# Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs.

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1 ECO 51 Research Methods 1. Data and Descriptive Statistics (Review) Data A variable - a characteristic of population or sample that is of interest for us. Data - the actual values of variables Quantitative data are numerical observations Qualitative data are categorical observations Type of analysis allowed for each type of data Quantitative data - arithmetic calculations Qualitative data - counting the number of observation in each category Ranked data - computations based on an ordering process 0 1 Cross-Sectional and Time-Series Data Cross sectional data is collected at a certain point in time Marketing survey (observe preferences by gender, age) Test score in a statistics course Starting salaries of an MBA program graduates Time series data is collected over successive points in time closing price of gold Amount of crude oil imported The data collected has different names depending on whether it contains all the data points of the variable (s) or only a subset of the data points. Population: A collection of data points of a variable. Sample: A of the population. This distinction is made because it can be very costly and sometimes impossible to collect population data: Does it make sense to measure the height of all,000 students at OWU in order to know the average height of OWU students? Does it make sense to test every missile produced in order to know their precision? Population vs. Sample 3 1

2 In transforming data to useful information, two levels of analysis can be utilized: Descriptive statistics: calculating summary characteristics of data. Inferential statistics: Using sample summary measures to estimate population characteristics. population Summarize the data Population Characteristics are unknown Descriptive Statistics Inferential Statistics Inference sample Summarize the data Sample: Find summarizing measures In descriptive statistics we summarize the data from a population or a sample of it. Data on population is OT available. We take a sample and use its summarizing measures to estimate the unknown population 4 characteristics. Population parameter vs. Sample statistic The summarizing numerical measures are called parameters in the case of a population and statistics in the case of a sample. Since we generally don t have access to population information, the parameters are unknown constants. Since samples are picked randomly from the population, the statistics are random variables. We pick a sample from the population, calculate the statistics and use it to estimate the unknown parameters. 5 Descriptive Statistics include measurements of central tendency dispersion linear association Population mean Central Tendency Arithmetic mean This is the most popular and useful measure of central location because it takes in to account all data points available and weights them equally. It is also the most influenced by extreme observations. Sample mean These measures are called if calculated for a sample and if calculated for a population. We use different notations for summary measures depending on whether they are parameter or statistics. 6 x μ = i= 1 i Population size x = n 1 i= xi n Sample size 7

3 Median The median of a set of measurements is the value that falls in the middle when the measurements are arranged in order of magnitude. When there are extreme numbers is more representative of a typical observation than the. Mode The mode of a set of measurements is the value that occurs most frequently. A set of data may have zero modes (non-modal), one mode, or two or more modes. 8 Relationship among Mean, Median, and Mode If a distribution is mound shaped and symmetrical, the mean, median and mode coincide If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution ( skewed to the right ) Mode Mean Median A negatively skewed distribution ( skewed to the left ) Mean Mode Median 9 Examples Measures of Central Tendency #1 Find the mean, median and mode for data: 1, -15, 1, -14, 18 How and why do the measures differ? # The manager of a men s store observes the waist size (in inches) of trousers sold yesterday: 3, 34, 33, 3, 8, 33, 30, 48, 3, 40 What are the mean, median and modal values? ow try using Excel Use Data Analysis go to Tools Add-ins check Analysis ToolPak go to Tools Data Analysis select Descriptive Statistics Is the distribution symmetrical? Dispersion Range The range of a set of measurements is the difference between the largest and smallest measurements. Its major shortcoming is its failure to provide information on the dispersion of the values between the two end points. Which makes the most sense to use?

4 Variance This measure of dispersion reflects the values of all the measurements. The drawback is that to obtain this measure, the differences have to be squared. The variance of a population of measurements x 1, x,,x having a mean μ is defined as i = 1 ( x i μ ) σ = The variance of a sample of n measurements x 1, x,,x n having a mean x is defined as s ( xi x ) n 1 n n i= 1 i = 1 = = x i nx n 1 1 Standard Deviation The standard deviation of a set of measurements is the square root of the variance. Is most commonly used because it includes all the data points, and is in the same terms as the original data. It is the measure of used in financial markets Can be compared across different data sets, and can be used to describe the general shape of a distribution population standard deviation σ = σ sample standard deviation s = s 13 If a sample of measurements has a symmetrical moundshaped distribution, the interval of +/- one standard deviation from the mean should contain approximately 68% of all data points, +/- two standard deviations should contain approximately 95% of all data points, and +/- three standard deviations should contain approximately 99.7% of all data points. ( x s, x + s) ( x s, x + s) ( x 3s,x + 3s) The empirical rule contains approximately 68% of the data points contains approximately 95% of the data points contains approximately 99.7% of the data points 14 Example The Empirical Rule You are interested in determining where your grade on the first Econ 51 midterm falls relative to your classmates. You know the average from the sample you took was 83%, the standard deviation was 5.5 and the variance was You received a 94%. How well did you do? (assume a symmetrical & mound shaped distribution) 83% - 5.5% = 77.5%; 83% + 5.5% = 88.5%, therefore 68% of students scored between 77.5% and 88.5% 83% - ()*5.5% = 7%; 83% + ()*5.5% = 94%, 95% of the students scored between 7% and 94% 83% - (3)* 5.5% = %; 83% + (3)* 5.5% = %, % of students scored between 15 4

5 Examples Measures of Dispersion #1 Return to examples 1 and used in Measures of Central Tendency. Calculate the range, variance and standard deviations of both data sets. What percentage of the observations are within 1, and 3 standard deviations of the mean? How does this compare with what the empirical rule predicted? # Last year, the rates of return on the common stocks in a large portfolio had an approximately mound shaped distribution, with a mean of 0% and a standard deviation of 10%. Use the empirical rule to answer the following questions: What proportion of the stocks had a return of between 10% and 30%? Between 10% and 50%? What proportion had a return that was either less than 10% or more than 30%? What proportion had a positive return? Linear Association Covariance An un-scaled measure of the linear co-movement of two variables. Identifies whether a linear association between the variables exists. Values range from - to + ; large negative values suggest negative linear association; large positive value suggests positive linear association. Values near zero suggest no linear association. Its weakness: what is a large number? what is close to zero? The answer depends upon the units being used. Hence it can t be used to compare different data sets. population covariance (x i μ x )( yi μ y ) COV(X, Y) = sample covariance (xi x)( yi y) cov(x, y) = n Correlation Coefficient (Coefficient of Correlation) This is closely related to the covariance, but is a scaled measure of the linear co-movement of two variables. It not only identifies whether a linear association between the variables exists, but also indicates the strength of that relationship. Values range from -1 to +1: 1 indicates a perfect negative linear association (all data points are exactly on the line; the slope is negative); as the number moves from 1 toward zero, the strength of the linear association is weakening (e.g. the data points move off the line to a increasingly loose cluster around the line; the slope of the line is still negative). population correlation coeff. COV ( X, Y) ρ = σ σ x y sample correlation coeff. cov( x, y) r = 19 s xs y 5

6 At zero, there is no linear association knowing the value of the x variable does not help you know the value of the y variable. This is consistent with a perfectly horizontal line being drawn through the data. A positive number indicates the slope of the line which defines the relationship between the variables is positive. If the number is positive, but near zero, the slope is positive, but the data points are only loosely clustered around the positive line. The nearer the correlation coefficient is to +1, the more closely the data points are to the line, until at +1, they are all exactly on the line. Remember: either covariance nor correlation coefficient define the precise slope of a line, they only indicate whether the slope is positive or negative. The strength of the association is a measure of how close the data points are around the line. 0 1 Examples Measures of Association #1 Find the covariance and correlation coefficient for the following sets of data: Covariance: Correlation Coefficient: What does each measure tell you? What doesn t it tell you? A B # Compute the covariance and the coefficient of correlation to measure how the number of commercials and sales level are related to one another. Calculate by hand from the data provided. ow try using Excel: Use Data Analysis go to Tools Data Analysis select Correlation Commercls sales Commercls 1 Sales? 1 Correlation matrix What are these numbers telling you? Commercls Sales

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