the right of the typical value. o The standard deviation is a single value used to summarize lots of information quickly. It measures the typical distance from the mean and variability of symmetric distributions.The Sample Mean: our first formula.  is the mean  or “sigma” means add all that follow X is representative for data values  N is representative of the sample size (how many)Here is an example: Find the mean age of the first 6 presidents. Their ages are: 51, 61, 57, 57, 58, 57  57+61+57+57+58+57 / 6 Sample mean 57.8 yearsStandard Deviation Formula S is representative of standard deviation Steps to solve:  Take the sum of each individual value (x) Subtract the sum from the mean ( ) Square this value  Add the number of values in the sample size (n) and subtract 1 Divide the squared value (the individual values – the mean) by n-1 Take the square root of this value What is Standard Deviation?  Standard deviation is the sum of the spread.  It measures the typical distance of an observation from the mean of all the observations in the sample.  In mound shaped distributions most values are going to fall in this range The square of standard deviation is variance:oUsing Fathom to find the mean and standard deviation  Open a “table” of data Click the “summary” icon and drag the variable you want analyzed to a row or colum  Fathom auto-compleates a man for a summary table  Click “summary” on the tio meanue bar and add the function for standard deviation Empirical Rule  Empirical rule: if a distribution is both unimodal and symmetric then the following are trueo 68% of the data is within one standard deviation of the mean (approximately 2/3)o 95% of the data is within two standard deviations of the meano Almost all of the data (99.7%) is with in 3 standard deviations of the mean Z-score The z-score measures the number of standard deviations a value is from the mean  The z-score uses standard units or standard deviation units which allow us to compare values with different units such as liters to inches  The z-score is only used when distributions are symmetric and mounded  Example: if z-score is 2 that means the value is 2 standard deviations away from the mean Z-score formula: o Take the x value and subtract it from the mean and divide by the standard deviation. Theresulting value is the z score.Skew-ness For a skewed distribution the mean gets pulled toward the tail ad likewise pulled toward the outliers  For skewed distributions the mean (balancing point) does not represent the typical value like it does in symmetric mound distributions  A better representation of a typical value in skewed distributions is the median What is the Median? The median is the middle value of a distribution  The median is the middle number or the average of the two middle most numbers The median cuts all the values in half leaving half above and bellow itHow to compute the median?first, sort all the numbers from small to large  Then if the number of values is oven take the two centermost values and average If the numbers are off extract the central value. Next time: Quartiles and the inter quartile