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PUAF 610 TA Session 2 1 Today • Class Review- summary statistics • STATA Introduction • Reminder: HW this week 2 Review: Two types of Statistics • Descriptive statistics summarize numerical information. • Inferential statistics uses a sample to infer the population. 3 Summary statistic • In descriptive statistics, summary statistics are used to summarize a set of observations. • Typically, – What is the central value? – How widely are values spread from the center? – Are there data that are very atypical? – …. 4 Summary statistic • a measure of location, or central tendency • a measure of statistical dispersion • a measure of the shape of the distribution 5 Central tendency • Central tendency relates to the way in which quantitative data tend to cluster around some value. • A measure of central tendency is any of a number of ways of specifying the “central value”. 6 Basic measures of central tendency • Mean • Median • Mode 7 Mean • the sum of all measurements divided by the number of observations in the data set • population mean () v. sample mean (“xbar”) 8 Example • Assume 4 people take PUAF 610, and their final exam scores are 95, 87, 93, 83. What’s the mean for exam score? 9 Example • Mean= (95+87+93+83)/4=89.5 10 Median • the middle observation, when data are ordered from smallest to largest • the point of a distribution that divides the bottom 50% from the top 50% of the data. The median is the 50th percentile. 11 Median • If there is an odd number of observations, the median is the middle observation • If there is an even number of observations, the median is the average of the two middle observations • If the dataset is arranged in increasing order the median is located at position (n+1)/2 12 Example • Calculate the sample median for the following observations: 1, 5, 2, 8, 7. • Start by sorting the values: 1, 2, 5, 7, 8. • The median is located at position (n+1)/2=3, thus it is 5. • An odd number of values. 13 Example • Calculate the sample median for the following observations: 1, 5, 2, 8, 7, 2. • Start by sorting the values: 1, 2, 2, 5, 7, 8. • The median is located at position (n+1)/2=3.5, Thus, it is the average of the two middlemost terms (2 + 5)/2 = 3.5. • An even number of values 14 Mode • the most frequent value in the data set • It is possible for a distribution to have more than one mode or not to have a mode at all. 15 Example • • • • The mode for the following data set (1) 1, 2, 2, 3, 4, 7, 9 (2) 12, 26, 26, 53, 84, 71, 71, 79 (3) 32, 46, 53, 94, 37, 29 16 Comparing of Mode, Median and Mean • Pros and Cons • For descriptive purposes we might use the measure that suits the data. • If we would like to infer from samples to populations, the mean is a measure of choice because it can be manipulated mathematically. 17 Summary statistic • a measure of location, or central tendency • a measure of statistical dispersion, or variation • a measure of the shape of the distribution 18 Measures of Variation • Variation is variability or spread in a variable • Measures of variation are lengths of intervals on the measurement scale that indicate the spread of values in a distribution. 19 Measures of Variation • • • • • Range Quartiles Interquartile range Variance Standard Deviation 20 Range • the length of the smallest interval which contains all the data • (highest value – lowest value) + 1 21 Quartiles • any of the three values which divide the sorted data set into four equal parts, so that each part represents one fourth of the sampled population. 22 Quartiles • first quartile (Q1) = lower quartile = cuts off lowest 25% of data = 25th percentile • second quartile (Q2) = median = cuts data set in half = 50th percentile • third quartile (Q3) = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile • * The difference between the upper and lower quartiles is called the interquartile range. 23 Variance • Describes how far values lie from the mean. • Use the absolute values or to square the deviation scores to get rid of the minus signs. • Averaging absolute values cannot be used in more advanced analyses. – By averaging the sum of squared deviations (sum of squares) we can get a measure that is susceptible to further algebraic manipulations that are difficult or impossible with absolute values. 24 Variance • Less intuitive and more difficult to interpret, because it is measured in squared units rather than original units • Do not use variance much • (in population) and (in sample) where μ is the mean and N is the number of population. 25 25 Standard deviation • A widely used measure of the variability or dispersion. • It shows how much variation there is from the "average“. • Standard deviation is obtained by taking a square root of the variance, i.e. (population) (sample) 26 26 Standard deviation • A low standard deviation indicates that the data points tend to be very close to the mean. • A high standard deviation indicates that the data is spread out over a large range of values. 27 Summary statistic • a measure of location, or central tendency • a measure of statistical dispersion, or variation • a measure of the shape of the distribution 28 Shape of the distribution • Skewness • Kurtosis 29 Skewness • a measure of the asymmetry of the distribution • The skewness value can be positive or negative, or even undefined. 30 Skewness • negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. It has relatively few low values. 31 Skewness • positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. It has relatively few high values. 32 Skewness • A zero value indicates that the values are relatively evenly distributed on both sides of the mean. 33 Kurtosis • a measure of the "peakedness" of the distribution • Higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations 34 That’s all for class review. So far so good? Let’s go to STATA! 35