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Introductory Statistics for Economists ECON 2500 – Winter 2011 – Xianghong Li Chapter 1 – Looking at Data – Distributions – Jan 4 New Concepts - Context of a data set, purpose of it. - Observations: individuals or firms. - Variable: any characteristic of an individual. - Quantitative variable: takes on numerical values. Unit of measurement, e.g. hourly wage in dollars. - Categorical variable: places an individual into one of several groups or categories. E.g. gender. Distributions - Not all observations (observed values of the variables) are the same – variation. - The pattern of variation of a variable is called its distribution. Distribution is a summary of the values a variable takes and how often it takes the different values. - Using R to take frequency. 1.1 Displays Distribution with Graphs How to Draw a Stemplot - Sort observations and rank them from the smallest to the largest. - Write down the stems according to the range of the data. - Add leaves. How to Draw a Histogram - Sort the observation. - Decide (equal width) intervals. - Make a summary table (counts and percent). o Choose counts for small data sets. o Choose percent for large data sets. - Graph histogram (no gap between columns). - Interval choice: your judgement, choose the one show, the overall pattern best. Examining Distributions - Overall pattern: shape, spread, center. - Deviations from the overall pattern, especially outliers. - Mode: major peak(s) of a distribution. - Symmetric: mirror images on each side of the midpoint (think of a Stemplot). - Skewed to the right: the right tail is much longer. - Skewed to the left: the left tail is much longer. Plot Distribution - Bar and pie charts: categorical variables. o Pie: require all the categories that make up a whole. o Bar: more flexible. - Stemplot: suitable for small data sets. - Histogram: suitable for large data sets. 1.2 Describing Distributions with Number - Math rep: summation operators. - Center of distribution: mean and median. - Spread of distribution: quantiles and standard deviation. About Mean - It is where the histogram balances. - It is not resistant to outliers or skewness of a distribution. o A simple example. o Example 5: The mean highway mileage for the two-seater without Honda insight (outlier). - R command. Measuring Center: Median - The middle of a distribution, how to get it: o Sort all observations. o If the number of observations is odd, the median is the center observation. o If the number of observations is even, the median is the mean of the two center observations. Mean Versus Median - For a symmetric distribution, the median and mean are the same. - Median is less sensitive to outliers and skewness of a distribution, while mean is very sensitive to both. - Mean often turns out to be a more meaningful measure, e.g. portfolio returns. - Suggestion: reporting both mean and median. The Quartiles Q and Q 1 3 - To calculate the quartiles: o Arrange the observations in increasing order and locate the median M in the ordered list of observations. o The first quartile Q is1the median of the observations whose position in the ordered list is to the left of the location of the overall median. o The third quartile Q is 3he median of the observations whose position in
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